Inverse Problems in Science and Engineering, ISSN 1741-5977, 03/2016, Volume 24, Issue 3, pp. 524 - 541

This paper focuses on prior information for improved sparsity reconstruction in electrical impedance tomography with partial data, i.e. Cauchy data measured on...

partial data | sparsity | inverse boundary value problem | electrical impedance tomography | ill-posed problem | BOUNDARY-VALUE PROBLEM | RECONSTRUCTION ALGORITHM | GLOBAL UNIQUENESS | EQUATIONS | CALDERON PROBLEM | 3D RECONSTRUCTIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | INVERSE CONDUCTIVITY PROBLEM | PARTIAL CAUCHY DATA | 65N21 | REGULARIZATION | 65N20 | Tomography | Boundary value problems | Inverse problems | Sparsity | Velocity | Reconstruction | Norms | Mathematical models | Boundaries | Flexibility | Optimization | Electrical impedance | Mathematics - Numerical Analysis

partial data | sparsity | inverse boundary value problem | electrical impedance tomography | ill-posed problem | BOUNDARY-VALUE PROBLEM | RECONSTRUCTION ALGORITHM | GLOBAL UNIQUENESS | EQUATIONS | CALDERON PROBLEM | 3D RECONSTRUCTIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | INVERSE CONDUCTIVITY PROBLEM | PARTIAL CAUCHY DATA | 65N21 | REGULARIZATION | 65N20 | Tomography | Boundary value problems | Inverse problems | Sparsity | Velocity | Reconstruction | Norms | Mathematical models | Boundaries | Flexibility | Optimization | Electrical impedance | Mathematics - Numerical Analysis

Journal Article

Journal of Scientific Computing, ISSN 0885-7474, 1/2019, Volume 78, Issue 1, pp. 226 - 245

Inspired by the method (Li, Ji and Zhou in J Sci Comput, 2018, https://doi.org/10.1007/s10915-018-0774-y ) using a dynamics of points on virtual geometric...

Equality constrained saddles | Computational Mathematics and Numerical Analysis | 58E05 | Constrained local minimax method | 35A15 | Theoretical, Mathematical and Computational Physics | Mathematics | Implementation | Convergence | Virtual geometric objects | Algorithms | Mathematical and Computational Engineering | 49M05 | 65N20

Equality constrained saddles | Computational Mathematics and Numerical Analysis | 58E05 | Constrained local minimax method | 35A15 | Theoretical, Mathematical and Computational Physics | Mathematics | Implementation | Convergence | Virtual geometric objects | Algorithms | Mathematical and Computational Engineering | 49M05 | 65N20

Journal Article

Journal of Scientific Computing, ISSN 0885-7474, 1/2019, Volume 78, Issue 1, pp. 202 - 225

By a dynamics of points on virtual geometric objects such as curves, surfaces, etc., with a flexible endpoint, this paper is to develop a new local minimax...

Computational Mathematics and Numerical Analysis | 58E05 | 35A15 | Theoretical, Mathematical and Computational Physics | Mathematics | Implementation | Local minimax method | Convergence | Virtual geometric objects | Algorithms | Mathematical and Computational Engineering | Saddles | 49M05 | 65N20

Computational Mathematics and Numerical Analysis | 58E05 | 35A15 | Theoretical, Mathematical and Computational Physics | Mathematics | Implementation | Local minimax method | Convergence | Virtual geometric objects | Algorithms | Mathematical and Computational Engineering | Saddles | 49M05 | 65N20

Journal Article

Journal of Inverse and Ill-posed Problems, ISSN 0928-0219, 02/2019, Volume 27, Issue 1, pp. 1 - 16

In this paper, we consider an inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach; that is, to...

