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General

• 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

• 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 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

• Summation:

• Kahan summation algorithm

• Pairwise summation — slightly worse than Kahan summation but cheaper

• Binary splitting

• 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 remainder of two numbers

• Exponentiation:

• 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

• Shifting nth root algorithm — similar to long division

• hypot — the function (x2 + y2)1/2

• 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 ex

• 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:

• 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

• 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 — study of numerical algorithms for linear algebra problems

• 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

• 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

• 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 – 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

• 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

• 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

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 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 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

• Monotone cubic interpolation

• Hermite spline

• Bézier spline

• Bézier curve

• De Casteljau's algorithm

• 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 2m entries used in many FFTs.

• Butterfly diagram

• Twiddle factor — the trigonometric constant coefficients that are multiplied by the data

• 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

• 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

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 Lp 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

• ErdÅ‘s–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 xk 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' and f''; 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

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

Basic concepts

• Active set

• Candidate solution

• Constraint (mathematics)

• 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 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

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 xT 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 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

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

• Caratheodory-π solution — generalized solution to an ordinary differential equation whose right-hand side is not differentiable

• 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

• Probabilistic-based design optimization

• 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

• 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 âŸ¨u, vâŸ© = 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

• 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

• 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 program — problem 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 (1 − x2)±1/2 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(−x2) 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

• 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

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

• 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

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

• 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 elemets:

• Bilinear quadrilateral element — also known as the Q4 element

• Constant strain triangle element (CST) — also known as the T3 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

• 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)

• 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 vertiÑes 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 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)

• 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

• Computational magnetohydrodynamics (CMHD) — studies electrically conducting fluids

• Climate model

• Numerical weather prediction

• Geodesic grid

• Celestial mechanics

• Numerical model of the Solar System

• 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

- All aspects covers interdisciplinary influence

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