Global Journal of Research In Engineering

Recent News

**Thermal Science: **State-of-the-art computational and experimental facilities are used in fundamental studies and applications of thermodynamics, fluid mechanics and heat transfer.

- Cryogenics & High Current.
- Engine Research
- HVAC Systems and Controls
- Industrial Refrigeration
- Powertrain Control, Diagnostics and Dynamic Modeling
- Shock Tube Laboratory
- Solar Energy
- Vapor Explosions

**Dynamics Vibrations and Acoustics: **Analytical, numerical and experimental methods applied to the characterization of mechanical components, structures, systems and materials. These activities intimately support product development, safety, weight minimization and component optimization for aerospace, automotive, electronics and general manufacturing. Current areas of emphasis include stress, strain and deformation analysis; modeling, testing and verification of kinematics and dynamic systems; applied finite elements; plate, shell, and pressure vessel characterization; composites; micromechanical design and analysis; photomechanics and optical techniques, and multi body problems.

- Experimental Mechanics and Mechanical Measurements
- Structural Dynamics and Vibrations

**Mechatronics, Robotics and Automation:**Mechatronics, Robotics and Automation research is conducted in a variety of areas

- Mechatronics Laboratory
- Robotics
- Sensors, Signal Processing and Real-time Controls Integration
- Space Automation and Robotics

**Design and Manufacturing:**Design and Manufacturing activities include the design and manufacturing of machines, systems, products, mechanisms and process.

- Fluid Power
- Engineering Representation and Simulation
- Laser-Assisted Manufacturing
- Mechanical Design
- Powertrain Control, Diagnostics and Dynamic Modeling

**Polymer Engineering:**The Polymer Engineering Center focuses on advancing technologies for a wide range of polymer and polymeric composite manufacturing processes.

- Engineering Polymer Industrial
- Polymer Engineering
- Rheology Research Center

**Biomechanical Engineering:**Development of fundamental and applied engineering knowledge related to biomechanical systems, and the application of engineering expertise towards the design and development of leading-edge rehabilitative, assistive, and adaptive technologies that allow those with disabilities to achieve greater independence.

- Biomechanics
- Musculoskeletal Research: Bone and Joint Group & Neuromuscular Biomechanics
- UW-CREATe

**Computer Aided Engineering:**The primary thrust of Computer-Aided Engineering research is to develop mathematically sound theories, computationally efficient algorithms, and next generation tools for modeling, design, and simulation of a wide range of engineering artifacts and processes. Focus areas include mechanical, micro/nano-mechanical, electro-mechanical, thermal, fluid, and other multi-disciplinary and multi-scale systems.

- Computational Mechanics
- Engineering Representation and Simulation

- U.S D.o.E GATE on Advanced Propulsions Systems (Sustainable Mobility)
- CAR EcoSystem (Sustainable Mobility)
- SMART@CAR - PHEV (Sustainable Mobility)
- Advanced Powertrain Systems
- Flow, Engine and Acoustics
- System Fault Diagnosis and Prognosis
- Intelligent Transportation Systems and Vehicle-to-Vehicle Networks
- Noise, Vibration and Dynamics
- Vehicle Dynamics
- Vehicle Duty Cycle and Terrain Characterization
- Concept Design
- Injury Biomechan

- Biochemical Engineering
- Catalysis
- Chemical Engineering (General)
- Chemical Health and Safety
- Fluid Flow / Transfer Processes
- Industrial Chemistry
- Materials Chemistry and Engineering
- Membranes and Separation Technology
- Particle Technology
- Petroleum and Fuel Technology
- Process Chemistry and Technology

- Aeroacoustics
- Aerodynamics
- Aeroelasticity
- Aerospace Information Technology
- Aerospace Systems Design and Simulation
- Astrodynamics
- Combustion and Propulsion
- Computational Fluid Dynamics (CFD)
- Dynamical Systems and Structural Dynamics
- Flight Vehicle Synthesis
- Lasers
- Materials and Structures
- Structural Mechanics
- Systems and Control

**Offshore Engineering**

- Innovative Structural Systems
- Jack-Up Platform and Floating Production Systems
- Marine Operations and Installation
- Very Large Floating Structures

**Protective Engineering**

- Advanced and New Protective Materials
- Airblast and Groundshock Effects, including Blast-Induced Liquefaction
- Hardening and Protective Measures for Structures, Personnels and Vehicles
- Rapidly Deployable Protective Structures

**Hazards, Risks and Mitigation**

- Design and Protection of Infrastructures against Natural and Manmade Hazards
- Disaster Prevention and Mitigation
- Earthquake Effects on Soils, Foundations and Structures
- Earthquake Tectonics
- Hazards Induced by Climate Change
- Risk Analysis and Management
- Tsunami Forecasting, Propagation and Run-Up

**Structural Engineering**

- High Strength, Lightweight and High-Performance Materials
- Novel Composite Structural Systems
- Repair and Strengthening
- Smart Materials and Structural Health Monitoring

**Geotechnical Engineering**

- Land reclamation and Coastal & Offshore Geotechnics
- Underground Construction

**Hydrology and Hydraulic Engineering**

- Coastal Engineering & Protection
- Modelling of Hydrodynamic and Transport Process
- Environmental Hydraulics
- Hydroinformatics
- Water Resources Planning and Management

**Infrastructure Systems**

- Intelligent Transportation Systems
- Transportation Logistics
- Infrastructure & Project Management
- Performance-based Asset Management

** Electronics System**

- Agri-Electronics
- Embedded Systems
- Digital Systems
- Power Electronics

** Electron Tubes**

- Gyrotron
- Klystron
- Magnetrons
- Plasma Devices
- Traveling Wave Tubes

** Semiconductor**

- Hybrid Mircrocircuits
- IC Design
- MEMS and Microsensors
- Sensors and Nanotechnology
- Photonics and Optoelectronics
- Semiconductor Materials and Technology

**Computational Electronics and Photonics**

- Laser Physics
- MicroElectroMechanical Systems (MEMS)
- Nanophotonics
- Nanotechnology
- Plasma Devices and Plasma Science
- Semiconductor Electronic Devices
- Semiconductor Lasers and Photonic Devices
- Semiconductor Materials
- Ultrafast Laser Spectroscopy

- Decision Science/Operations Research
- Health Systems
- Human Factors and Ergonomics
- Manufacturing and Production Systems
- Quality Engineering

- Adaptive control
- Aerial robotics
- Anthrobotics
- Artificial intelligence
- Autonomous car
- Autonomous research robotics
- Bayesian network
- BEAM robotics
- Behavior-based robotics
- Biomimetic
- Biomorphic robotics .
- Bionics
- Biorobotics
- Cognitive robotics
- Clustering
- Computational neuroscience
- Robot control
- Robotics conventions
- Data mining
- Degrees of freedom
- Developmental Robotics
- Digital control
- Digital image processing
- Dimensionality reduction
- Distributed robotics
- Electronic Stability Control
- Evolutionary computation
- Evolutionary robotics
- Extended Kalman filter
- Flexible Distribution functions
- Feedback control and Regulation
- Human–computer interaction
- Human robot interaction
- Kinematics
- Laboratory robotics
- Robot learning
- Manifold learning
- Nanorobotics
- Artificial neural networks
- Passive dynamics
- Swarm robotics
- Telepresence
- Computer vision
- Green nanotechnology
- Nanoengineering
- Wet nanotechnology
- Nanobiotechnology
- Ceramic engineering
- Materials science
- Nanoarchitectonics
- Nanoelectronics
- Nanomechanics
- Nanophotonics

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