Topological quantum field theory: Difference between revisions

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This is a '''list of [[numerical analysis]] topics'''.
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==General==
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*[[Iterative method]]
 
*[[Rate of convergence]] — the speed at which a convergent sequence approaches its limit
  <li>[http://cerisier.info/spip.php?article20/ http://cerisier.info/spip.php?article20/]</li>
**[[Order of accuracy]] — rate at which numerical solution of differential equation converges to exact solution
 
*[[Series acceleration]] — methods to accelerate the speed of convergence of a series
  <li>[http://web.zaiwww.com/news/html/?172554.html http://web.zaiwww.com/news/html/?172554.html]</li>
**[[Aitken's delta-squared process]] — most useful for linearly converging sequences
 
**[[Minimum polynomial extrapolation]] — for vector sequences
  <li>[http://www.histoirepassion.eu/spip.php?article7/ http://www.histoirepassion.eu/spip.php?article7/]</li>
**[[Richardson extrapolation]]
 
**[[Shanks transformation]] — similar to Aitken's delta-squared process, but applied to the partial sums
  <li>[http://www.metransparent.com/spip.php?article20626&lang=ar&id_forum=34132/ http://www.metransparent.com/spip.php?article20626&lang=ar&id_forum=34132/]</li>
**[[Van Wijngaarden transformation]] — for accelerating the convergence of an alternating series
 
*[[Abramowitz and Stegun]] — book containing formulas and tables of many special functions
  <li>[http://www.metransparent.net/spip.php?article2&lang=ar&id_forum=8554/ http://www.metransparent.net/spip.php?article2&lang=ar&id_forum=8554/]</li>
**[[Digital Library of Mathematical Functions]] — successor of book by Abramowitz and Stegun
 
*[[Curse of dimensionality]]
</ul>
*[[Local convergence]] and global convergence — whether you need a good initial guess to get convergence
*[[Superconvergence]]
*[[Discretization]]
*[[Difference quotient]]
*Complexity:
**[[Computational complexity of mathematical operations]]
**[[Smoothed analysis]] — measuring the expected performance of algorithms under slight random perturbations of worst-case inputs
*[[Symbolic-numeric computation]] — combination of symbolic and numeric methods
*Cultural and historical aspects:
**[[History of numerical solution of differential equations using computers]]
**[[Hundred-dollar, Hundred-digit Challenge problems]] — list of ten problems proposed by Nick Trefethen in 2002
**[[International Workshops on Lattice QCD and Numerical Analysis]]
**[[Timeline of numerical analysis after 1945]]
*General classes of methods:
**[[Collocation method]] — discretizes a continuous equation by requiring it only to hold at certain points
**[[Level set method]]
***[[Level set (data structures)]] — data structures for representing level sets
**[[Sinc numerical methods]] — methods based on the sinc function, sinc(''x'') = sin(''x'') / ''x''
**[[ABS methods]]


==Error==
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[[Error analysis]]
*[[Approximation]]
*[[Approximation error]]
*[[Condition number]]
*[[Discretization error]]
*[[Floating point]] number
**[[Guard digit]] — extra precision introduced during a computation to reduce round-off error
**[[Truncation]] — rounding a floating-point number by discarding all digits after a certain digit
**[[Round-off error]]
***[[Numeric precision in Microsoft Excel]]
**[[Arbitrary-precision arithmetic]]
*[[Interval arithmetic]] — represent every number by two floating-point numbers guaranteed to have the unknown number between them
**[[Interval contractor]] — maps interval to subinterval which still contains the unknown exact answer
**[[Interval propagation]] — contracting interval domains without removing any value consistent with the constraints
***See also: [[Interval boundary element method]], [[Interval finite element]]
*[[Loss of significance]]
*[[Numerical error]]
*[[Numerical stability]]
*Error propagation:
**[[Propagation of uncertainty]]
***[[List of uncertainty propagation software]]
**[[Significance arithmetic]]
** [[Residual (numerical analysis)]]
*[[Relative change and difference]] — the relative difference between ''x'' and ''y'' is |''x'' − ''y''| / max(|''x''|, |''y''|)
*[[Significant figures]]
**[[False precision]] — giving more significant figures than appropriate
*[[Truncation error]] — error committed by doing only a finite numbers of steps
*[[Well-posed problem]]
*[[Affine arithmetic]]


==Elementary and special functions==
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*Summation:
 
**[[Kahan summation algorithm]]
  <li>[http://www.philatelie-france-russie.fr/spip.php?article51/ http://www.philatelie-france-russie.fr/spip.php?article51/]</li>
**[[Pairwise summation]] — slightly worse than Kahan summation but cheaper
 
**[[Binary splitting]]
  <li>[http://www.clasife.com.ar/index.php?page=item&id=59355 http://www.clasife.com.ar/index.php?page=item&id=59355]</li>
*Multiplication:
 
**[[Multiplication algorithm]] — general discussion, simple methods
  <li>[http://tesomarket.com/activity/p/68221/ http://tesomarket.com/activity/p/68221/]</li>
**[[Karatsuba algorithm]] — the first algorithm which is faster than straightforward multiplication
 
**[[Toom–Cook multiplication]] — generalization of Karatsuba multiplication
  <li>[http://www.dailyqr.com/blog_entry.php?user=1109099&blogentry_id=15880474 http://www.dailyqr.com/blog_entry.php?user=1109099&blogentry_id=15880474]</li>
**[[Schönhage–Strassen algorithm]] — based on Fourier transform, asymptotically very fast
 
**[[Fürer's algorithm]] — asymptotically slightly faster than Schönhage–Strassen
  <li>[http://co-at-work.zib.de/phorum/read.php?3,442226 http://co-at-work.zib.de/phorum/read.php?3,442226]</li>
*[[Division algorithm]] — for computing quotient and remainder of two numbers
 
*Exponentiation:
</ul>
**[[Exponentiation by squaring]]
**[[Addition-chain exponentiation]]
*Polynomials:
**[[Horner's method]]
**[[Estrin's scheme]] — modification of the Horner scheme with more possibilities for parallellization
**[[Clenshaw algorithm]]
**[[De Casteljau's algorithm]]
*Square roots and other roots:
**[[Integer square root]]
**[[Methods of computing square roots]]
**[[Nth root algorithm|''n''th root algorithm]]
**[[Shifting nth root algorithm|Shifting ''n''th root algorithm]] — similar to long division
**[[hypot]] — the function (''x''<sup>2</sup> + ''y''<sup>2</sup>)<sup>1/2</sup>
**[[Alpha max plus beta min algorithm]] — approximates hypot(x,y)
**[[Fast inverse square root]] — calculates 1 / √''x'' using details of the IEEE floating-point system
*Elementary functions (exponential, logarithm, trigonometric functions):
**[[Trigonometric tables]] — different methods for generating them
**[[CORDIC]] — shift-and-add algorithm using a table of arc tangents
**[[BKM algorithm]] — shift-and-add algorithm using a table of logarithms and complex numbers
*Gamma function:
**[[Lanczos approximation]]
**[[Spouge's approximation]] — modification of Stirling's approximation; easier to apply than Lanczos
*[[AGM method]] — computes arithmetic–geometric mean; related methods compute special functions
*[[FEE method]] (Fast E-function Evaluation) — fast summation of series like the power series for e<sup>''x''</sup>
*[[Gal's accurate tables]] — table of function values with unequal spacing to reduce round-off error
*[[Spigot algorithm]] — algorithms that can compute individual digits of a real number
*[[Approximations of π|Approximations of {{pi}}]]:
**[[Liu Hui's π algorithm]] — first algorithm that can compute π to arbitrary precision
**[[Leibniz formula for π]] — alternating series with very slow convergence
**[[Wallis product]] — infinite product converging slowly to π/2
**[[Viète's formula]] — more complicated infinite product which converges faster
**[[Gauss–Legendre algorithm]] — iteration which converges quadratically to π, based on arithmetic–geometric mean
**[[Borwein's algorithm]] — iteration which converges quartically to 1/π, and other algorithms
**[[Chudnovsky algorithm]] — fast algorithm that calculates a hypergeometric series
**[[Bailey–Borwein–Plouffe formula]] — can be used to compute individual hexadecimal digits of π
**[[Bellard's formula]] — faster version of Bailey–Borwein–Plouffe formula
**[[List of formulae involving π]]


==Numerical linear algebra==
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[[Numerical linear algebra]] — study of numerical algorithms for linear algebra problems


===Basic concepts===
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*Types of matrices appearing in numerical analysis:
 
**[[Sparse matrix]]
  <li>[http://tu.linuxaudiosystems.com/node/6#comment-60525141 http://tu.linuxaudiosystems.com/node/6#comment-60525141]</li>
***[[Band matrix]]
 
***[[Bidiagonal matrix]]
  <li>[http://www.proyectoalba.com.ar/spip.php?article66/&quot;/ http://www.proyectoalba.com.ar/spip.php?article66/&quot;/]</li>
***[[Tridiagonal matrix]]
 
***[[Pentadiagonal matrix]]
  <li>[http://www.frel.univ-paris-diderot.fr/spip.php?article1 http://www.frel.univ-paris-diderot.fr/spip.php?article1]</li>
***[[Skyline matrix]]
 
**[[Circulant matrix]]
  <li>[http://www.achicourtautrement.fr/spip.php?article451/ http://www.achicourtautrement.fr/spip.php?article451/]</li>
**[[Triangular matrix]]
 
**[[Diagonally dominant matrix]]
  <li>[http://bbs.mouro.cn/forum.php?mod=viewthread&tid=136328 http://bbs.mouro.cn/forum.php?mod=viewthread&tid=136328]</li>
**[[Block matrix]] — matrix composed of smaller matrices
 