conjugate gradientalgorithm | 65N21 | uniqueness | Tikhonov regularization | Inverse spatial source problem | 65N20 | conjugate gradient algorithm | MATHEMATICS | MATHEMATICS, APPLIED | TRANSPORT | IDENTIFY | Regularization methods | Regularization

conjugate gradientalgorithm | 65N21 | uniqueness | Tikhonov regularization | Inverse spatial source problem | 65N20 | conjugate gradient algorithm | MATHEMATICS | MATHEMATICS, APPLIED | TRANSPORT | IDENTIFY | Regularization methods | Regularization

Journal Article

Numerische Mathematik, ISSN 0029-599X, 10/2018, Volume 140, Issue 2, pp. 449 - 478

In this paper we consider the iteratively regularized Gauss–Newton method (IRGNM) in its classical Tikhonov version as well as two further—Ivanov type and...

Mathematical Methods in Physics | 65F22 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | 65N20 | ELLIPTIC CONTROL-PROBLEMS | EQUATIONS | MATHEMATICS, APPLIED | ILL-POSED PROBLEMS | PARAMETER

Mathematical Methods in Physics | 65F22 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematical and Computational Engineering | Mathematics, general | Mathematics | Numerical and Computational Physics, Simulation | 65N20 | ELLIPTIC CONTROL-PROBLEMS | EQUATIONS | MATHEMATICS, APPLIED | ILL-POSED PROBLEMS | PARAMETER

Journal Article

Journal of Inverse and Ill-posed Problems, ISSN 0928-0219, 10/2019, Volume 27, Issue 5, pp. 657 - 669

A method based on least squares support vector machines (LS-SVM) is proposed to solve the source inverse problem of wave equations. Contrary to the most...

68T99 | least squares support vector machines | Wave equations | 34A55 | approximate solutions | source inverse problem | 35L20 | 65M32 | 65N20 | MATHEMATICS | MATHEMATICS, APPLIED | BOUNDARY-VALUE-PROBLEMS | NEURAL-NETWORK METHOD | Support vector machines | Inverse problems | Approximation | Robustness (mathematics) | Initial conditions | Boundary conditions

68T99 | least squares support vector machines | Wave equations | 34A55 | approximate solutions | source inverse problem | 35L20 | 65M32 | 65N20 | MATHEMATICS | MATHEMATICS, APPLIED | BOUNDARY-VALUE-PROBLEMS | NEURAL-NETWORK METHOD | Support vector machines | Inverse problems | Approximation | Robustness (mathematics) | Initial conditions | Boundary conditions

Journal Article

Numerische Mathematik, ISSN 0029-599X, 9/2015, Volume 131, Issue 1, pp. 33 - 57

In this paper we analyze a second order method for regularizing nonlinear inverse problems, which comes from adapting Halley’s method to the ill-posed...

Mathematical Methods in Physics | 65F22 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Numerical and Computational Physics | Mathematics, general | Mathematics | 65N20 | MATHEMATICS, APPLIED | CONVERGENCE

Mathematical Methods in Physics | 65F22 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Numerical and Computational Physics | Mathematics, general | Mathematics | 65N20 | MATHEMATICS, APPLIED | CONVERGENCE

Journal Article

Numerical algorithms, ISSN 1572-9265, 2018, Volume 79, Issue 3, pp. 825 - 851

This paper focuses on efficient computational approaches to compute approximate solutions of a linear inverse problem that is contaminated with mixed...

Preconditioner | Poisson–Gaussian model | Numeric Computing | Theory of Computation | Image restoration | Weighted least squares | Algorithms | Algebra | Numerical Analysis | Computer Science | 49M15 | Robust regression | 62F35 | 65N20 | Poisson-Gaussian model | MATHEMATICS, APPLIED | NOISE | PRECONDITIONERS | IMAGE | ITERATIVE METHODS | Computer science | Models | Analysis | Gaussian processes | Mathematics - Numerical Analysis

