List of numerical analysis topics
From Wikipedia, the free encyclopedia
This is a list of numerical analysis topics, by Wikipedia page.
[edit] General
- Iterative method
- Sequence transformations — methods to accelerate the speed of convergence
- Richardson extrapolation
- Van Wijngaarden transformation — for accelerating the convergence of an alternating series
- Level set method
- Level set (data structures) — data structures for representing level sets
- 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
- Collocation method — discretizes a continuous equation by requiring it only to hold at certain points
- Difference quotient
- Computational complexity of mathematical operations
- International Workshops on Lattice QCD and Numerical Analysis
- ABS methods
[edit] Error
- Approximation
- Approximation error
- Arithmetic precision
- Condition number
- Discretization error
- Floating point number
- Guard digit — extra precision introduced during a computation to reduce round-off error
- Arbitrary-precision arithmetic
- Truncation — rounding a floating-point number by discarding all digits after a certain digit
- Loss of significance
- Numerical error
- Numerical stability
- Error propagation:
- Relative difference — the relative difference between x and y is |x − y| / max(|x|, |y|)
- Round-off error
- 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
[edit] Elementary and special functions
- Summation:
- 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
- Exponentiation:
- Polynomials:
- Horner scheme
- Estrin's scheme — modification of the Horner scheme with more possibilities for parallellization
- Clenshaw algorithm
- De Casteljau's algorithm
- Square roots and other roots:
- Elementary functions (exponential, logarithm, trigonometric functions):
- Generating trigonometric tables
- 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
- Spigot algorithm — algorithms that can compute individual digits of a real number
- Computation of π:
- Gauss-Legendre algorithm — iteration which converges quadratically to π, based on arithmetic-geometric mean
- Bailey-Borwein-Plouffe formula — can be used to compute individual hexadecimal digits of π
- Borwein's algorithm — iteration which converges quartically to 1/π, and other algorithms
- Chudnovsky algorithm — fast algorithm that calculates a hypergeometric series
[edit] Numerical linear algebra
Numerical linear algebra — study of numerical algorithms for linear algebra problems
[edit] Basic concepts
- Types of matrices appearing in numerical analysis:
- Sparse matrix
- Circulant matrix
- Triangular matrix
- Diagonally dominant matrix
- Hilbert matrix — example of a matrix which is extremely ill-conditioned (and thus difficult to handle)
- 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
- Freivald's algorithm — a randomized algorithm for checking the result of a multiplication
[edit] Solving systems of linear equations
- Gaussian elimination
- Row echelon form — matrix in which all entries below a nonzero entry are zero
- Gauss–Jordan elimination — variant in which the entries below the pivot are also zeroed
- Montante's method — 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
- Frontal solver — for sparse matrices; used in finite element methods
- Levinson recursion — for Toeplitz matrices
- Iterative methods:
- Jacobi method
- Gauss–Seidel method
- Successive over-relaxation (SOR) — a technique to accelerate the Gauss–Seidel method
- Modified Richardson iteration
- Conjugate gradient method (CG) — assumes that the matrix is positive definite
- Preconditioned conjugate gradient method (PCG)
- Nonlinear conjugate gradient method — generalization for nonlinear optimization problems
- Biconjugate gradient method (BiCG)
- Generalized minimal residual method (GMRES) — based on the Arnoldi iteration
- Stone's method (SIP - Srongly Implicit Procedure) — uses an incomplete LU decomposition
- Kaczmarz method
- Preconditioner
- 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)
[edit] 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
- Divide-and-conquer eigenvalue algorithm
- Folded spectrum method
[edit] Other concepts and algorithms
- Orthogonalization algorithms:
- QR decomposition
- Krylov subspace
- Block matrix pseudoinverse
- Bidiagonalization
- 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
[edit] Interpolation
- Nearest-neighbor interpolation — takes the value of the nearest neighbor
[edit] 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
- Newton polynomial
- 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
[edit] 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
- Cardinal 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
- Truncated power function
- De Boor's algorithm — generalizes De Casteljau's algorithm
- Nonuniform rational B-spline (NURBS)
- Kochanek–Bartels spline
- Catmull-Rom spline
- Blossom (mathematics) — a unique, affine, symmetric map associated to a polynomial or spline
- See also: List of numerical computational geometry topics
[edit] Trigonometric interpolation
Trigonometric interpolation — interpolation by trigonometric polynomials
- Discrete Fourier transform — can be viewed as trigonometric interpolation at equidistant points
- Fast Fourier transform — 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
- Butterfly diagram
- Twiddle factor — the trigonometric constant coefficients that are multiplied by the data
- Fast cosine transform — fast methods for computing the discrete cosine transform
- Methods for computing discrete convulations with finite impulse response filters using the FFT:
- Sigma approximation