**[[Stieltjes matrix]] — symmetric positive definite with non-positive off-diagonal entries
</ul>
**[[Hilbert matrix]] — example of a matrix which is extremely ill-conditioned (and thus difficult to handle)
**[[Wilkinson matrix]] — example of a symmetric tridiagonal matrix with pairs of nearly, but not exactly, equal eigenvalues
**[[Convergent matrix]] – square matrix whose successive powers approach the zero matrix
*Algorithms for matrix multiplication:
**[[Strassen algorithm]]
**[[Coppersmith–Winograd algorithm]]
**[[Cannon's algorithm]] — a distributed algorithm, especially suitable for processors laid out in a 2d grid
**[[Freivalds' algorithm]] — a randomized algorithm for checking the result of a multiplication
*[[Matrix decomposition]]s:
**[[LU decomposition]] — lower triangular times upper triangular
**[[QR decomposition]] — orthogonal matrix times triangular matrix
***[[RRQR factorization]] — rank-revealing QR factorization, can be used to compute rank of a matrix
**[[Polar decomposition]] — unitary matrix times positive-semidefinite Hermitian matrix
**Decompositions by similarity:
***[[Eigendecomposition of a matrix|Eigendecomposition]] — decomposition in terms of eigenvectors and eigenvalues
***[[Jordan normal form]] — bidiagonal matrix of a certain form; generalizes the eigendecomposition
****[[Weyr canonical form]] — permutation of Jordan normal form
***[[Jordan–Chevalley decomposition]] — sum of commuting nilpotent matrix and diagonalizable matrix
***[[Schur decomposition]] — similarity transform bringing the matrix to a triangular matrix
**[[Singular value decomposition]] — unitary matrix times diagonal matrix times unitary matrix
*[[Matrix splitting]] – expressing a given matrix as a sum or difference of matrices


===Solving systems of linear equations===
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*[[Gaussian elimination]]
**[[Row echelon form]] — matrix in which all entries below a nonzero entry are zero
**[[Bareiss algorithm]] — variant which ensures that all entries remain integers if the initial matrix has integer entries
**[[Tridiagonal matrix algorithm]] — simplified form of Gaussian elimination for tridiagonal matrices
*[[LU decomposition]] — write a matrix as a product of an upper- and a lower-triangular matrix
**[[Crout matrix decomposition]]
**[[LU reduction]] — a special parallelized version of a LU decomposition algorithm
*[[Block LU decomposition]]
*[[Cholesky decomposition]] — for solving a system with a positive definite matrix
**[[Minimum degree algorithm]]
**[[Symbolic Cholesky decomposition]]
*[[Iterative refinement]] — procedure to turn an inaccurate solution in a more accurate one
*Direct methods for sparse matrices:
**[[Frontal solver]] — used in finite element methods
**[[Nested dissection]] — for symmetric matrices, based on graph partitioning
*[[Levinson recursion]] — for Toeplitz matrices
*[[SPIKE algorithm]] — hybrid parallel solver for narrow-banded matrices
*[[Cyclic reduction]] — eliminate even or odd rows or columns, repeat
*Iterative methods:
**[[Jacobi method]]
**[[Gauss–Seidel method]]
***[[Successive over-relaxation]] (SOR) — a technique to accelerate the Gauss–Seidel method
****[[Symmetric successive overrelaxation]] (SSOR) — variant of SOR for symmetric matrices
***[[Backfitting algorithm]] — iterative procedure used to fit a generalized additive model, often equivalent to Gauss–Seidel
**[[Modified Richardson iteration]]
**[[Conjugate gradient method]] (CG) — assumes that the matrix is positive definite
***[[Derivation of the conjugate gradient method]]
***[[Nonlinear conjugate gradient method]] — generalization for nonlinear optimization problems
**[[Biconjugate gradient method]] (BiCG)
***[[Biconjugate gradient stabilized method]] (BiCGSTAB) — variant of BiCG with better convergence
**[[Conjugate residual method]] — similar to CG but only assumed that the matrix is symmetric
**[[Generalized minimal residual method]] (GMRES) — based on the Arnoldi iteration
**[[Chebyshev iteration]] — avoids inner products but needs bounds on the spectrum
**[[Stone method|Stone's method]] (SIP – Srongly Implicit Procedure) — uses an incomplete LU decomposition
**[[Kaczmarz method]]
**[[Preconditioner]]
***[[Incomplete Cholesky factorization]] — sparse approximation to the Cholesky factorization
***[[Incomplete LU factorization]] — sparse approximation to the LU factorization
**[[Uzawa iteration]] — for saddle node problems
*Underdetermined and overdetermined systems (systems that have no or more than one solution):
**[[Kernel (matrix)#Numerical computation of null space|Numerical computation of null space]] — find all solutions of an underdetermined system
**[[Moore–Penrose pseudoinverse]] — for finding solution with smallest 2-norm (for underdetermined systems) or smallest residual
**[[Sparse approximation]] — for finding the sparsest solution (i.e., the solution with as many zeros as possible)


===Eigenvalue algorithms===
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[[Eigenvalue algorithm]] — a numerical algorithm for locating the eigenvalues of a matrix
 
*[[Power iteration]]
  <li>[http://www.x-poser.com/ http://www.x-poser.com/]</li>
*[[Inverse iteration]]
 
*[[Rayleigh quotient iteration]]
  <li>[http://www.jamiatou.com/spip.php?article21 http://www.jamiatou.com/spip.php?article21]</li>
*[[Arnoldi iteration]] — based on Krylov subspaces
 
*[[Lanczos algorithm]] — Arnoldi, specialized for positive-definite matrices
  <li>[http://www.think-group.cn/VK/blog/article.php?type=blog&cid=2&itemid=1677723 http://www.think-group.cn/VK/blog/article.php?type=blog&cid=2&itemid=1677723]</li>
**[[Block Lanczos algorithm]] — for when matrix is over a finite field
 
*[[QR algorithm]]
  <li>[http://enseignement-lsf.com/spip.php?article64#forum17985775 http://enseignement-lsf.com/spip.php?article64#forum17985775]</li>
*[[Jacobi eigenvalue algorithm]] — select a small submatrix which can be diagonalized exactly, and repeat
 
**[[Jacobi rotation]] — the building block, almost a Givens rotation
  <li>[http://enseignement-lsf.com/spip.php?article64#forum17951155 http://enseignement-lsf.com/spip.php?article64#forum17951155]</li>
**[[Jacobi method for complex Hermitian matrices]]
 
*[[Divide-and-conquer eigenvalue algorithm]]
</ul>
*[[Folded spectrum method]]
*[[LOBPCG]] — Locally Optimal Block Preconditioned Conjugate Gradient Method
*[[Eigenvalue perturbation]] — stability of eigenvalues under perturbations of the matrix


===Other concepts and algorithms===
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*[[Orthogonalization]] algorithms:
**[[Gram–Schmidt process]]
**[[Householder transformation]]
***[[Householder operator]] — analogue of Householder transformation for general inner product spaces
**[[Givens rotation]]
*[[Krylov subspace]]
*[[Block matrix pseudoinverse]]
*[[Bidiagonalization]]
*[[Cuthill–McKee algorithm]] — permutes rows/columns in sparse matrix to yield a narrow band matrix
*[[In-place matrix transposition]] — computing the transpose of a matrix without using much additional storage
*[[Pivot element]] — entry in a matrix on which the algorithm concentrates
*[[Matrix-free methods]] — methods that only access the matrix by evaluating matrix-vector products


==Interpolation and approximation==
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[[Interpolation]] — construct a function going through some given data points
 
*[[Nearest-neighbor interpolation]] — takes the value of the nearest neighbor
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</ul>


===Polynomial interpolation===
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[[Polynomial interpolation]] — interpolation by polynomials
*[[Linear interpolation]]
*[[Runge's phenomenon]]
*[[Vandermonde matrix]]
*[[Chebyshev polynomials]]
*[[Chebyshev nodes]]
*[[Lebesgue constant (interpolation)]]
*Different forms for the interpolant:
**[[Newton polynomial]]
***[[Divided differences]]
***[[Neville's algorithm]] — for evaluating the interpolant; based on the Newton form
**[[Lagrange polynomial]]
**[[Bernstein polynomial]] — especially useful for approximation
**[[Brahmagupta's interpolation formula]] — seventh-century formula for quadratic interpolation
*Extensions to multiple dimensions:
**[[Bilinear interpolation]]
**[[Trilinear interpolation]]
**[[Bicubic interpolation]]
**[[Tricubic interpolation]]
**[[Padua points]] — set of points in '''R'''<sup>2</sup> with unique polynomial interpolant and minimal growth of Lebesgue constant
*[[Hermite interpolation]]
*[[Birkhoff interpolation]]
*[[Abel–Goncharov interpolation]]


===Spline interpolation===
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[[Spline interpolation]] — interpolation by piecewise polynomials
 
*[[Spline (mathematics)]] — the piecewise polynomials used as interpolants
  <li>[http://netburst.org/index.php?site=polls&pollID=3 http://netburst.org/index.php?site=polls&pollID=3]</li>
*[[Perfect spline]] — polynomial spline of degree ''m'' whose ''m''th derivate is &plusmn;1
 
*[[Cubic Hermite spline]]
  <li>[http://www.xjxg.cn/thread-221316-1-1.html http://www.xjxg.cn/thread-221316-1-1.html]</li>
**[[Centripetal Catmull–Rom spline]] — special case of cubic Hermite splines without self-intersections or cusps
 
*[[Monotone cubic interpolation]]
  <li>[http://www.proyectoalba.com.ar/spip.php?article66/ http://www.proyectoalba.com.ar/spip.php?article66/]</li>
*[[Hermite spline]]
 
*[[Bézier spline]]
  <li>[http://enseignement-lsf.com/spip.php?article64#forum17657115 http://enseignement-lsf.com/spip.php?article64#forum17657115]</li>
**[[Bézier curve]]
 
**[[De Casteljau's algorithm]]
  <li>[http://bmd78.comyr.com/forum.php?mod=viewthread&tid=307410&fromuid=13437 http://bmd78.comyr.com/forum.php?mod=viewthread&tid=307410&fromuid=13437]</li>
**Generalizations to more dimensions:
 