Preconditioner | Poisson–Gaussian model | Numeric Computing | Theory of Computation | Image restoration | Weighted least squares | Algorithms | Algebra | Numerical Analysis | Computer Science | 49M15 | Robust regression | 62F35 | 65N20 | Poisson-Gaussian model | MATHEMATICS, APPLIED | NOISE | PRECONDITIONERS | IMAGE | ITERATIVE METHODS | Computer science | Models | Analysis | Gaussian processes | Mathematics - Numerical Analysis

Journal Article

9.
Full Text
A comparison of error estimates at a point and on a set when solving ill-posed problems

Journal of Inverse and Ill-posed Problems, ISSN 0928-0219, 08/2018, Volume 26, Issue 4, pp. 541 - 550

The problem of correlating the error estimates at a point and on a correctness class is of interest to many mathematicians. Since the desired solution to a...

Ill-posed problem | error estimate on a set | error estimate at a point | regularization | 65N20 | MATHEMATICS | MATHEMATICS, APPLIED | Fuzzy sets | Mathematical research | Functions, Inverse | Error functions | Set theory | Research | Points (Geometry) | Ill-posed problems (mathematics) | Error correction

Ill-posed problem | error estimate on a set | error estimate at a point | regularization | 65N20 | MATHEMATICS | MATHEMATICS, APPLIED | Fuzzy sets | Mathematical research | Functions, Inverse | Error functions | Set theory | Research | Points (Geometry) | Ill-posed problems (mathematics) | Error correction

Journal Article

Journal of Inverse and Ill-posed Problems, ISSN 0928-0219, 08/2018, Volume 26, Issue 4, pp. 493 - 499

This article presents the solution of a special inverse elastography problem: knowing vertical displacements of compressed biological tissue to find a...

Young’s modulus | inverse problem | 65N12 | quasisolutions | a posteriori error estimates | 65N21 | Elastography | 65N30 | 65N20 | Young's modulus | MATHEMATICS | MATHEMATICS, APPLIED | Mathematical research | Functions, Inverse | Elasticity | Models | Research | Tissues | Properties | Parameter modification | Inverse problems | Differential equations | Compacts | Modulus of elasticity | Inclusions | Strain

Young’s modulus | inverse problem | 65N12 | quasisolutions | a posteriori error estimates | 65N21 | Elastography | 65N30 | 65N20 | Young's modulus | MATHEMATICS | MATHEMATICS, APPLIED | Mathematical research | Functions, Inverse | Elasticity | Models | Research | Tissues | Properties | Parameter modification | Inverse problems | Differential equations | Compacts | Modulus of elasticity | Inclusions | Strain

Journal Article

Computational Methods in Applied Mathematics, ISSN 1609-4840, 04/2019, Volume 19, Issue 2, pp. 323 - 339

We are concerned with the wave propagation in a homogeneous 2D or 3D membrane Ω of finite size. We assume that either the membrane is initially at rest or we...

65N15 | Satisfier Function | 65N12 | Inverse Wave Problem | 65N21 | Hyperbolic Equation | 65N20 | MATHEMATICS, APPLIED | HEAT-SOURCE | ELASTIC FORCE | INVERSE PROBLEMS | OPTIMIZATION | BOUNDARY CONTROL | FUNDAMENTAL-SOLUTIONS | Time dependence | Wave propagation | Inverse problems | Estimation | Displacement

65N15 | Satisfier Function | 65N12 | Inverse Wave Problem | 65N21 | Hyperbolic Equation | 65N20 | MATHEMATICS, APPLIED | HEAT-SOURCE | ELASTIC FORCE | INVERSE PROBLEMS | OPTIMIZATION | BOUNDARY CONTROL | FUNDAMENTAL-SOLUTIONS | Time dependence | Wave propagation | Inverse problems | Estimation | Displacement

Journal Article

Numerische Mathematik, ISSN 0029-599X, 11/2015, Volume 131, Issue 3, pp. 517 - 540

In this paper we study the impact of two types of preconditioning on the numerical solution of large sparse augmented linear systems. The first preconditioning...

