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Structured backward error analysis of linearized structured polynomial eigenvalue problems

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 05/2019, Volume 88, Issue 317, pp. 1189 - 1228

We start by introducing a new class of structured matrix polynomials, namely, the class of \mathbf {M}_A-structured matrix polynomials, to provide a common...

Structured backward error analysis | matrix pertubation theory | MATHEMATICS, APPLIED | complete polynomial eigenproblems | dual minimal bases | Mobius transformations | EIGENSTRUCTURE | FIEDLER PENCILS | PERTURBATION | ALGORITHM | HERMITIAN MATRIX POLYNOMIALS | MINIMAL BASES | VECTOR-SPACES | RECOVERY | structure-preserving linearizations | structured matrix polynomials | EVEN

Structured backward error analysis | matrix pertubation theory | MATHEMATICS, APPLIED | complete polynomial eigenproblems | dual minimal bases | Mobius transformations | EIGENSTRUCTURE | FIEDLER PENCILS | PERTURBATION | ALGORITHM | HERMITIAN MATRIX POLYNOMIALS | MINIMAL BASES | VECTOR-SPACES | RECOVERY | structure-preserving linearizations | structured matrix polynomials | EVEN

Journal Article

BIT Numerical Mathematics, ISSN 0006-3835, 12/2015, Volume 55, Issue 4, pp. 1057 - 1103

We consider numerical approximations of stochastic Langevin equations by implicit methods. We show a weak backward error analysis result in the sense that the...

Weak error | Computational Mathematics and Numerical Analysis | 37M25 | Numeric Computing | Backward error analysis | Mathematics | Exponential mixing | Numerical scheme | 65C30 | Langevin equation | Mathematics, general | Kolmogorov equation | 60H35 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | MODIFIED EQUATIONS | Probability | Numerical Analysis

Weak error | Computational Mathematics and Numerical Analysis | 37M25 | Numeric Computing | Backward error analysis | Mathematics | Exponential mixing | Numerical scheme | 65C30 | Langevin equation | Mathematics, general | Kolmogorov equation | 60H35 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | MODIFIED EQUATIONS | Probability | Numerical Analysis

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 1/2012, Volume 50, Issue 3, pp. 1735 - 1752

We consider long time numerical approximations of stochastic differential equations (SDEs) by the Euler method. In the case where the SDE is elliptic or...

Error rates | Error analysis | Approximation | Numerical integration | Differential equations | Vector fields | Markov processes | Polynomials | Numerical schemes | Weak error | Backward error analysis | Stochastic differential equations | Numerical scheme | Kolmogorov equation | Exponential mixing | weak error | MATHEMATICS, APPLIED | STOCHASTIC DIFFERENTIAL-EQUATIONS | NUMERICAL INTEGRATORS | numerical scheme | CONVERGENCE | backward error analysis | stochastic differential equations | exponential mixing | Probability | Numerical Analysis | Mathematics

Error rates | Error analysis | Approximation | Numerical integration | Differential equations | Vector fields | Markov processes | Polynomials | Numerical schemes | Weak error | Backward error analysis | Stochastic differential equations | Numerical scheme | Kolmogorov equation | Exponential mixing | weak error | MATHEMATICS, APPLIED | STOCHASTIC DIFFERENTIAL-EQUATIONS | NUMERICAL INTEGRATORS | numerical scheme | CONVERGENCE | backward error analysis | stochastic differential equations | exponential mixing | Probability | Numerical Analysis | Mathematics

Journal Article

BIT Numerical Mathematics, ISSN 0006-3835, 9/2019, Volume 59, Issue 3, pp. 613 - 646

This paper presents a study of the approximation error corresponding to a symplectic scheme of weak order one for a stochastic autonomous Hamiltonian system. A...

Computational Mathematics and Numerical Analysis | 65C30 | 37M15 | Numeric Computing | Mathematics, general | Backward error analysis | Weak symplectic scheme | Mathematics | Kolmogorov equation | Stochastic Hamiltonian systems | 60H35 | NUMERICAL-METHODS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | SYMPLECTIC SCHEMES | DIFFERENTIAL-EQUATIONS | SDES | TIME | STRONG-CONVERGENCE

Computational Mathematics and Numerical Analysis | 65C30 | 37M15 | Numeric Computing | Mathematics, general | Backward error analysis | Weak symplectic scheme | Mathematics | Kolmogorov equation | Stochastic Hamiltonian systems | 60H35 | NUMERICAL-METHODS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | SYMPLECTIC SCHEMES | DIFFERENTIAL-EQUATIONS | SDES | TIME | STRONG-CONVERGENCE

Journal Article

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2016, Volume 38, Issue 3, pp. A1639 - A1661

The Leja method is a polynomial interpolation procedure that can be used to compute matrix functions. In particular, computing the action of the matrix...