***[[Bézier triangle]] — maps a triangle to '''R'''<sup>3</sup>
</ul>
***[[Bézier surface]] — maps a square to '''R'''<sup>3</sup>
*[[B-spline]]
**[[Box spline]] — multivariate generalization of B-splines
**[[Truncated power function]]
**[[De Boor's algorithm]] — generalizes De Casteljau's algorithm
*[[Non-uniform rational B-spline]] (NURBS)
**[[T-spline]] — can be thought of as a NURBS surface for which a row of control points is allowed to terminate
*[[Kochanek–Bartels spline]]
*[[Coons patch]] — type of manifold parametrization used to smoothly join other surfaces together
*[[M-spline]] — a non-negative spline
*[[I-spline]] — a monotone spline, defined in terms of M-splines
*[[Smoothing spline]] — a spline fitted smoothly to noisy data
*[[Blossom (functional)]] — a unique, affine, symmetric map associated to a polynomial or spline
*See also: [[List of numerical computational geometry topics]]


===Trigonometric interpolation===
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[[Trigonometric interpolation]] — interpolation by trigonometric polynomials
*[[Discrete Fourier transform]] — can be viewed as trigonometric interpolation at equidistant points
**[[Relations between Fourier transforms and Fourier series]]
*[[Fast Fourier transform]] (FFT) — a fast method for computing the discrete Fourier transform
**[[Bluestein's FFT algorithm]]
**[[Bruun's FFT algorithm]]
**[[Cooley–Tukey FFT algorithm]]
**[[Split-radix FFT algorithm]] — variant of Cooley–Tukey that uses a blend of radices 2 and 4
**[[Goertzel algorithm]]
**[[Prime-factor FFT algorithm]]
**[[Rader's FFT algorithm]]
**[[Bit-reversal permutation]] — particular permutation of vectors with 2<sup>''m''</sup> entries used in many FFTs.
**[[Butterfly diagram]]
**[[Twiddle factor]] — the trigonometric constant coefficients that are multiplied by the data
**[[Cyclotomic fast Fourier transform]] — for FFT over finite fields
**Methods for computing discrete convolutions with finite impulse response filters using the FFT:
***[[Overlap–add method]]
***[[Overlap–save method]]
*[[Sigma approximation]]
*[[Dirichlet kernel]] — convolving any function with the Dirichlet kernel yields its trigonometric interpolant
*[[Gibbs phenomenon]]


===Other interpolants===
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*[[Simple rational approximation]]
 
**[[Polynomial and rational function modeling]] — comparison of polynomial and rational interpolation
  <li>[http://livinginplaya.net/read.php?2,152826 http://livinginplaya.net/read.php?2,152826]</li>
*[[Wavelet]]
 
**[[Continuous wavelet]]
  <li>[http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/ http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/]</li>
**[[Transfer matrix]]
 
**See also: [[List of functional analysis topics]], [[List of wavelet-related transforms]]
  <li>[http://www.hfdbsw.com/bbs/forum.php?mod=viewthread&tid=24569 http://www.hfdbsw.com/bbs/forum.php?mod=viewthread&tid=24569]</li>
*[[Inverse distance weighting]]
 
*[[Radial basis function]] (RBF) — a function of the form ƒ(''x'') = ''φ''(|''x''−''x''<sub>0</sub>|)
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**[[Polyharmonic spline]] — a commonly used radial basis function
 
**[[Thin plate spline]] — a specific polyharmonic spline: ''r''<sup>2</sup> log ''r''
  <li>[http://lightasic.com/forum.php?mod=viewthread&tid=1829504 http://lightasic.com/forum.php?mod=viewthread&tid=1829504]</li>
**[[Hierarchical RBF]]
 
*[[Subdivision surface]] — constructed by recursively subdividing a piecewise linear interpolant
</ul>
**[[Catmull–Clark subdivision surface]]
**[[Doo–Sabin subdivision surface]]
**[[Loop subdivision surface]]
*[[Slerp]] (spherical linear interpolation) — interpolation between two points on a sphere
**[[Generalized quaternion interpolation]] — generalizes slerp for interpolation between more than two quaternions
*[[Irrational base discrete weighted transform]]
*[[Nevanlinna–Pick interpolation]] — interpolation by analytic functions in the unit disc subject to a bound
**[[Pick matrix]] — the Nevanlinna–Pick interpolation has a solution if this matrix is positive semi-definite
*[[Multivariate interpolation]] — the function being interpolated depends on more than one variable
**[[Barnes interpolation]] — method for two-dimensional functions using Gaussians common in meteorology
**[[Coons surface]] — combination of linear interpolation and bilinear interpolation
**[[Lanczos resampling]] — based on convolution with a sinc function
**[[Natural neighbor]] interpolation
**[[Nearest neighbor value interpolation]]
**[[PDE surface]]
**[[Transfinite interpolation]] — constructs function on planar domain given its values on the boundary
**[[Trend surface analysis]] — based on low-order polynomials of spatial coordinates; uses scattered observations
**Method based on polynomials are listed under ''Polynomial interpolation''


===Approximation theory===
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[[Approximation theory]]
*[[Orders of approximation]]
*[[Lebesgue's lemma]]
*[[Curve fitting]]
**[[Vector field reconstruction]]
*[[Modulus of continuity]] — measures smoothness of a function
*[[Least squares (function approximation)]] — minimizes the error in the L<sup>2</sup>-norm
*[[Minimax approximation algorithm]] — minimizes the maximum error over an interval (the L<sup>∞</sup>-norm)
**[[Equioscillation theorem]] — characterizes the best approximation in the L<sup>∞</sup>-norm
*[[Unisolvent point set]] — function from given function space is determined uniquely by values on such a set of points
*[[Stone–Weierstrass theorem]] — continuous functions can be approximated uniformly by polynomials, or certain other function spaces
*Approximation by polynomials:
**[[Linear approximation]]
**[[Bernstein polynomial]] — basis of polynomials useful for approximating a function
**[[Bernstein's constant]] — error when approximating |''x''| by a polynomial
**[[Remez algorithm]] — for constructing the best polynomial approximation in the L<sup>∞</sup>-norm
**[[Bernstein's inequality (mathematical analysis)]] — bound on maximum of derivative of polynomial in unit disk
**[[Mergelyan's theorem]] — generalization of Stone–Weierstrass theorem for polynomials
**[[Müntz–Szász theorem]] — variant of Stone–Weierstrass theorem for polynomials if some coefficients have to be zero
**[[Bramble–Hilbert lemma]] — upper bound on L<sup>p</sup> error of polynomial approximation in multiple dimensions
**[[Discrete Chebyshev polynomials]] — polynomials orthogonal with respect to a discrete measure
**[[Favard's theorem]] — polynomials satisfying suitable 3-term recurrence relations are orthogonal polynomials
*Approximation by Fourier series / trigonometric polynomials:
**[[Jackson's inequality]] — upper bound for best approximation by a trigonometric polynomial
***[[Bernstein's theorem (approximation theory)]] — a converse to Jackson's inequality
**[[Fejér's theorem]] — Cesàro means of partial sums of Fourier series converge uniformly for continuous periodic functions
**[[Erdős–Turán inequality]] — bounds distance between probability and Lebesgue measure in terms of Fourier coefficients
*Different approximations:
**[[Moving least squares]]
**[[Padé approximant]]
***[[Padé table]] — table of Padé approximants
**[[Hartogs–Rosenthal theorem]] — continuous functions can be approximated uniformly by rational functions on a set of Lebesgue measure zero
**[[Szász–Mirakyan operator]] — approximation by e<sup>&minus;''n''</sup> ''x''<sup>''k''</sup> on a semi-infinite interval
**[[Szász–Mirakjan–Kantorovich operator]]
**[[Baskakov operator]] — generalize Bernstein polynomials, Szász–Mirakyan operators, and Lupas operators
**[[Favard operator]] — approximation by sums of Gaussians
*[[Surrogate model]] — application: replacing a function that is hard to evaluate by a simpler function
*[[Constructive function theory]] — field that studies connection between degree of approximation and smoothness
*[[Universal differential equation]] — differential–algebraic equation whose solutions can approximate any continuous function
*[[Fekete problem]] — find ''N'' points on a sphere that minimize some kind of energy
*[[Carleman's condition]] — condition guaranteeing that a measure is uniquely determined by its moments
*[[Krein's condition]] — condition that exponential sums are dense in weighted L<sup>2</sup> space
*[[Lethargy theorem]] — about distance of points in a metric space from members of a sequence of subspaces
*[[Wirtinger's representation and projection theorem]]
*Journals:
**[[Constructive Approximation]]
**[[Journal of Approximation Theory]]


===Miscellaneous===
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*[[Extrapolation]]
 
**[[Linear predictive analysis]] — linear extrapolation
  <li>[http://www.promo-grimpe.com/spip.php?article124/ http://www.promo-grimpe.com/spip.php?article124/]</li>
*[[Unisolvent functions]] — functions for which the interpolation problem has a unique solution
 
*[[Regression analysis]]
  <li>[http://huanxun9999.com/news/html/?59489.html http://huanxun9999.com/news/html/?59489.html]</li>
**[[Isotonic regression]]
 
*[[Curve-fitting compaction]]
  <li>[http://ad.il24.net/~ura/cgi-bin/ape-shoushin/apeboard_plus.cgi/ http://ad.il24.net/~ura/cgi-bin/ape-shoushin/apeboard_plus.cgi/]</li>
*[[Interpolation (computer graphics)]]
 
  <li>[http://www.emil86.fr/spip.php?article1/ http://www.emil86.fr/spip.php?article1/]</li>
 
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</ul>


==Finding roots of nonlinear equations==
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:''See [[#Numerical linear algebra]] for linear equations''


[[Root-finding algorithm]] — algorithms for solving the equation ''f''(''x'') = 0
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*General methods:
 