Mathematical Methods in Physics | 65F10 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Numerical and Computational Physics | Mathematics, general | Mathematics | 65N20 | CR:5.13 | HERMITIAN SPLITTING METHODS | ALGEBRAIC MULTIGRID METHOD | MATHEMATICS, APPLIED | SOR | ALGORITHM | CONVERGENCE | INEXACT | SMOOTHERS | Linear systems | Comparative analysis | Methods

Mathematical Methods in Physics | 65F10 | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Numerical and Computational Physics | Mathematics, general | Mathematics | 65N20 | CR:5.13 | HERMITIAN SPLITTING METHODS | ALGEBRAIC MULTIGRID METHOD | MATHEMATICS, APPLIED | SOR | ALGORITHM | CONVERGENCE | INEXACT | SMOOTHERS | Linear systems | Comparative analysis | Methods

Journal Article

Journal of Inverse and Ill-posed Problems, ISSN 0928-0219, 12/2018, Volume 26, Issue 6, pp. 835 - 857

We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary...

ill-posed problems | regularization | 65N60 | 35J15 | non-local boundary value problems | very weak solution | finite difference scheme | Cauchy problem | elliptic equation | 65N20 | MATHEMATICS | MATHEMATICS, APPLIED | INVERSE | Viscosity | Boundary value problems | Cauchy problems | Mathematical analysis | Elliptic functions | Well posed problems | Finite difference method

ill-posed problems | regularization | 65N60 | 35J15 | non-local boundary value problems | very weak solution | finite difference scheme | Cauchy problem | elliptic equation | 65N20 | MATHEMATICS | MATHEMATICS, APPLIED | INVERSE | Viscosity | Boundary value problems | Cauchy problems | Mathematical analysis | Elliptic functions | Well posed problems | Finite difference method

Journal Article

BIT Numerical Mathematics, ISSN 0006-3835, 9/2016, Volume 56, Issue 3, pp. 919 - 949

This paper proposes a new approach for choosing the regularization parameters in multi-parameter regularization methods when applied to approximate the...

Arnoldi–Tikhonov method | Computational Mathematics and Numerical Analysis | Ill-posed problems | Discrepancy principle | 65F10 | 65F22 | Numeric Computing | Mathematics, general | Multi-parameter Tikhonov method | Mathematics | 65N21 | 65N20 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | TIKHONOV REGULARIZATION | MATHEMATICS, APPLIED | Arnoldi-Tikhonov method | L-CURVE | PARAMETER

Arnoldi–Tikhonov method | Computational Mathematics and Numerical Analysis | Ill-posed problems | Discrepancy principle | 65F10 | 65F22 | Numeric Computing | Mathematics, general | Multi-parameter Tikhonov method | Mathematics | 65N21 | 65N20 | COMPUTER SCIENCE, SOFTWARE ENGINEERING | TIKHONOV REGULARIZATION | MATHEMATICS, APPLIED | Arnoldi-Tikhonov method | L-CURVE | PARAMETER

Journal Article

Applicable Analysis: Multiscale Inverse Problems, ISSN 0003-6811, 01/2018, Volume 97, Issue 1, pp. 3 - 12

Despite the strong focus of regularization on ill-posed problems, the general construction of such methods has not been fully explored. Moreover, many previous...

error estimate | mild solution | Ill-posed problems | Inverse problems | filter regularization | well-posedness | MATHEMATICS, APPLIED | ELLIPTIC EQUATION | 47A52 | 46E20 | 47J06 | 65N21 | 65N20 | Operators | Nonlinear equations | Well posed problems | Cauchy problem | Problems | Mathematical analysis | Chaos theory | Construction methods | Nonlinear evolution equations | Hilbert space | Ill-posed problems (mathematics) | Nonlinear programming | Regularization | Quantum theory

error estimate | mild solution | Ill-posed problems | Inverse problems | filter regularization | well-posedness | MATHEMATICS, APPLIED | ELLIPTIC EQUATION | 47A52 | 46E20 | 47J06 | 65N21 | 65N20 | Operators | Nonlinear equations | Well posed problems | Cauchy problem | Problems | Mathematical analysis | Chaos theory | Construction methods | Nonlinear evolution equations | Hilbert space | Ill-posed problems (mathematics) | Nonlinear programming | Regularization | Quantum theory