Leja interpolation | Backward error analysis | Taylor series | Action of matrix exponential | φ functions | Exponential integrators | INTERPOLATION | MATHEMATICS, APPLIED | exponential integrators | phi functions | backward error analysis | action of matrix exponential | Mathematics - Numerical Analysis

Leja interpolation | Backward error analysis | Taylor series | Action of matrix exponential | φ functions | Exponential integrators | INTERPOLATION | MATHEMATICS, APPLIED | exponential integrators | phi functions | backward error analysis | action of matrix exponential | Mathematics - Numerical Analysis

Journal Article

BIT Numerical Mathematics, ISSN 0006-3835, 03/2019, Volume 59, Issue 1, pp. 271 - 297

When polynomial roots vary widely in order of magnitude, severe numerical instability problem may occur due to deflation schemes. Peters and Wilkinson (IMA J...

Backward error analysis | Polynomial deflation | Tropical polynomial | EIGENVALUE | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | Deflation (Finance) | Analysis

Backward error analysis | Polynomial deflation | Tropical polynomial | EIGENVALUE | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | Deflation (Finance) | Analysis

Journal Article

Mathematical Programming, ISSN 0025-5610, 2019, Volume 180, Issue 1-2, pp. 371 - 416

This paper reveals that a common and central role, played in many error bound (EB) conditions and a variety of gradient-type methods, is a residual measure...

Dual objective function | Gradient descent | Residual measure operator | Linear convergence | Proximal alternating linearized minimization | Error bound | Forward–backward splitting algorithm | Nesterov’s acceleration | Proximal point algorithm | Algorithms | Ascent | Measurement methods | Convexity | Optimization | Linearization | Convergence

Dual objective function | Gradient descent | Residual measure operator | Linear convergence | Proximal alternating linearized minimization | Error bound | Forward–backward splitting algorithm | Nesterov’s acceleration | Proximal point algorithm | Algorithms | Ascent | Measurement methods | Convexity | Optimization | Linearization | Convergence

Journal Article

SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, 2016, Volume 37, Issue 1, pp. 123 - 144

We perform a backward error analysis of polynomial eigenvalue problems solved via linearization. Through the use of dual minimal bases, we unify the...

Backward error | Stability | Dual minimal basis | Polynomial eigenvalue problem | Linearization | Strong linearization | MATRIX | MATHEMATICS, APPLIED | strong linearization | backward error | dual minimal basis | polynomial eigenvalue problem | EIGENPROBLEMS | stability | linearization

Backward error | Stability | Dual minimal basis | Polynomial eigenvalue problem | Linearization | Strong linearization | MATRIX | MATHEMATICS, APPLIED | strong linearization | backward error | dual minimal basis | polynomial eigenvalue problem | EIGENPROBLEMS | stability | linearization

Journal Article

JOURNAL OF SCIENTIFIC COMPUTING, ISSN 0885-7474, 08/2019, Volume 80, Issue 2, pp. 1171 - 1194

Strong approximation errors of both finite element semi-discretization and spatio-temporal full discretization are analyzed for the stochastic Allen-Cahn...

Additive noise | Strong approximation | Finite element method | MATHEMATICS, APPLIED | Stochastic Allen-Cahn equation | DISCRETIZATION | APPROXIMATION | DIFFERENTIAL-EQUATIONS | TIME | Backward Euler scheme | STRONG-CONVERGENCE | Analysis | Methods

Additive noise | Strong approximation | Finite element method | MATHEMATICS, APPLIED | Stochastic Allen-Cahn equation | DISCRETIZATION | APPROXIMATION | DIFFERENTIAL-EQUATIONS | TIME | Backward Euler scheme | STRONG-CONVERGENCE | Analysis | Methods

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 07/2018, Volume 76, Issue 2, pp. 340 - 360

We present an a posteriori error analysis of the fully discretized time-dependent Darcy and Stokes equations, that models laminar fluid flow over a porous...

