List of numerical analysis topics

This is a list of numerical analysis topics.

General

• Validated numerics
• Iterative method
• Rate of convergence — the speed at which a convergent sequence approaches its limit
• 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
• Aitken's delta-squared process — most useful for linearly converging sequences
• Minimum polynomial extrapolation — for vector sequences
• Richardson extrapolation
• Shanks transformation — similar to Aitken's delta-squared process, but applied to the partial sums
• Van Wijngaarden transformation — for accelerating the convergence of an alternating series
• Abramowitz and Stegun — book containing formulas and tables of many special functions
• Digital Library of Mathematical Functions — successor of book by Abramowitz and Stegun
• Curse of dimensionality
• 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
• 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

Error analysis (mathematics)

• Approximation
• Approximation error
• Catastrophic cancellation
• 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
• Residual (numerical analysis)
• Relative change and difference — the relative difference between x and y is |x − y| / max(|x|, |y|)
• Significant figures
• Artificial precision — when a numerical value or semantic is expressed with more precision than was initially provided from measurement or user input
• False precision — giving more significant figures than appropriate
• Sterbenz lemma
• Truncation error — error committed by doing only a finite numbers of steps
• Well-posed problem
• Affine arithmetic

Elementary and special functions

• Unrestricted algorithm
• Summation:
• Kahan summation algorithm
• Pairwise summation — slightly worse than Kahan summation but cheaper
• Binary splitting
• 2Sum
• Multiplication:
• Multiplication algorithm — general discussion, simple methods
• Karatsuba algorithm — the first algorithm which is faster than straightforward multiplication
• Toom–Cook multiplication — generalization of Karatsuba multiplication
• Schönhage–Strassen algorithm — based on Fourier transform, asymptotically very fast
• Fürer's algorithm — asymptotically slightly faster than Schönhage–Strassen
• Division algorithm — for computing quotient and/or remainder of two numbers
• Long division
• Restoring division
• Non-restoring division
• SRT division
• Newton–Raphson division: uses Newton's method to find the reciprocal of D, and multiply that reciprocal by N to find the final quotient Q.
• Goldschmidt division
• Exponentiation:
• Exponentiation by squaring
• Addition-chain exponentiation
• Multiplicative inverse Algorithms: for computing a number's multiplicative inverse (reciprocal).
• Newton's method
• Polynomials:
• Horner's method
• Estrin's scheme — modification of the Horner scheme with more possibilities for parallelization
• Clenshaw algorithm
• De Casteljau's algorithm
• Square roots and other roots:
• Integer square root
• Methods of computing square roots
• nth root algorithm
• hypot — the function (x² + y²)^1/2
• Alpha max plus beta min algorithm — approximates hypot(x,y)
• Fast inverse square root — calculates 1 / 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^x
• 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:
• 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

Numerical linear algebra — study of numerical algorithms for linear algebra problems

Basic concepts

• Types of matrices appearing in numerical analysis:
• Sparse matrix
• Band matrix
• Bidiagonal matrix
• Tridiagonal matrix
• Pentadiagonal matrix
• Skyline matrix
• Circulant matrix
• Triangular matrix
• Diagonally dominant matrix
• Block matrix — matrix composed of smaller matrices
• Stieltjes matrix — symmetric positive definite with non-positive off-diagonal entries
• 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 decompositions:
• 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 — 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

• 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 over-relaxation (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's method (SIP — Strongly 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):
• 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

Eigenvalue algorithm — a numerical algorithm for locating the eigenvalues of a matrix

• Power iteration
• Inverse iteration
• Rayleigh quotient iteration
• Arnoldi iteration — based on Krylov subspaces
• Lanczos algorithm — Arnoldi, specialized for positive-definite matrices
• Block Lanczos algorithm — for when matrix is over a finite field
• QR algorithm
• Jacobi eigenvalue algorithm — select a small submatrix which can be diagonalized exactly, and repeat
• Jacobi rotation — the building block, almost a Givens rotation
• Jacobi method for complex Hermitian matrices
• Divide-and-conquer eigenvalue algorithm
• 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

• 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

Interpolation — construct a function going through some given data points

• Nearest-neighbor interpolation — takes the value of the nearest neighbor

Polynomial interpolation

Polynomial interpolation — interpolation by polynomials

• Linear interpolation
• Runge's phenomenon
• Vandermonde matrix
• Chebyshev polynomials
• Chebyshev nodes
• Lebesgue constants
• 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 R² with unique polynomial interpolant and minimal growth of Lebesgue constant
• Hermite interpolation
• Birkhoff interpolation
• Abel–Goncharov interpolation