**[[Bisection method]] — simple and robust; linear convergence
  <li>[http://mba-zone.com/forum.php?mod=viewthread&tid=243418 http://mba-zone.com/forum.php?mod=viewthread&tid=243418]</li>
***[[Lehmer–Schur algorithm]] — variant for complex functions
 
**[[Fixed-point iteration]]
  <li>[http://cerisier.info/spip.php?article20/ http://cerisier.info/spip.php?article20/]</li>
**[[Newton's method]] — based on linear approximation around the current iterate; quadratic convergence
 
***[[Kantorovich theorem]] — gives a region around solution such that Newton's method converges
  <li>[http://www.sociland.com/index.php/forums/topic/4456/andrew-abercrombie-melbourne-as/view/post_id/4449 http://www.sociland.com/index.php/forums/topic/4456/andrew-abercrombie-melbourne-as/view/post_id/4449]</li>
***[[Newton fractal]] — indicates which initial condition converges to which root under Newton iteration
 
***[[Quasi-Newton method]] — uses an approximation of the Jacobian:
  <li>[http://cerisier.info/spip.php?article25/ http://cerisier.info/spip.php?article25/]</li>
****[[Broyden's method]] — uses a rank-one update for the Jacobian
 
****[[Symmetric rank-one]] — a symmetric (but not necessarily positive definite) rank-one update of the Jacobian
  <li>[http://www.jinxi.gd.cn/news/html/?39004.html http://www.jinxi.gd.cn/news/html/?39004.html]</li>
****[[Davidon–Fletcher–Powell formula]] — update of the Jacobian in which the matrix remains positive definite
 
****[[Broyden–Fletcher–Goldfarb–Shanno algorithm]] — rank-two update of the Jacobian in which the matrix remains positive definite
</ul>
****[[Limited-memory BFGS]] method — truncated, matrix-free variant of BFGS method suitable for large problems
***[[Steffensen's method]] — uses divided differences instead of the derivative
**[[Secant method]] — based on linear interpolation at last two iterates
**[[False position method]] — secant method with ideas from the bisection method
**[[Muller's method]] — based on quadratic interpolation at last three iterates
**[[Sidi's generalized secant method]] — higher-order variants of secant method
**[[Inverse quadratic interpolation]] — similar to Muller's method, but interpolates the inverse
**[[Brent's method]] — combines bisection method, secant method and inverse quadratic interpolation
**[[Ridders' method]] — fits a linear function times an exponential to last two iterates and their midpoint
**[[Halley's method]] — uses ''f'', ''f''<nowiki>'</nowiki> and ''f''<nowiki>''</nowiki>; achieves the cubic convergence
**[[Householder's method]] — uses first ''d'' derivatives to achieve order ''d'' + 1; generalizes Newton's and Halley's method
*Methods for polynomials:
**[[Aberth method]]
**[[Bairstow's method]]
**[[Durand–Kerner method]]
**[[Graeffe's method]]
**[[Jenkins–Traub algorithm]] — fast, reliable, and widely used
**[[Laguerre's method]]
**[[Splitting circle method]]
*Analysis:
**[[Wilkinson's polynomial]]
*[[Numerical continuation]] — tracking a root as one parameters in the equation changes
**[[Piecewise linear continuation]]


==Optimization==
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[[Mathematical optimization]] — algorithm for finding maxima or minima of a given function


===Basic concepts===
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*[[Active set]]
 
*[[Candidate solution]]
  <li>[http://bbs.0746.so/forum.php?mod=viewthread&tid=342398&extra= http://bbs.0746.so/forum.php?mod=viewthread&tid=342398&extra=]</li>
*[[Constraint (mathematics)]]
 
**[[Constrained optimization]] — studies optimization problems with constraints
  <li>[http://ciarcr.org/spip.php?article310/ http://ciarcr.org/spip.php?article310/]</li>
**[[Binary constraint]] — a constraint that involves exactly two variables
 
*[[Corner solution]]
  <li>[http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/ http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/]</li>
*[[Feasible region]] — contains all solutions that satisfy the constraints but may not be optimal
 
*[[Global optimum]] and [[Local optimum]]
  <li>[http://www.shaffaf.net/spip.php?article929&lang=ar&id_forum=27372/ http://www.shaffaf.net/spip.php?article929&lang=ar&id_forum=27372/]</li>
*[[Maxima and minima]]
 
*[[Slack variable]]
  <li>[http://advertsbook.net/forum/read.php?44,210222 http://advertsbook.net/forum/read.php?44,210222]</li>
*[[Continuous optimization]]
 
*[[Discrete optimization]]
</ul>


===Linear programming===
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[[Linear programming]] (also treats ''integer programming'') — objective function and constraints are linear
* Algorithms for linear programming:
**[[Simplex algorithm]]
***[[Bland's rule]] — rule to avoid cycling in the simplex method
***[[Klee–Minty cube]] — perturbed (hyper)cube; simplex method has exponential complexity on such a domain
***[[Criss-cross algorithm]] — similar to the simplex algorithm
***[[Big M method]] — variation of simplex algorithm for problems with both "less than" and "greater than" constraints
**[[Interior point method]]
***[[Ellipsoid method]]
***[[Karmarkar's algorithm]]
***[[Mehrotra predictor–corrector method]]
**[[Column generation]]
**[[k-approximation of k-hitting set]] — algorithm for specific LP problems (to find a weighted hitting set)
*[[Linear complementarity problem]]
*Decompositions:
**[[Benders' decomposition]]
**[[Dantzig–Wolfe decomposition]]
**[[Theory of two-level planning]]
**[[Variable splitting]]
*[[Basic solution (linear programming)]] — solution at vertex of feasible region
*[[Fourier–Motzkin elimination]]
*[[Hilbert basis (linear programming)]] — set of integer vectors in a convex cone which generate all integer vectors in the cone
*[[LP-type problem]]
*[[Linear inequality]]
*[[Vertex enumeration problem]] — list all vertices of the feasible set


===Convex optimization===
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[[Convex optimization]]
 
*[[Quadratic programming]]
  <li>[http://www.hhwf.net/bbs/forum.php?mod=viewthread&tid=25764&fromuid=6805 http://www.hhwf.net/bbs/forum.php?mod=viewthread&tid=25764&fromuid=6805]</li>
**[[Linear least squares (mathematics)]]
 
**[[Total least squares]]
  <li>[http://453300.biz/thread-302271-1-1.html http://453300.biz/thread-302271-1-1.html]</li>
**[[Frank–Wolfe algorithm]]
 
**[[Sequential minimal optimization]] — breaks up large QP problems into a series of smallest possible QP problems
  <li>[http://general.assembly.codesria.org/spip.php?article87&lang=pt/ http://general.assembly.codesria.org/spip.php?article87&lang=pt/]</li>
**[[Bilinear program]]
 
*[[Basis pursuit]] — minimize L<sub>1</sub>-norm of vector subject to linear constraints
  <li>[http://enseignement-lsf.com/spip.php?article369#forum18169041 http://enseignement-lsf.com/spip.php?article369#forum18169041]</li>
**[[Basis pursuit denoising]] (BPDN) — regularized version of basis pursuit
 
***[[In-crowd algorithm]] — algorithm for solving basis pursuit denoising
  <li>[http://www.vlpa.org/Forum/default.aspx?g=posts&t=435711 http://www.vlpa.org/Forum/default.aspx?g=posts&t=435711]</li>
*[[Linear matrix inequality]]
 
*[[Conic optimization]]
</ul>
**[[Semidefinite programming]]
**[[Second-order cone programming]]
**[[Sum-of-squares optimization]]
**Quadratic programming (see above)
*[[Bregman method]] — row-action method for strictly convex optimization problems
*[[Proximal Gradient Methods]] — use splitting of objective function in sum of possible non-differentiable pieces
*[[Subgradient method]] — extension of steepest descent for problems with a non-differentiable objective function
*[[Biconvex optimization]] — generalization where objective function and constraint set can be biconvex


===Nonlinear programming===
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[[Nonlinear programming]] — the most general optimization problem in the usual framework
*Special cases of nonlinear programming:
**See ''Linear programming'' and ''Convex optimization'' above
**[[Geometric programming]] — problems involving signomials or posynomials
***[[Signomial]] — similar to polynomials, but exponents need not be integers
***[[Posynomial]] — a signomial with positive coefficients
**[[Quadratically constrained quadratic program]]
**[[Linear-fractional programming]] — objective is ratio of linear functions, constraints are linear
***[[Fractional programming]] — objective is ratio of nonlinear functions, constraints are linear
**[[Nonlinear complementarity problem]] (NCP) — find ''x'' such that ''x'' &ge; 0, ''f''(''x'') &ge; 0 and ''x''<sup>T</sup> ''f''(''x'') = 0
**[[Least squares]] — the objective function is a sum of squares
***[[Non-linear least squares]]
***[[Gauss–Newton algorithm]]
****[[BHHH algorithm]] — variant of Gauss–Newton in econometrics
****[[Generalized Gauss–Newton method]] — for constrained nonlinear least-squares problems
***[[Levenberg–Marquardt algorithm]]
***[[Iteratively reweighted least squares]] (IRLS) — solves a weigted least-squares problem at every iteration
***[[Partial least squares]] — statistical techniques similar to principal components analysis
****[[Non-linear iterative partial least squares]] (NIPLS)
**[[Mathematical programming with equilibrium constraints]] — constraints include variational inequalities or complementarities
**Univariate optimization:
***[[Golden section search]]
***[[Successive parabolic interpolation]] — based on quadratic interpolation through the last three iterates
*General algorithms:
**Concepts:
***[[Descent direction]]
***[[Guess value]] — the initial guess for a solution with which an algorithm starts
***[[Line search]]
****[[Backtracking line search]]
****[[Wolfe conditions]]
**[[Gradient method]] — method that uses the gradient as the search direction
***[[Gradient descent]]
****[[Stochastic gradient descent]]
***[[Landweber iteration]] — mainly used for ill-posed problems
**[[Successive linear programming]] (SLP) — replace problem by a linear programming problem, solve that, and repeat
**[[Sequential quadratic programming]] (SQP) — replace problem by a quadratic programming problem, solve that, and repeat
**[[Newton's method in optimization]]
***See also under ''Newton algorithm'' in the [[#Finding roots of nonlinear equations|section ''Finding roots of nonlinear equations'']]
**[[Nonlinear conjugate gradient method]]
**Derivative-free methods
***[[Coordinate descent]] — move in one of the coordinate directions
****[[Adaptive coordinate descent]] — adapt coordinate directions to objective function
****[[Random coordinate descent]] — randomized version
***[[Nelder–Mead method]]
***[[Pattern search (optimization)]]
***[[Powell's method]] — based on conjugate gradient descent
***[[Rosenbrock methods]] — derivative-free method, similar to Nelder–Mead but with guaranteed convergence
**[[Augmented Lagrangian method]] — replaces contrained problems by unconstrained problems with a term added to the objective function
**[[Ternary search]]
**[[Tabu search]]
**[[Guided Local Search]] — modification of search algorithms which builds up penalties during a search
**[[Reactive search optimization]] (RSO) — the algorithm adapts its parameters automatically
**[[MM algorithm]] — majorize-minimization, a wide framework of methods
**[[Least absolute deviations]]
***[[Expectation–maximization algorithm]]
****[[Ordered subset expectation maximization]]
**[[Adaptive projected subgradient method]]
**[[Nearest neighbor search]]
**[[Space mapping]] — uses "coarse" (ideal or low-fidelity) and "fine" (practical or high-fidelity) models