A posteriori error analysis | Finite elements | Darcy equations | Backward Euler scheme | Stokes equations | POROUS-MEDIA FLOW | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | FLUID-FLOW | MODEL | FINITE-ELEMENT-METHOD

A posteriori error analysis | Finite elements | Darcy equations | Backward Euler scheme | Stokes equations | POROUS-MEDIA FLOW | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | FLUID-FLOW | MODEL | FINITE-ELEMENT-METHOD

Journal Article

ESAIM: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, 03/2018, Volume 52, Issue 2, pp. 773 - 801

In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a...

Convolution quadrature | Optimal error estimate | Subdiffusion | Fractional order evolution equation | Finite volume element method | Backward Euler and second-order backward difference methods | Laplace transform | Smooth and nonsmooth data | DIFFUSION-WAVE EQUATIONS | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | VOLUME ELEMENT METHOD | CONVOLUTION QUADRATURE | PARABOLIC EQUATIONS | APPROXIMATIONS | CALCULUS | TIME DISCRETIZATION | LAPLACE TRANSFORMATION | FEM | Finite element method | Triangulation | Error analysis | Convolution | Mathematical analysis | Norms | Evolution | Galerkin method | Laplace transforms

Convolution quadrature | Optimal error estimate | Subdiffusion | Fractional order evolution equation | Finite volume element method | Backward Euler and second-order backward difference methods | Laplace transform | Smooth and nonsmooth data | DIFFUSION-WAVE EQUATIONS | MATHEMATICS, APPLIED | NUMERICAL-SOLUTION | VOLUME ELEMENT METHOD | CONVOLUTION QUADRATURE | PARABOLIC EQUATIONS | APPROXIMATIONS | CALCULUS | TIME DISCRETIZATION | LAPLACE TRANSFORMATION | FEM | Finite element method | Triangulation | Error analysis | Convolution | Mathematical analysis | Norms | Evolution | Galerkin method | Laplace transforms

Journal Article

IMA Journal of Numerical Analysis, ISSN 0272-4979, 03/2015, Volume 35, Issue 2, pp. 583 - 614

We consider an overdamped Langevin stochastic differential equation and show a weak backward error analysis result for its numerical approximations defined by...

backward error analysis | implicit numerical scheme | Kolmogorov equation | overdamped Langevin equation | weak error; exponential mixing | weak error | MATHEMATICS, APPLIED | STOCHASTIC DIFFERENTIAL-EQUATIONS | TIME | exponential mixing | STRONG-CONVERGENCE | Asymptotic expansions | Error analysis | Approximation | Mathematical analysis | Differential equations | Implicit methods | Mathematical models | Invariants | Probability | Numerical Analysis | Mathematics

backward error analysis | implicit numerical scheme | Kolmogorov equation | overdamped Langevin equation | weak error; exponential mixing | weak error | MATHEMATICS, APPLIED | STOCHASTIC DIFFERENTIAL-EQUATIONS | TIME | exponential mixing | STRONG-CONVERGENCE | Asymptotic expansions | Error analysis | Approximation | Mathematical analysis | Differential equations | Implicit methods | Mathematical models | Invariants | Probability | Numerical Analysis | Mathematics

Journal Article

Applied Numerical Mathematics, ISSN 0168-9274, 02/2019, Volume 136, pp. 23 - 45

This paper is concerned with residual type a posteriori error estimates of fully discrete finite element approximations for parabolic optimal control problems...

A posteriori error estimates | Measure data in time | Parabolic optimal control problem | Measure data in space | The backward Euler scheme | Finite element approximations | MATHEMATICS, APPLIED | PRIORI | EQUATIONS

A posteriori error estimates | Measure data in time | Parabolic optimal control problem | Measure data in space | The backward Euler scheme | Finite element approximations | MATHEMATICS, APPLIED | PRIORI | EQUATIONS

Journal Article

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From error bounds to the complexity of first-order descent methods for convex functions

Mathematical Programming, ISSN 0025-5610, 10/2017, Volume 165, Issue 2, pp. 471 - 507

This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a...