- Dirichlet kernel — convolving any function with the Dirichlet kernel yields its trigonometric interpolant
- Gibbs phenomenon
[edit] Other interpolants
- Simple rational approximation
- Wavelet
- Inverse distance weighting
- Cascade algorithm — iterative algorithm to compute wavelets
- Radial basis function
- Polyharmonic spline — a commonly used radial basis function
- Thin plate spline — a specific polyharmonic spline: r2 log r
- Radial basis function network — neural network using radial basis functions as activation functions
- Subdivision surface — constructed by recursively subdividing a piecewise linear interpolant
- Slerp
- 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
- Pareto interpolation
- Multivariate interpolation — the function being interpolated depends on more than one variable
- Lanczos resampling — based on convolution with a sinc function
- Natural neighbor interpolation
- PDE surface
- Method based on polynomials are listed under Polynomial interpolation
[edit] Approximation theory
- Orders of approximation
- Lebesgue's lemma
- Curve fitting
- Modulus of continuity — measures smoothness of a function
- Minimax approximation algorithm — minimizes the maximum error over an interval (the L∞-norm)
- Approximation by polynomials:
- Linear approximation
- Bernstein polynomial — basis of polynomials useful for approximating a function
- Remez algorithm — for constructing the best polynomial approximation in the L∞-norm
- Bramble-Hilbert lemma — upper bound on Lp error of polynomial approximation in multiple dimensions
- Bernstein's constant — error when approximating |x| by a polynomial
- Surrogate model — application: replacing a function that is hard to evaluate by a simpler function
- Jackson's inequality — upper bound for best approximation by a trigonometric polynomial
- Different approximations:
- Moving least squares
- Padé approximant
- Padé table — table of Padé approximants
- Szász-Mirakyan operator — approximation by e−n xk on a semi-infinite interval
- Szász-Mirakyan-Kantorovich operator
- Baskakov operator — generalize Bernstein polynomials, Szász-Mirakyan operators, and Lupas operators
- Favard operator — approximation by sums of Gaussians
[edit] Miscellaneous
- Extrapolation
- Linear predictive analysis — linear extrapolation
- Regression analysis
- Curve-fitting compaction
- Interpolation (computer programming) — interpolation in the context of computer graphics
[edit] 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
- Newton fractal
- Quasi-Newton method — uses an approximation of the Jacobian:
- Broyden's method — uses a rank-one update for the Jacobian
- SR1 formula — 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
- BFGS method — rank-two update of the Jacobian in which the matrix remains positive definite
- 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
- Müller's method — based on quadratic interpolation at last three iterates
- Inverse quadratic interpolation — similar to Müller'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
- Bisection method — simple and robust; linear convergence
- 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
- Methods for other special cases:
- Analysis:
- Numerical continuation — tracking a root as one parameters in the equation changes
[edit] Optimization
Optimization (mathematics) — algorithm for finding maxima or minima of a given function
[edit] Basic concepts
- Active set
- Candidate solution
- Constraint (mathematics)
- Corner solution
- Fitness function — (esp. in genetic algorithms) an approximation to the objective function that is easier to evaluate
- Global optimum and Local optimum
- Maxima and minima
- Slack variable
- Surplus variable
- Continuous optimization
- Discrete optimization
[edit] 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
- Interior point method
- Delayed column generation
- k-approximation of k-hitting set — algorithm for specific LP problems (to find a weighted hitting set)
- Simplex algorithm
- Algorithms for integer programming:
- Linear complementarity problem
- Fourier–Motzkin elimination
[edit] Nonlinear programming
Nonlinear programming — the most general optimization problem in the usual framework
- Special cases of nonlinear programming:
- Quadratic programming
- Convex optimization
- Linear matrix inequality
- Conic optimization
- Semidefinite programming
- Second order cone programming
- Quadratic programming (see above)
- Subgradient method — extension of steepest descent for problems with a nondifferentiable objective function
- Geometric programming — problems involving posynomials
- Posynomial — similar to polynomials, but coefficients have to be positive while exponents need not be integers
- Quadratically constrained quadratic program
- Least squares — the objective function is a sum of squares
- Univariate optimization:
- Golden section search
- Successive parabolic interpolation — based on quadratic interpolation through the last three iterates
- General algorithms:
- Concepts:
- Gradient descent
- Successive linear programming (SLP) — replace problem by a linear 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
- Nelder-Mead method
- Ternary search
- Tabu search
- Guided Local Search — modification of search algorithms which builds up penalties during a search
- Least absolute deviations
- Nearest neighbor search
- Mixed complementarity problem
[edit] Uncertainty and randomness
- Approaches to deal with uncertainty:
- Random optimization algorithms:
- Simulated annealing
- Adaptive simulated annealing — variant in which the algorithm parameters are adjusted during the computation.