===Optimal control and infinite-dimensional optimization===
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[[Optimal control]]
 
*[[Pontryagin's minimum principle]] — infinite-dimensional version of Lagrange multipliers
  <li>[http://bbs.yapaifs.cn/forum.php?mod=viewthread&tid=855903 http://bbs.yapaifs.cn/forum.php?mod=viewthread&tid=855903]</li>
**[[Costate equations]] — equation for the "Lagrange multipliers" in Pontryagin's minimum principle
 
**[[Hamiltonian (control theory)]] — minimum principle says that this function should be minimized
  <li>[http://bbs.shenghuoqu.com.cn/forum.php?mod=viewthread&tid=1237008 http://bbs.shenghuoqu.com.cn/forum.php?mod=viewthread&tid=1237008]</li>
*Types of problems:
 
**[[Linear-quadratic regulator]] — system dynamics is a linear differential equation, objective is quadratic
  <li>[http://www.frel.univ-paris-diderot.fr/spip.php?article1 http://www.frel.univ-paris-diderot.fr/spip.php?article1]</li>
**[[Linear-quadratic-Gaussian control]] (LQG) — system dynamics is a linear SDE with additive noise, objective is quadratic
 
***[[Optimal projection equations]] — method for reducing dimension of LQG control problem
  <li>[http://www.think-group.cn/VK/blog/article.php?type=blog&cid=5&itemid=1696587 http://www.think-group.cn/VK/blog/article.php?type=blog&cid=5&itemid=1696587]</li>
*[[Algebraic Riccati equation]] — matrix equation occurring in many optimal control problems
 
*[[Bang–bang control]] — control that switches abruptly between two states
  <li>[http://enseignement-lsf.com/spip.php?article64#forum18028665 http://enseignement-lsf.com/spip.php?article64#forum18028665]</li>
*[[Covector mapping principle]]
 
*[[Differential dynamic programming]] — uses locally-quadratic models of the dynamics and cost functions
</ul>
*[[DNSS point]] — initial state for certain optimal control problems with multiple optimal solutions
*[[Legendre–Clebsch condition]] — second-order condition for solution of optimal control problem
*[[Pseudospectral optimal control]]
**[[Bellman pseudospectral method]] — based on Bellman's principle of optimality
**[[Chebyshev pseudospectral method]] — uses Chebyshev polynomials (of the first kind)
**[[Flat pseudospectral method]] — combines Ross–Fahroo pseudospectral method with differential flatness
**[[Gauss pseudospectral method]] — uses collocation at the Legendre–Gauss points
**[[Legendre pseudospectral method]] — uses Legendre polynomials
**[[Pseudospectral knotting method]] — generalization of pseudospectral methods in optimal control
**[[Ross–Fahroo pseudospectral method]] — class of pseudospectral method including Chebyshev, Legendre and knotting
*[[Ross–Fahroo lemma]] — condition to make discretization and duality operations commute
*[[Ross' π lemma]] — there is fundamental time constant within which a control solution must be computed for controllability and stability
*[[Sethi model]] — optimal control problem modelling advertising


[[Infinite-dimensional optimization]]
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*[[Semi-infinite programming]] — infinite number of variables and finite number of constraints, or other way around
*[[Shape optimization]], [[Topology optimization]] — optimization over a set of regions
**[[Topological derivative]] — derivative with respect to changing in the shape
*[[Generalized semi-infinite programming]] — finite number of variables, infinite number of constraints


===Uncertainty and randomness===
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*Approaches to deal with uncertainty:
 
**[[Markov decision process]]
  <li>[http://neoplanning.net/demo/modules/d3blog/details.php?bid=8/ http://neoplanning.net/demo/modules/d3blog/details.php?bid=8/]</li>
**[[Partially observable Markov decision process]]
 
**[[Probabilistic-based design optimization]]
  <li>[http://gift.xueersi.org/home.php?mod=space&uid=23553&do=blog&quickforward=1&id=4887075 http://gift.xueersi.org/home.php?mod=space&uid=23553&do=blog&quickforward=1&id=4887075]</li>
**[[Robust optimization]]
 
***[[Wald's maximin model]]
  <li>[http://www.middleeasttransparent.com/spip.php?article9467&lang=ar&id_forum=10581/ http://www.middleeasttransparent.com/spip.php?article9467&lang=ar&id_forum=10581/]</li>
**[[Scenario optimization]] — constraints are uncertain
 
**[[Stochastic approximation]]
  <li>[http://www.m-shell.net/framehelper.aspx?g=posts&t=368299 http://www.m-shell.net/framehelper.aspx?g=posts&t=368299]</li>
**[[Stochastic optimization]]
 
**[[Stochastic programming]]
  <li>[http://www.chcmp.com/bbs/forum.php?mod=viewthread&tid=132184 http://www.chcmp.com/bbs/forum.php?mod=viewthread&tid=132184]</li>
**[[Stochastic gradient descent]]
 
*[[Random optimization]] algorithms:
</ul>
**[[Random search]] — choose a point randomly in ball around current iterate
**[[Simulated annealing]]
***[[Adaptive simulated annealing]] — variant in which the algorithm parameters are adjusted during the computation.
***[[Great Deluge algorithm]]
***[[Mean field annealing]] — deterministic variant of simulated annealing
**[[Bayesian optimization]] — treats objective function as a random function and places a prior over it
**[[Evolutionary algorithm]]
***[[Differential evolution]]
***[[Evolutionary programming]]
***[[Genetic algorithm]], [[Genetic programming]]
****[[Genetic algorithms in economics]]
***[[MCACEA]] (Multiple Coordinated Agents Coevolution Evolutionary Algorithm) — uses an evolutionary algorithm for every agent
***[[Simultaneous perturbation stochastic approximation]] (SPSA)
**[[Luus–Jaakola]]
**[[Particle swarm optimization]]
**[[Stochastic tunneling]]
**[[Harmony search]] — mimicks the improvisation process of musicians
**see also the section ''Monte Carlo method''


===Theoretical aspects===
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*[[Convex analysis]] — function ''f'' such that ''f''(''tx'' + (1 − ''t'')''y'') ≥ ''tf''(''x'') + (1 − ''t'')''f''(''y'') for ''t'' ∈ [0,1]
**[[Pseudoconvex function]] — function ''f'' such that ∇''f'' · (''y'' − ''x'') ≥ 0 implies ''f''(''y'') ≥ ''f''(''x'')
**[[Quasiconvex function]] — function ''f'' such that ''f''(''tx'' + (1 − ''t'')''y'') ≤ max(''f''(''x''), ''f''(''y'')) for ''t'' ∈ [0,1]
**[[Subderivative]]
**[[Geodesic convexity]] — convexity for functions defined on a Riemannian manifold
*[[Duality (optimization)]]
**[[Weak duality]] — dual solution gives a bound on the primal solution
**[[Strong duality]] — primal and dual solutions are equivalent
**[[Shadow price]]
**[[Dual cone and polar cone]]
**[[Duality gap]] — difference between primal and dual solution
**[[Fenchel's duality theorem]] — relates minimization problems with maximization problems of convex conjugates
**[[Perturbation function]] — any function which relates to primal and dual problems
**[[Slater's condition]] — sufficient condition for strong duality to hold in a convex optimization problem
**[[Total dual integrality]] — concept of duality for integer linear programming
**[[Wolfe duality]] — for when objective function and constraints are differentiable
*[[Farkas' lemma]]
*[[Karush–Kuhn–Tucker conditions]] (KKT) — sufficient conditions for a solution to be optimal
**[[Fritz John conditions]] — variant of KKT conditions
*[[Lagrange multiplier]]
**[[Lagrange multipliers on Banach spaces]]
*[[Semi-continuity]]
*[[Complementarity theory]] — study of problems with constraints of the form &lang;''u'', ''v''&rang; = 0
**[[Mixed complementarity problem]]
***[[Mixed linear complementarity problem]]
***[[Lemke's algorithm]] — method for solving (mixed) linear complementarity problems
*[[Danskin's theorem]] — used in the analysis of minimax problems
*[[Maximum theorem]] — the maximum and maximizer are continuous as function of parameters, under some conditions
*[[No free lunch in search and optimization]]
*[[Relaxation (approximation)]] — approximating a given problem by an easier problem by relaxing some constraints
**[[Lagrangian relaxation]]
**[[Linear programming relaxation]] — ignoring the integrality constraints in a linear programming problem
*[[Self-concordant function]]
*[[Reduced cost]] — cost for increasing a variable by a small amount
*[[Hardness of approximation]] — computational complexity of getting an approximate solution