65K05 | Theoretical, Mathematical and Computational Physics | 90C06 | Mathematics | Forward-backward method | Convex minimization | Mathematical Methods in Physics | Complexity of first-order methods | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | LASSO | Error bounds | KL inequality | Combinatorics | 90C60 | Compressed sensing | POLYNOMIAL SYSTEMS | MATHEMATICS, APPLIED | INEQUALITIES | STABILITY | THRESHOLDING ALGORITHM | ASYMPTOTIC CONVERGENCE | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | OPTIMIZATION | PROJECTION ALGORITHMS | Errors | Equivalence | Globalization | Inequalities | Constants | Shrinkage | Iterative methods | Regularization | Convex analysis | Descent | Methods | Complexity

65K05 | Theoretical, Mathematical and Computational Physics | 90C06 | Mathematics | Forward-backward method | Convex minimization | Mathematical Methods in Physics | Complexity of first-order methods | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | 90C25 | Numerical Analysis | LASSO | Error bounds | KL inequality | Combinatorics | 90C60 | Compressed sensing | POLYNOMIAL SYSTEMS | MATHEMATICS, APPLIED | INEQUALITIES | STABILITY | THRESHOLDING ALGORITHM | ASYMPTOTIC CONVERGENCE | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MINIMIZATION | OPTIMIZATION | PROJECTION ALGORITHMS | Errors | Equivalence | Globalization | Inequalities | Constants | Shrinkage | Iterative methods | Regularization | Convex analysis | Descent | Methods | Complexity

Journal Article

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Backward error analysis for linearizations in heavily damped quadratic eigenvalue problem

Numerical Linear Algebra with Applications, ISSN 1070-5325, 08/2019, Volume 26, Issue 4, p. n/a

Summary Heavily damped quadratic eigenvalue problem (QEP) is a special type of QEPs. It has a large gap between small and large eigenvalues in absolute value....

growth factor | heavily damped QEP | linearizations | backward error | MATHEMATICS | MATHEMATICS, APPLIED | PERTURBATION | ALGORITHM | Eigenvalues | Error analysis | Upper bounds | Growth factors | Eigen values

growth factor | heavily damped QEP | linearizations | backward error | MATHEMATICS | MATHEMATICS, APPLIED | PERTURBATION | ALGORITHM | Eigenvalues | Error analysis | Upper bounds | Growth factors | Eigen values

Journal Article

Inverse Problems, ISSN 0266-5611, 03/2017, Volume 33, Issue 4, p. 44001

Many problems in science and engineering involve, as part of their solution process, the consideration of a separable function which is the sum of two convex...

error terms | FISTA | decay rate | minimization problem | superiorization | accelerated proximal forward-backward algorithm | inexactness | MATHEMATICS, APPLIED | CONVEX FEASIBILITY | IMAGE-RECONSTRUCTION | THRESHOLDING ALGORITHM | PHYSICS, MATHEMATICAL | SUBGRADIENT PROJECTION ALGORITHMS | EXTRAGRADIENT METHOD | OPTIMIZATION | POINT ALGORITHM | MONOTONE-OPERATORS | INEXACT INFINITE PRODUCTS

error terms | FISTA | decay rate | minimization problem | superiorization | accelerated proximal forward-backward algorithm | inexactness | MATHEMATICS, APPLIED | CONVEX FEASIBILITY | IMAGE-RECONSTRUCTION | THRESHOLDING ALGORITHM | PHYSICS, MATHEMATICAL | SUBGRADIENT PROJECTION ALGORITHMS | EXTRAGRADIENT METHOD | OPTIMIZATION | POINT ALGORITHM | MONOTONE-OPERATORS | INEXACT INFINITE PRODUCTS

Journal Article

International Journal of Computer Mathematics: A special collection of papers relating to computational linear algebra and nonlinear equation solvers, ISSN 0020-7160, 08/2019, Volume 96, Issue 8, pp. 1603 - 1622

In this paper, we consider the backward error and condition number of the indefinite linear least squares (ILS) problem. For the normwise backward error of...

componentwise perturbation | power method | 15A12 | normwise backward error | 65F35 | linearization estimate | 15A09 | Indefinite least squares | condition number | MATHEMATICS, APPLIED | PERTURBATION BOUNDS | COMPONENTWISE CONDITION NUMBERS | ALGORITHM | INERTIA | MOORE-PENROSE INVERSE | Algorithms | Error analysis | Mathematical analysis | Upper bounds | Least squares method | Matrix methods | Linearization | Linear functions

componentwise perturbation | power method | 15A12 | normwise backward error | 65F35 | linearization estimate | 15A09 | Indefinite least squares | condition number | MATHEMATICS, APPLIED | PERTURBATION BOUNDS | COMPONENTWISE CONDITION NUMBERS | ALGORITHM | INERTIA | MOORE-PENROSE INVERSE | Algorithms | Error analysis | Mathematical analysis | Upper bounds | Least squares method | Matrix methods | Linearization | Linear functions