Spline interpolation

Spline interpolation — interpolation by piecewise polynomials

• Spline (mathematics) — the piecewise polynomials used as interpolants
• Perfect spline — polynomial spline of degree m whose mth derivate is ±1
• Cubic Hermite spline
• Centripetal Catmull–Rom spline — special case of cubic Hermite splines without self-intersections or cusps
• Monotone cubic interpolation
• Hermite spline
• Bézier curve
• De Casteljau's algorithm
• composite Bézier curve
• Generalizations to more dimensions:
• Bézier triangle — maps a triangle to R³
• Bézier surface — maps a square to R³
• 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

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^m 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

• Simple rational approximation
• Polynomial and rational function modeling — comparison of polynomial and rational interpolation
• Wavelet
• Continuous wavelet
• Transfer matrix
• See also: List of functional analysis topics, List of wavelet-related transforms
• Inverse distance weighting
• Radial basis function (RBF) — a function of the form ƒ(x) = φ(|x−x₀|)
• Polyharmonic spline — a commonly used radial basis function
• Thin plate spline — a specific polyharmonic spline: r² log r
• Hierarchical RBF
• Subdivision surface — constructed by recursively subdividing a piecewise linear interpolant
• 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
• 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

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 L²-norm
• Minimax approximation algorithm — minimizes the maximum error over an interval (the L^∞-norm)
• Equioscillation theorem — characterizes the best approximation in the L^∞-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^∞-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^p 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^−n x^k 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 L² 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

• Extrapolation
• Linear predictive analysis — linear extrapolation
• Unisolvent functions — functions for which the interpolation problem has a unique solution
• Regression analysis
• Isotonic regression
• Curve-fitting compaction
• Interpolation (computer graphics)

Finding roots of nonlinear equations

See #Numerical linear algebra for linear equations

Root-finding algorithm — algorithms for solving the equation f(x) = 0

• General methods:
• Bisection method — simple and robust; linear convergence
• Lehmer–Schur algorithm — variant for complex functions
• Fixed-point iteration
• 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
• Newton fractal — indicates which initial condition converges to which root under Newton iteration
• Quasi-Newton method — uses an approximation of the Jacobian:
• 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
• 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
• 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 parameter in the equation changes
• Piecewise linear continuation

Optimization

Mathematical optimization — algorithm for finding maxima or minima of a given function

Basic concepts

• Active set
• Candidate solution
• Constraint (mathematics)
• Constrained optimization — studies optimization problems with constraints
• Binary constraint — a constraint that involves exactly two variables
• Corner solution
• Feasible region — contains all solutions that satisfy the constraints but may not be optimal
• Global optimum and Local optimum
• Maxima and minima
• Slack variable
• Continuous optimization
• Discrete optimization

Linear programming

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

Convex optimization

• Quadratic programming
• Linear least squares (mathematics)
• Total least squares
• Frank–Wolfe algorithm
• Sequential minimal optimization — breaks up large QP problems into a series of smallest possible QP problems
• Bilinear program
• Basis pursuit — minimize L₁-norm of vector subject to linear constraints
• Basis pursuit denoising (BPDN) — regularized version of basis pursuit
• In-crowd algorithm — algorithm for solving basis pursuit denoising
• Linear matrix inequality
• Conic optimization
• 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 method — 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

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 ≥ 0, f(x) ≥ 0 and x^T 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 weighted 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 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 constrained 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
• Nearest neighbor search
• Space mapping — uses "coarse" (ideal or low-fidelity) and "fine" (practical or high-fidelity) models

Optimal control and infinite-dimensional optimization

Optimal control

• Pontryagin's minimum principle — infinite-dimensional version of Lagrange multipliers
• Costate equations — equation for the "Lagrange multipliers" in Pontryagin's minimum principle
• Hamiltonian (control theory) — minimum principle says that this function should be minimized
• Types of problems:
• Linear-quadratic regulator — system dynamics is a linear differential equation, objective is quadratic
• 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
• Algebraic Riccati equation — matrix equation occurring in many optimal control problems
• Bang–bang control — control that switches abruptly between two states
• Covector mapping principle
• Differential dynamic programming — uses locally-quadratic models of the dynamics and cost functions
• 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