===Applications===
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*In geometry:
 
**[[Geometric median]] — the point minimizing the sum of distances to a given set of points
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**[[Chebyshev center]] — the centre of the smallest ball containing a given set of points
 
*In statistics:
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**[[Iterated conditional modes]] — maximizing joint probability of Markov random field
 
**[[Response surface methodology]] — used in the design of experiments
  <li>[http://ggcsw.com/forum.php?mod=viewthread&tid=115107 http://ggcsw.com/forum.php?mod=viewthread&tid=115107]</li>
*[[Automatic label placement]]
 
*[[Compressed sensing]] — reconstruct a signal from knowledge that it is sparse or compressible
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*[[Cutting stock problem]]
 
*[[Demand optimization]]
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*[[Destination dispatch]] — an optimization technique for dispatching elevators
 
*[[Energy minimization]]
</ul>
*[[Entropy maximization]]
*[[Highly optimized tolerance]]
*[[Hyperparameter optimization]]
*[[Inventory control problem]]
**[[Newsvendor model]]
**[[Extended newsvendor model]]
**[[Assemble-to-order system]]
*[[Linear programming decoding]]
*[[Linear search problem]] — find a point on a line by moving along the line
*[[Low-rank approximation]] — find best approximation, constraint is that rank of some matrix is smaller than a given number
*[[Meta-optimization]] — optimization of the parameters in an optimization method
*[[Multidisciplinary design optimization]]
*[[Optimal computing budget allocation]] — maximize the overall simulation efficiency for finding an optimal decision
*[[Paper bag problem]]
*[[Process optimization]]
*[[Recursive economics]] — individuals make a series of two-period optimization decisions over time.
*[[Stigler diet]]
*[[Space allocation problem]]
*[[Stress majorization]]
*[[Trajectory optimization]]
*[[Transportation theory (mathematics)|Transportation theory]]
*[[Wing-shape optimization]]


===Miscellaneous===
== Timberlands Boots Australia  and behavioral targeting ==
*[[Combinatorial optimization]]
*[[Dynamic programming]]
**[[Bellman equation]]
**[[Hamilton–Jacobi–Bellman equation]] — continuous-time analogue of Bellman equation
**[[Backward induction]] — solving dynamic programming problems by reasoning backwards in time
**[[Optimal stopping]] — choosing the optimal time to take a particular action
***[[Odds algorithm]]
***[[Robbins' problem]]
*[[Global optimization]]:
**[[BRST algorithm]]
**[[MCS algorithm]]
*[[Multi-objective optimization]] — there are multiple conflicting objectives
**[[Benson's algorithm]] — for linear [[vector optimization]] problems
*[[Bilevel optimization]] — studies problems in which one problem is embedded in another
*[[Optimal substructure]]
*[[Dykstra's projection algorithm]] — finds a point in intersection of two convex sets
*Algorithmic concepts:
**[[Barrier function]]
**[[Penalty method]]
**[[Trust region]]
*[[Test functions for optimization]]:
**[[Rosenbrock function]] — two-dimensional function with a banana-shaped valley
**[[Himmelblau's function]] — two-dimensional with four local minima, defined by <math>f(x, y) = (x^2+y-11)^2 + (x+y^2-7)^2</math>
**[[Rastrigin function]] — two-dimensional function with many local minima
**[[Shekel function]] — multimodal and multidimensional
*[[Mathematical Optimization Society]]


==Numerical quadrature (integration)==
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[[Numerical integration]] — the numerical evaluation of an integral
 
*[[Rectangle method]] — first-order method, based on (piecewise) constant approximation
  <li>[http://www.jamiatou.com/spip.php?article21 http://www.jamiatou.com/spip.php?article21]</li>
*[[Trapezoidal rule]] — second-order method, based on (piecewise) linear approximation
 
*[[Simpson's rule]] — fourth-order method, based on (piecewise) quadratic approximation
  <li>[http://cerisier.info/spip.php?article20/ http://cerisier.info/spip.php?article20/]</li>
**[[Adaptive Simpson's method]]
 
*[[Boole's rule]] — sixth-order method, based on the values at five equidistant points
  <li>[http://coalition.movementcamp.org/circle/content/tim-rayners-shout-out-invite-movementcamp#comment-17465546 http://coalition.movementcamp.org/circle/content/tim-rayners-shout-out-invite-movementcamp#comment-17465546]</li>
*[[Newton–Cotes formulas]] — generalizes the above methods
 
*[[Romberg's method]] — Richardson extrapolation applied to trapezium rule
  <li>[http://www.histoirepassion.eu/spip.php?article1895/ http://www.histoirepassion.eu/spip.php?article1895/]</li>
*[[Gaussian quadrature]] — highest possible degree with given number of points
 
**[[Chebyshev–Gauss quadrature]] — extension of Gaussian quadrature for integrals with weight {{nowrap|(1 − ''x''<sup>2</sub>)<sup>±1/2</sup>}} on [−1, 1]
  <li>[http://verdamilio.net/tonio/spip.php?article1970/ http://verdamilio.net/tonio/spip.php?article1970/]</li>
**[[Gauss–Hermite quadrature]] — extension of Gaussian quadrature for integrals with weight exp(−''x''<sup>2</sub>) on [−∞, ∞]
 
**[[Gauss–Jacobi quadrature]] — extension of Gaussian quadrature for integrals with weight (1 − ''x'')<sup>''α''</sup> (1 + ''x'')<sup>''β''</sup> on [−1, 1]
</ul>
**[[Gauss–Laguerre quadrature]] — extension of Gaussian quadrature for integrals with weight exp(−''x'') on [0, ∞]
**[[Gauss–Kronrod quadrature formula]] — nested rule based on Gaussian quadrature
**[[Gaussian quadrature|Gauss–Kronrod rules]]
*[[Tanh-sinh quadrature]] — variant of Gaussian quadrature which works well with singularities at the end points
*[[Clenshaw–Curtis quadrature]] — based on expanding the integrand in terms of Chebyshev polynomials
*[[Adaptive quadrature]] — adapting the subintervals in which the integration interval is divided depending on the integrand
*[[Monte Carlo integration]] — takes random samples of the integrand
**''See also [[#Monte Carlo method]]''
*[[Quantized state systems method]] (QSS) — based on the idea of state quantization
*[[Lebedev quadrature]] — uses a grid on a sphere with octahedral symmetry
*[[Sparse grid]]
*[[Coopmans approximation]]
*[[Numerical differentiation]] — for fractional-order integrals
**[[Numerical smoothing and differentiation]]
**[[Adjoint state method]] — approximates gradient of a function in an optimization problem
*[[Euler–Maclaurin formula]]


==Numerical methods for ordinary differential equations==
== Fake Ugg Australia  New York ==
[[Numerical methods for ordinary differential equations]] — the numerical solution of ordinary differential equations (ODEs)
*[[Euler method]] — the most basic method for solving an ODE
*[[Explicit and implicit methods]] — implicit methods need to solve an equation at every step
*[[Backward Euler method]] — implicit variant of the Euler method
*[[Trapezoidal rule (differential equations)|Trapezoidal rule]] — second-order implicit method
*[[Runge–Kutta methods]] — one of the two main classes of methods for initial-value problems
**[[Midpoint method]] — a second-order method with two stages
**[[Heun's method]] — either a second-order method with two stages, or a third-order method with three stages
**[[Bogacki–Shampine method]] — a third-order method with four stages (FSAL) and an embedded fourth-order method
**[[Cash–Karp method]] — a fifth-order method with six stages and an embedded fourth-order method
**[[Dormand–Prince method]] — a fifth-order method with seven stages (FSAL) and an embedded fourth-order method
**[[Runge–Kutta–Fehlberg method]] — a fifth-order method with six stages and an embedded fourth-order method
**[[Gauss–Legendre method]] — family of A-stable method with optimal order based on Gaussian quadrature
**[[Butcher group]] — algebraic formalism involving rooted trees for analysing Runge–Kutta methods
**[[List of Runge–Kutta methods]]
*[[Linear multistep method]] — the other main class of methods for initial-value problems
**[[Backward differentiation formula]] — implicit methods of order 2 to 6; especially suitable for stiff equations
**[[Numerov's method]] — fourth-order method for equations of the form <math>y'' = f(t,y)</math>
**[[Predictor–corrector method]] — uses one method to approximate solution and another one to increase accuracy
*[[General linear methods]] — a class of methods encapsulating linear multistep and Runge-Kutta methods
*[[Bulirsch–Stoer algorithm]] — combines the midpoint method with Richardson extrapolation to attain arbitrary order
*[[Exponential integrator]] — based on splitting ODE in a linear part, which is solved exactly, and a nonlinear part
*Methods designed for the solution of ODEs from classical physics:
**[[Newmark-beta method]] — based on the extended mean-value theorem
**[[Verlet integration]] — a popular second-order method
**[[Leapfrog integration]] — another name for Verlet integration
**[[Beeman's algorithm]] — a two-step method extending the Verlet method
**[[Dynamic relaxation]]
*[[Geometric integrator]] — a method that preserves some geometric structure of the equation
**[[Symplectic integrator]] — a method for the solution of Hamilton's equations that preserves the symplectic structure
***[[Variational integrator]] — symplectic integrators derived using the underlying variational principle
***[[Semi-implicit Euler method]] — variant of Euler method which is symplectic when applied to separable Hamiltonians
**[[Energy drift]] — phenomenon that energy, which should be conserved, drifts away due to numerical errors
*Other methods for initial value problems (IVPs):
**[[Bi-directional delay line]]
**[[Partial element equivalent circuit]]
*Methods for solving two-point boundary value problems (BVPs):
**[[Shooting method]]
**[[Direct multiple shooting method]] — divides interval in several subintervals and applies the shooting method on each subinterval
*Methods for solving differential-algebraic equations (DAEs), i.e., ODEs with constraints:
**[[Constraint algorithm]] — for solving Newton's equations with constraints
**[[Pantelides algorithm]] — for reducing the index of a DEA
*Methods for solving stochastic differential equations (SDEs):
**[[Euler–Maruyama method]] — generalization of the Euler method for SDEs
**[[Milstein method]] — a method with strong order one
**[[Runge–Kutta method (SDE)]] — generalization of the family of Runge–Kutta methods for SDEs
*Methods for solving integral equations:
**[[Nyström method]] — replaces the integral with a quadrature rule
*Analysis:
**[[Truncation error (numerical integration)]] — local and global truncation errors, and their relationships
***[[Lady Windermere's Fan (mathematics)]] — telescopic identity relating local and global truncation errors
*[[Stiff equation]] — roughly, an ODE for which unstable methods need a very short step size, but stable methods do not
**[[L-stability]] — method is A-stable and stability function vanishes at infinity
**[[Dynamic errors of numerical methods of ODE discretization]] — logarithm of stability function
*[[Adaptive stepsize]] — automatically changing the step size when that seems advantageous