Journal Article

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LOWER BOUNDS ON APPROXIMATION ERRORS TO NUMERICAL SOLUTIONS OF DYNAMIC ECONOMIC MODELS

Econometrica, ISSN 0012-9682, 5/2017, Volume 85, Issue 3, pp. 991 - 1012

We propose a novel methodology for evaluating the accuracy of numerical solutions to dynamic economic models. It consists in constructing a lower bound on the...

NOTES AND COMMENTS | numerical solution | approximate solution | error bound | Euler equation residuals | new Keynesian model | accuracy | Approximation errors | backward error analysis | forward error analysis | upper error bound | lower error bound | TESTS | PERTURBATION | STATISTICS & PROBABILITY | SIMULATION | POLICY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | SOCIAL SCIENCES, MATHEMATICAL METHODS | ECONOMICS | Economic models | Accuracy | Error analysis | Growth models

NOTES AND COMMENTS | numerical solution | approximate solution | error bound | Euler equation residuals | new Keynesian model | accuracy | Approximation errors | backward error analysis | forward error analysis | upper error bound | lower error bound | TESTS | PERTURBATION | STATISTICS & PROBABILITY | SIMULATION | POLICY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | SOCIAL SCIENCES, MATHEMATICAL METHODS | ECONOMICS | Economic models | Accuracy | Error analysis | Growth models

Journal Article

BIT Numerical Mathematics, ISSN 0006-3835, 12/2018, Volume 58, Issue 4, pp. 907 - 935

We describe how to perform the backward error analysis for the approximation of $$\exp (A)v$$ exp(A)v by $$p(s^{-1}A)^sv$$ p(s-1A)sv , for any given polynomial...

Computational Mathematics and Numerical Analysis | 65F60 | 65F30 | Numeric Computing | Leja–Hermite interpolation | 65D05 | Mathematics, general | Backward error analysis | Taylor series | Mathematics | Action of matrix exponential | INTERPOLATION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | ALGORITHM | DIVIDED DIFFERENCES | COMPUTATION | Leja-Hermite interpolation | Computer science | Analysis

Computational Mathematics and Numerical Analysis | 65F60 | 65F30 | Numeric Computing | Leja–Hermite interpolation | 65D05 | Mathematics, general | Backward error analysis | Taylor series | Mathematics | Action of matrix exponential | INTERPOLATION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | ALGORITHM | DIVIDED DIFFERENCES | COMPUTATION | Leja-Hermite interpolation | Computer science | Analysis

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 05/2016, Volume 437, Issue 2, pp. 1283 - 1304

We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive...

Finite element method | Convolution quadrature | Strong and weak convergence | Malliavin regularity | Stochastic Volterra equations | Backward Euler | MATHEMATICS, APPLIED | APPROXIMATION | MATHEMATICS | PARTIAL-DIFFERENTIAL-EQUATIONS | EVOLUTION EQUATION | CONVERGENCE | DISCRETIZED OPERATIONAL CALCULUS | Mathematics - Numerical Analysis | Stochastic Volterra equation; Finite element method; Backward Euler; Convolution quadrature; Strong and weak convergence; Malliavin calculus; Regularity; Duality | Beräkningsmatematik | Computational Mathematics | Sannolikhetsteori och statistik | Probability Theory and Statistics

Finite element method | Convolution quadrature | Strong and weak convergence | Malliavin regularity | Stochastic Volterra equations | Backward Euler | MATHEMATICS, APPLIED | APPROXIMATION | MATHEMATICS | PARTIAL-DIFFERENTIAL-EQUATIONS | EVOLUTION EQUATION | CONVERGENCE | DISCRETIZED OPERATIONAL CALCULUS | Mathematics - Numerical Analysis | Stochastic Volterra equation; Finite element method; Backward Euler; Convolution quadrature; Strong and weak convergence; Malliavin calculus; Regularity; Duality | Beräkningsmatematik | Computational Mathematics | Sannolikhetsteori och statistik | Probability Theory and Statistics

Journal Article

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