• 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

• Approaches to deal with uncertainty:
• Markov decision process
• Partially observable Markov decision process
• Robust optimization
• Wald's maximin model
• Scenario optimization — constraints are uncertain
• Stochastic approximation
• Stochastic optimization
• Stochastic programming
• Stochastic gradient descent
• Random optimization algorithms:
• 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

• 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

• In geometry:
• Geometric median — the point minimizing the sum of distances to a given set of points
• Chebyshev center — the centre of the smallest ball containing a given set of points
• In statistics:
• Iterated conditional modes — maximizing joint probability of Markov random field
• Response surface methodology — used in the design of experiments
• Automatic label placement
• Compressed sensing — reconstruct a signal from knowledge that it is sparse or compressible
• Cutting stock problem
• Demand optimization
• Destination dispatch — an optimization technique for dispatching elevators
• Energy minimization
• 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
• Wing-shape optimization

Miscellaneous

• 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
• Rastrigin function — two-dimensional function with many local minima
• Shekel function — multimodal and multidimensional
• Mathematical Optimization Society

Numerical quadrature (integration)

Numerical integration — the numerical evaluation of an integral

• Rectangle method — first-order method, based on (piecewise) constant approximation
• Trapezoidal rule — second-order method, based on (piecewise) linear approximation
• Simpson's rule — fourth-order method, based on (piecewise) quadratic approximation
• Adaptive Simpson's method
• Boole's rule — sixth-order method, based on the values at five equidistant points
• Newton–Cotes formulas — generalizes the above methods
• Romberg's method — Richardson extrapolation applied to trapezium rule
• Gaussian quadrature — highest possible degree with given number of points
• Chebyshev–Gauss quadrature — extension of Gaussian quadrature for integrals with weight on [−1, 1]
• Gauss–Hermite quadrature — extension of Gaussian quadrature for integrals with weight exp(−x²) on [−∞, ∞]
• Gauss–Jacobi quadrature — extension of Gaussian quadrature for integrals with weight (1 − x)^α (1 + x)^β on [−1, 1]
• 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
• 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

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 — 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
• 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
• Adaptive stepsize — automatically changing the step size when that seems advantageous
• Parareal -- a parallel-in-time integration algorithm

Numerical methods for partial differential equations

Numerical partial differential equations — the numerical solution of partial differential equations (PDEs)

Finite difference methods

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, gradient discretisation methods

Finite element method — based on a discretization of the space of solutions gradient discretisation method — based on both the discretization of the solution and of its gradient

• Finite element method in structural mechanics — a physical approach to finite element methods
• 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
• Rayleigh–Ritz method — a finite element method based on variational principles
• Spectral element method — high-order finite element methods
• hp-FEM — variant in which both the size and the order of the elements are automatically adapted
• Examples of finite elements:
• Bilinear quadrilateral element — also known as the Q4 element
• Constant strain triangle element (CST) — also known as the T3 element
• Quadratic quadrilateral element — also known as the Q8 element
• Barsoum elements
• Direct stiffness method — a particular implementation of the finite element method, often used in structural analysis
• Trefftz method
• Finite element updating
• Extended finite element method — puts functions tailored to the problem in the approximation space
• Functionally graded elements — 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
• Loubignac iteration
• 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
• Variational multiscale 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:
• 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
• Balancing domain decomposition by constraints (BDDC) — further development of BDD
• 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 — 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 vertices 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, ..., 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 dispersion
• 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
• Walk-on-spheres method — to generate exit-points of Brownian motion from bounded domains
• 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 a large list of software, see the list of numerical-analysis software.

Journals

• Acta Numerica
• Mathematics of Computation (published by the American Mathematical Society)
• Journal of Computational and Applied Mathematics
• BIT Numerical Mathematics
• Numerische Mathematik
• Journals from the Society for Industrial and Applied Mathematics
• SIAM Journal on Numerical Analysis
• SIAM Journal on Scientific Computing

Researchers

• Cleve Moler
• Gene H. Golub
• James H. Wilkinson
• Margaret H. Wright
• Nicholas J. Higham
• Nick Trefethen
• Peter Lax
• Richard S. Varga
• Ulrich W. Kulisch
• Vladik Kreinovich
• Chi-Wang Shu

References