==Numerical methods for partial differential equations==
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[[Numerical partial differential equations]] — the numerical solution of partial differential equations (PDEs)
 
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===Finite difference methods===
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[[Finite difference method]] — based on approximating differential operators with difference operators
*[[Finite difference]] — the discrete analogue of a differential operator
**[[Finite difference coefficient]] — table of coefficients of finite-difference approximations to derivatives
**[[Discrete Laplace operator]] — finite-difference approximation of the Laplace operator
***[[Eigenvalues and eigenvectors of the second derivative]] — includes eigenvalues of discrete Laplace operator
***[[Kronecker sum of discrete Laplacians]] — used for Laplace operator in multiple dimensions
**[[Discrete Poisson equation]] — discrete analogue of the Poisson equation using the discrete Laplace operator
*[[Stencil (numerical analysis)]] — the geometric arrangements of grid points affected by a basic step of the algorithm
**[[Compact stencil]] — stencil which only uses a few grid points, usually only the immediate and diagonal neighbours
***[[Higher-order compact finite difference scheme]]
**[[Non-compact stencil]] — any stencil that is not compact
**[[Five-point stencil]] — two-dimensional stencil consisting of a point and its four immediate neighbours on a rectangular grid
*Finite difference methods for heat equation and related PDEs:
**[[FTCS scheme]] (forward-time central-space) — first-order explicit
**[[Crank–Nicolson method]] — second-order implicit
*Finite difference methods for hyperbolic PDEs like the wave equation:
**[[Lax–Friedrichs method]] — first-order explicit
**[[Lax–Wendroff method]] — second-order explicit
**[[MacCormack method]] — second-order explicit
**[[Upwind scheme]]
***[[Upwind differencing scheme for convection]] — first-order scheme for convection–diffusion problems
**[[Lax–Wendroff theorem]] — conservative scheme for hyperbolic system of conservation laws converges to the weak solution
*[[Alternating direction implicit method]] (ADI) — update using the flow in ''x''-direction and then using flow in ''y''-direction
*[[Nonstandard finite difference scheme]]
*Specific applications:
**[[Finite difference methods for option pricing]]
**[[Finite-difference time-domain method]] — a finite-difference method for electrodynamics


===Finite element methods===
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[[Finite element method]] — based on a discretization of the space of solutions
 
*[[Finite element method in structural mechanics]] — a physical approach to finite element methods
  <li>[http://cerisier.info/spip.php?article20/ http://cerisier.info/spip.php?article20/]</li>
*[[Galerkin method]] — a finite element method in which the residual is orthogonal to the finite element space
 
**[[Discontinuous Galerkin method]] — a Galerkin method in which the approximate solution is not continuous
  <li>[http://enseignement-lsf.com/spip.php?article64#forum17973022 http://enseignement-lsf.com/spip.php?article64#forum17973022]</li>
*[[Rayleigh–Ritz method]] — a finite element method based on variational principles
 
*[[Spectral element method]] — high-order finite element methods
  <li>[http://ldsbee.com/index.php?page=item&id=2475590 http://ldsbee.com/index.php?page=item&id=2475590]</li>
*[[hp-FEM]] — variant in which both the size and the order of the elements are automatically adapted
 
*Examples of finite elemets:
  <li>[http://www.wxdemir.com/news/html/?750572.html http://www.wxdemir.com/news/html/?750572.html]</li>
**[[Bilinear quadrilateral element]] — also known as the Q4 element
 
**[[Constant strain triangle element]] (CST) — also known as the T3 element
  <li>[http://www.philatelie-france-russie.fr/spip.php?article51/ http://www.philatelie-france-russie.fr/spip.php?article51/]</li>
**[[Barsoum elements]]
 
*[[Direct stiffness method]] — a particular implementation of the finite element method, often used in structural analysis
</ul>
*[[Trefftz method]]
*[[Finite element updating]]
*[[Extended finite element method]] — puts functions tailored to the problem in the approximation space
*[[Functionally graded element]]s — elements for describing functionally graded materials
*[[Superelement]] — particular grouping of finite elements, employed as a single element
*[[Interval finite element]] method — combination of finite elements with interval arithmetic
*[[Discrete exterior calculus]] — discrete form of the exterior calculus of differential geometry
*[[Modal analysis using FEM]] — solution of eigenvalue problems to find natural vibrations
*[[Céa's lemma]] — solution in the finite-element space is an almost best approximation in that space of the true solution
*[[Patch test (finite elements)]] — simple test for the quality of a finite element
*[[MAFELAP]] (MAthematics of Finite ELements and APplications) — international conference held at Brunel University
*[[NAFEMS]] — not-for-profit organisation that sets and maintains standards in computer-aided engineering analysis
*[[Multiphase topology optimisation]] — technique based on finite elements for determining optimal composition of a mixture
*[[Interval finite element]]
*[[Applied element method]] — for simulation of cracks and structural collapse
*[[Wood–Armer method]] — structural analysis method based on finite elements used to design reinforcement for concrete slabs
*[[Isogeometric analysis]] — integrates finite elements into conventional NURBS-based CAD design tools
*[[Stiffness matrix]] — finite-dimensional analogue of differential operator
*Combination with meshfree methods:
**[[Weakened weak form]] — form of a PDE that is weaker than the standard weak form
**[[G space]] — functional space used in formulating the weakened weak form
**[[Smoothed finite element method]]
*[[List of finite element software packages]]
 
===Other methods===
*[[Spectral method]] — based on the Fourier transformation
**[[Pseudo-spectral method]]
*[[Method of lines]] — reduces the PDE to a large system of ordinary differential equations
*[[Boundary element method]] (BEM) — based on transforming the PDE to an integral equation on the boundary of the domain
**[[Interval boundary element method]] — a version using interval arithmetics
*[[Analytic element method]] — similar to the boundary element method, but the integral equation is evaluated analytically
*[[Finite volume method]] — based on dividing the domain in many small domains; popular in computational fluid dynamics
**[[Godunov's scheme]] — first-order conservative scheme for fluid flow, based on piecewise constant approximation
**[[MUSCL scheme]] — second-order variant of Godunov's scheme
**[[AUSM]] — advection upstream splitting method
**[[Flux limiter]] — limits spatial derivatives (fluxes) in order to avoid spurious oscillations
**[[Riemann solver]] — a solver for Riemann problems (a conservation law with piecewise constant data)
**[[Properties of discretization schemes]] — finite volume methods can be conservative, bounded, etc.
*[[Discrete element method]] — a method in which the elements can move freely relative to each other
**[[Extended discrete element method]] — adds properties such as strain to each particle
**[[Movable cellular automaton]] — combination of cellular automata with discrete elements
*[[Meshfree methods]] — does not use a mesh, but uses a particle view of the field
**[[Discrete least squares meshless method]] — based on minimization of weighted summation of the squared residual
**[[Diffuse element method]]
**[[Finite pointset method]] — represent continuum by a point cloud
**[[Moving Particle Semi-implicit Method]]
**[[Method of fundamental solutions]] (MFS) — represents solution as linear combination of fundamental solutions
**Variants of MFS with source points on the physical boundary:
***[[Boundary knot method]] (BKM)
***[[Boundary particle method]] (BPM)
***[[Regularized meshless method]] (RMM)
***[[Singular boundary method]] (SBM)
*Methods designed for problems from electromagnetics:
**[[Finite-difference time-domain method]] — a finite-difference method
**[[Rigorous coupled-wave analysis]] — semi-analytical Fourier-space method based on Floquet's theorem
**[[Transmission-line matrix method]] (TLM) — based on analogy between electromagnetic field and mesh of transmission lines
**[[Uniform theory of diffraction]] — specifically designed for scattering problems
*[[Particle-in-cell]] — used especially in fluid dynamics
**[[Multiphase particle-in-cell method]] — considers solid particles as both numerical particles and fluid
*[[High-resolution scheme]]
*[[Shock capturing method]]
*[[Vorticity confinement]] — for vortex-dominated flows in fluid dynamics, similar to shock capturing
*[[Split-step method]]
*[[Fast marching method]]
*[[Orthogonal collocation]]
*[[Lattice Boltzmann methods]] — for the solution of the Navier-Stokes equations
*[[Roe solver]] — for the solution of the Euler equation
*[[Relaxation (iterative method)]] — a method for solving elliptic PDEs by converting them to evolution equations
*Broad classes of methods:
**[[Mimesis (mathematics)|Mimetic methods]] — methods that respect in some sense the structure of the original problem
**[[Multiphysics]] — models consisting of various submodels with different physics
**[[Immersed boundary method]] — for simulating elastic structures immersed within fluids
*[[Multisymplectic integrator]] — extension of symplectic integrators, which are for ODEs
*[[Stretched grid method]] — for problems solution that can be related to an elastic grid behavior.
 