- Great Deluge algorithm
- Evolutionary algorithm, Evolution strategy
- Memetic algorithm
- Particle swarm optimization
- Cooperative optimization
- Stochastic tunneling
- Harmony search - mimicks the improvisation process of musicians
- see also the section Monte Carlo method
- Simulated annealing
[edit] Theoretical aspects
- Convex analysis
- Duality:
- Farkas' lemma
- Karush-Kuhn-Tucker conditions
- Lagrange multipliers
- Danskin's theorem — used in the analysis of minimax problems
- No free lunch in search and optimization
- Relaxation technique (mathematics)
- 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
[edit] 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
- Automatic label placement
- Cutting stock problem
- Demand optimization
- Energy minimization
- Entropy maximization
- Expenditure minimization problem
- Inventory control problem
- Job-shop problem
- Multidisciplinary design optimization
- Optimal classification
- Paper bag problem
- Process optimization
- Stigler diet
- Stress majorization
- Trajectory optimization
- Utility maximization problem
- Wing shape optimization
[edit] Miscellaneous
- Combinatorial optimization
- Dynamic programming
- Bellman equation
- Backward induction — solving dynamic programming problems by reasoning backwards in time
- Optimal stopping — choosing the optimal time to take a particular action
- Global optimization:
- Infinite-dimensional optimization
- Optimal control
- Pontryagin's minimum principle — infinite-dimensional version of Lagrange multipliers
- Shape optimization, Topology optimization — optimization over a set of regions
- Topological derivative — derivative with respect to changing in the shape
- Optimal control
- Optimal substructure
- Algorithmic concepts:
- Famous 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 f(x,y) = (x2 + y − 11)2 + (x + y2 − 7)2
- Shekel function — multimodal and multidimensional
- Mathematical Programming Society
[edit] Numerical quadrature
Numerical integration — the numerical evaluation of an integral
- Rectangle method
- Trapezium rule
- Simpson's rule
- Newton–Cotes formulas
- Romberg's method - Richardson extrapolation applied to Trapezium rule
- Gaussian quadrature - highest possible degree with given number of points
- 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
- T-integration — a non-standard method
- Lebedev grid — grid on a sphere with octahedral symmetry
- Sparse grid
- Numerical differentiation
- Euler–Maclaurin formula
[edit] Numerical ordinary differential equations
Numerical 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
- 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
- Cash–Karp method — a fifth-order method with six stages and an embedded fourth-order method
- Dormand–Prince method — another fifth-order method with six stages and an embedded fourth-order method
- Runge–Kutta–Fehlberg method — another fifth-order method with six stages and an embedded fourth-order method
- List of Runge-Kutta methods
- Linear multistep method — the other main class of methods for initial-value problems
- 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
- 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
- Euler–Cromer algorithm — variant of Euler method which is symplectic when applied to separable Hamiltonians
- Symplectic integrator — a method for the solution of Hamilton's equations that preserves the symplectic structure
- Adaptive stepsize — automatically changing the step size when that seems advantageous
- Stiff equation — roughly, an ODE for which the unstable methods needs a very short step size, but stable methods do not.
- Methods for solving two-point boundary value problems (BVPs):
- Methods for solving differential-algebraic equations (DAEs), i.e., ODEs with constraints:
- Constraint algorithm — for solving Newton's equations with constraints
- 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
- Bi-directional delay line
- History of numerical solution of differential equations using computers
[edit] Numerical partial differential equations
Numerical partial differential equations — the numerical solution of partial differential equations (PDEs)
[edit] Finite difference methods
Finite difference method — based on approximating differential operators with difference operators
- Finite difference — the discrete analogue of a differential operator
- Difference operator — the numerator of a finite difference
- Discrete Laplace operator — finite-difference approximation of the Laplace operator
- Discrete Poisson equation — discrete analogue of the Poisson equation using the discrete Laplace operator
- Five-point stencil — standard finite-difference approximation of the Laplace operator in two dimensions
- Crank–Nicolson method — second-order implicit method for heat equation and related PDEs
- Alternating direction implicit — second-order method for heat equation and related PDEs
- Lax–Wendroff method — second-order explicit method for hyperbolic PDEs
- MacCormack method — second-order explicit method for hyperbolic PDEs
- Upwind scheme — method for hyperbolic PDEs
- Finite-difference time-domain method — a finite-difference method for electrodynamics
[edit] Finite element methods
Finite element method, finite element analysis — 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
- Direct stiffness method — a particular implementation of the finite element method, often used in structural analysis
- Trefftz method
- Finite element updating
- XFEM — extended finite element methods, which put functions tailored to the problem in the approximation space
- Functionally graded elements — elements for describing functionally graded materials
- 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
- NAFEMS — not-for-profit organisation that sets and maintains standards in computer-aided engineering analysis
[edit] Other methods
- Spectral method — based on the Fourier transformation
- Method of lines — reduces the PDE to a large system of ordinary differential equations
- Boundary element method — based on transforming the PDE to an integral equation on the boundary of the domain
- 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
- Discrete element method — a method in which the elements can move freely relative to each other
- Meshfree methods — does not use a mesh, but uses a particle view of the field
- Uniform theory of diffraction — specifically designed for scattering problems in electromagnetics
- Particle-in-cell — used especially in fluid dynamics
- High-resolution scheme
- Shock capturing methods
- Split-step method
- Fast marching method
- Lattice Boltzmann methods — for the solution of the Navier-Stokes equations
- Roe solver — for the solution of the Euler equation
- Relaxation 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
[edit] 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 (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
- 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
- Schur complement method — early and basic method on subdomains that do not overlap
- Schwarz alternating method — early and basic method on subdomains that overlap
- Adaptive mesh refinement — uses the computed solution to refine the mesh only where necessary
- Fast Multipole Method — hierarchical method for evaluating particle-particle interactions
[edit] Miscellaneous
- 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
- Numerical diffusion — diffusion introduced by the numerical method, above to that which is naturally present
- 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
- Grids and meshes:
- Geodesic grid — isotropic grid on a sphere
- Parallel mesh generation
- Spatial twist continuum — dual representation of a mesh consisting of hexahedra
- Perfectly matched layer — artificial absorbing layer for wave equations, used to implement absorbing boundary conditions
[edit] Monte Carlo method
- Variants of the Monte Carlo method:
- Sampling methods:
- Inverse transform sampling — general and straightforward method but computationally expensive
- Rejection sampling — sample from a simpler distribution but reject some of the samples
- For sampling from a normal distribution:
- Low-discrepancy sequence
- Event generator
- Parallel tempering
- Umbrella sampling — improves sampling in physical systems with significant energy barriers
- Variance reduction techniques:
- Ensemble Kalman filter — recursive filter suitable for problems with a large number of variables
- Applications:
- Metropolis light transport
- Monte Carlo method for photon transport
- Monte Carlo methods in finance
- Monte Carlo project — research project modelling of human exposure to food chemicals and nutrients
- 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
- 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
[edit] Applications
- Climate model
- Numerical weather prediction
- Computational chemistry
- Computational electromagnetics
- Computational fluid dynamics
- Large eddy simulation
- Smoothed particle hydrodynamics
- Acoustic analogy — used in numerical aeroacoustics to reduce sound sources to simple emitter types
- Computational physics
- Computational statistics
- Celestial mechanics
[edit] Software
For software, see the list of numerical analysis software.