===Techniques for improving these methods===
*[[Multigrid method]] — uses a hierarchy of nested meshes to speed up the methods
*[[Domain decomposition methods]] — divides the domain in a few subdomains and solves the PDE on these subdomains
**[[Additive Schwarz method]]
**[[Abstract additive Schwarz method]] — abstract version of additive Schwarz without reference to geometric information
**[[Balancing domain decomposition method]] (BDD) — preconditioner for symmetric positive definite matrices
**[[BDDC|Balancing domain decomposition by constraints]] (BDDC) — further development of BDD
**[[FETI|Finite element tearing and interconnect]] (FETI)
**[[FETI-DP]] — further development of FETI
**[[Fictitious domain method]] — preconditioner constructed with a structured mesh on a fictitious domain of simple shape
**[[Mortar methods]] — meshes on subdomain do not mesh
**[[Neumann–Dirichlet method]] — combines Neumann problem on one subdomain with Dirichlet problem on other subdomain
**[[Neumann–Neumann methods]] — domain decomposition methods that use Neumann problems on the subdomains
**[[Poincaré–Steklov operator]] — maps tangential electric field onto the equivalent electric current
**[[Schur complement method]] — early and basic method on subdomains that do not overlap
**[[Schwarz alternating method]] — early and basic method on subdomains that overlap
*[[Coarse space (numerical analysis)|Coarse space]] — variant of the problem which uses a discretization with fewer degrees of freedom
*[[Adaptive mesh refinement]] — uses the computed solution to refine the mesh only where necessary
*[[Fast multipole method]] — hierarchical method for evaluating particle-particle interactions
*[[Perfectly matched layer]] — artificial absorbing layer for wave equations, used to implement absorbing boundary conditions
 
===Grids and meshes===
*[[Grid classification]] / [[Types of mesh]]:
**[[Polygon mesh]] — consists of polygons in 2D or 3D
**[[Triangle mesh]] — consists of triangles in 2D or 3D
***[[Triangulation (geometry)]] — subdivision of given region in triangles, or higher-dimensional analogue
***[[Nonobtuse mesh]] — mesh in which all angles are less than or equal to 90°
***[[Point set triangulation]] — triangle mesh such that given set of point are all a vertex of a triangle
***[[Polygon triangulation]] — triangle mesh inside a polygon
***[[Delaunay triangulation]] — triangulation such that no vertex is inside the circumcentre of a triangle
***[[Constrained Delaunay triangulation]] — generalization of the Delaunay triangulation that forces certain required segments into the triangulation
***[[Pitteway triangulation]] — for any point, triangle containing it has nearest neighbour of the point as a vertex
***[[Minimum-weight triangulation]] — triangulation of minimum total edge length
***[[Kinetic triangulation]] — a triangulation that moves over time
***[[Triangulated irregular network]]
***[[Quasi-triangulation]] — subdivision into simplices, where vertiсes are not points but arbitrary sloped line segments
**[[Volume mesh]] — consists of three-dimensional shapes
**[[Regular grid]] — consists of congruent parallelograms, or higher-dimensional analogue
**[[Unstructured grid]]
**[[Geodesic grid]] — isotropic grid on a sphere
*[[Mesh generation]]
**[[Image-based meshing]] — automatic procedure of generating meshes from 3D image data
**[[Marching cubes]] — extracts a polygon mesh from a scalar field
**[[Parallel mesh generation]]
**[[Ruppert's algorithm]] — creates quality Delauney triangularization from piecewise linear data
*Subdivisions:
*[[Apollonian network]] — undirected graph formed by recursively subdividing a triangle
*[[Barycentric subdivision]] —  standard way of dividing arbitrary convex polygons into triangles, or the higher-dimensional analogue
*Improving an existing mesh:
**[[Chew's second algorithm]] — improves Delauney triangularization by refining poor-quality triangles
**[[Laplacian smoothing]] — improves polynomial meshes by moving the vertices
*[[Jump-and-Walk algorithm]] — for finding triangle in a mesh containing a given point
*[[Spatial twist continuum]] — dual representation of a mesh consisting of hexahedra
*[[Pseudotriangle]] — simply connected region between any three mutually tangent convex sets
*[[Simplicial complex]] — all vertices, line segments, triangles, tetrahedra, &hellip;, making up a mesh
 
===Analysis===
*[[Lax equivalence theorem]] — a consistent method is convergent if and only if it is stable
*[[Courant–Friedrichs–Lewy condition]] — stability condition for hyperbolic PDEs
*[[Von Neumann stability analysis]] — all Fourier components of the error should be stable
*[[Numerical diffusion]] — diffusion introduced by the numerical method, above to that which is naturally present
**[[False diffusion]]
*[[Numerical resistivity]] — the same, with resistivity instead of diffusion
*[[Weak formulation]] — a functional-analytic reformulation of the PDE necessary for some methods
*[[Total variation diminishing]] — property of schemes that do not introduce spurious oscillations
*[[Godunov's theorem]] — linear monotone schemes can only be of first order
*[[Motz's problem]] — benchmark problem for singularity problems
 
==[[Monte Carlo method]]==
*Variants of the Monte Carlo method:
**[[Direct simulation Monte Carlo]]
**[[Quasi-Monte Carlo method]]
**[[Markov chain Monte Carlo]]
***[[Metropolis–Hastings algorithm]]
****[[Multiple-try Metropolis]] — modification which allows larger step sizes
****[[Wang and Landau algorithm]] — extension of Metropolis Monte Carlo
****[[Equation of State Calculations by Fast Computing Machines]] — 1953 article proposing the Metropolis Monte Carlo algorithm
****[[Multicanonical ensemble]] — sampling technique that uses Metropolis–Hastings to compute integrals
***[[Gibbs sampling]]
***[[Coupling from the past]]
***[[Reversible-jump Markov chain Monte Carlo]]
**[[Dynamic Monte Carlo method]]
***[[Kinetic Monte Carlo]]
***[[Gillespie algorithm]]
**[[Particle filter]]
***[[Auxiliary particle filter]]
**[[Reverse Monte Carlo]]
**[[Demon algorithm]]
*[[Pseudo-random number sampling]]
**[[Inverse transform sampling]] — general and straightforward method but computationally expensive
**[[Rejection sampling]] — sample from a simpler distribution but reject some of the samples
***[[Ziggurat algorithm]] — uses a pre-computed table covering the probability distribution with rectangular segments
**For sampling from a normal distribution:
***[[Box–Muller transform]]
***[[Marsaglia polar method]]
**[[Convolution random number generator]] — generates a random variable as a sum of other random variables
**[[Indexed search]]
*[[Variance reduction]] techniques:
**[[Antithetic variates]]
**[[Control variates]]
**[[Importance sampling]]
**[[Stratified sampling]]
**[[VEGAS algorithm]]
*[[Low-discrepancy sequence]]
**[[Constructions of low-discrepancy sequences]]
*[[Event generator]]
*[[Parallel tempering]]
*[[Umbrella sampling]] — improves sampling in physical systems with significant energy barriers
*[[Hybrid Monte Carlo]]
*[[Ensemble Kalman filter]] — recursive filter suitable for problems with a large number of variables
*[[Transition path sampling]]
*Applications:
**[[Ensemble forecasting]] — produce multiple numerical predictions from slightly initial conditions or parameters
**[[Bond fluctuation model]] — for simulating the conformation and dynamics of polymer systems
**[[Iterated filtering]]
**[[Metropolis light transport]]
**[[Monte Carlo localization]] — estimates the position and orientation of a robot
**[[Monte Carlo methods for electron transport]]
**[[Monte Carlo method for photon transport]]
**[[Monte Carlo methods in finance]]
***[[Monte Carlo methods for option pricing]]
***[[Quasi-Monte Carlo methods in finance]]
**[[Monte Carlo molecular modeling]]
***[[Path integral molecular dynamics]] — incorporates Feynman path integrals
**[[Quantum Monte Carlo]]
***[[Diffusion Monte Carlo]] — uses a Green function to solve the Schrödinger equation
***[[Gaussian quantum Monte Carlo]]
***[[Path integral Monte Carlo]]
***[[Reptation Monte Carlo]]
***[[Variational Monte Carlo]]
**Methods for simulating the Ising model:
***[[Swendsen–Wang algorithm]] — entire sample is divided into equal-spin clusters
***[[Wolff algorithm]] — improvement of the Swendsen–Wang algorithm
***[[Metropolis–Hastings algorithm]]
**[[Auxiliary field Monte Carlo]] — computes averages of operators in many-body quantum mechanical problems
**[[Cross-entropy method]] — for multi-extremal optimization and importance sampling
*Also see the [[list of statistics topics]]
 
==Applications==
*[[Computational physics]]
**[[Computational electromagnetics]]
**[[Computational fluid dynamics]] (CFD)
***[[Numerical methods in fluid mechanics]]
***[[Large eddy simulation]]
***[[Smoothed-particle hydrodynamics]]
***[[Aeroacoustic analogy]] — used in numerical aeroacoustics to reduce sound sources to simple emitter types
***[[Stochastic Eulerian Lagrangian method]] — uses Eulerian description for fluids and Lagrangian for structures
***[[Explicit algebraic stress model]]
**[[Computational magnetohydrodynamics]] (CMHD) — studies electrically conducting fluids
**[[Climate model]]
**[[Numerical weather prediction]]
***[[Geodesic grid]]
**[[Celestial mechanics]]
***[[Numerical model of the Solar System]]
**[[Quantum jump method]] — used for simulating open quantum systems, operates on wave function
**[[Dynamic Design Analysis Method]] (DDAM) — for evaluating effect of underwater explosions on equipment
*[[Computational chemistry]]
**[[Cell lists]]
**[[Coupled cluster]]
**[[Density functional theory]]
**[[DIIS]] — direct inversion in (or of) the iterative subspace
*[[Computational sociology]]
*[[Computational statistics]]
 
==Software==
For software, see the [[list of numerical analysis software]].
 
[[Category:Numerical analysis|*Topics]]
[[Category:Mathematics-related lists|Numerical analysis topics]]
[[Category:Outlines]]

Revision as of 06:14, 23 February 2014

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