Journal Article

Bulletin of the Malaysian Mathematical Sciences Society, ISSN 0126-6705, 10/2017, Volume 40, Issue 4, pp. 1493 - 1522

In this paper, a Cauchy problem for the Helmholtz equation is investigated. It is well known that this problem is severely ill-posed in the sense that the...

65N12 | 35R30 | Error estimate | Helmholtz equation | Mathematics, general | Mathematics | Ill-posed problem | Applications of Mathematics | Regularization | Cauchy problem | 65N20 | MATHEMATICS | REGULARIZATION METHODS | DOMAINS | FUNDAMENTAL-SOLUTIONS | ELLIPTIC-OPERATORS | Helmholtz equations | Cauchy problems | Regularization methods | Stability analysis | Ill-posed problems (mathematics)

65N12 | 35R30 | Error estimate | Helmholtz equation | Mathematics, general | Mathematics | Ill-posed problem | Applications of Mathematics | Regularization | Cauchy problem | 65N20 | MATHEMATICS | REGULARIZATION METHODS | DOMAINS | FUNDAMENTAL-SOLUTIONS | ELLIPTIC-OPERATORS | Helmholtz equations | Cauchy problems | Regularization methods | Stability analysis | Ill-posed problems (mathematics)

Journal Article

Numerische Mathematik, ISSN 0945-3245, 2002, Volume 93, Issue 2, pp. 333 - 359

In this paper, we consider some nonlinear inexact Uzawa methods for iteratively solving linear saddle-point problems. By means of a new technique, we first...

Mathematics Subject Classification: 65F10, 65N20 | MATHEMATICS, APPLIED | CONVERGENCE | SYSTEMS | PRECONDITIONER

Mathematics Subject Classification: 65F10, 65N20 | MATHEMATICS, APPLIED | CONVERGENCE | SYSTEMS | PRECONDITIONER

Journal Article

18.
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Multilevel Preconditioners for Reaction-Diffusion Problems with Discontinuous Coefficients

Journal of Scientific Computing, ISSN 0885-7474, 4/2016, Volume 67, Issue 1, pp. 324 - 350

In this paper, we extend some of the multilevel convergence results obtained by Xu and Zhu in [Xu and Zhu, M3AS 2008], to the case of second order linear...

Computational Mathematics and Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematics | BPX | Algorithms | 65F10 | Multigrid | Multilevel preconditioners | Appl.Mathematics/Computational Methods of Engineering | Robust solver | Reaction-diffusion equations | Discontinuous coefficients | 65N30 | 65N20 | Linear systems | Approximation | Mathematical analysis | Multilevel | Diffusion | Diffusion coefficient | Convergence | MATHEMATICS AND COMPUTING

Computational Mathematics and Numerical Analysis | Theoretical, Mathematical and Computational Physics | Mathematics | BPX | Algorithms | 65F10 | Multigrid | Multilevel preconditioners | Appl.Mathematics/Computational Methods of Engineering | Robust solver | Reaction-diffusion equations | Discontinuous coefficients | 65N30 | 65N20 | Linear systems | Approximation | Mathematical analysis | Multilevel | Diffusion | Diffusion coefficient | Convergence | MATHEMATICS AND COMPUTING

Journal Article

Computational and Applied Mathematics, ISSN 0101-8205, 5/2018, Volume 37, Issue 2, pp. 1507 - 1523

In this paper, we use a new special generalized Hermitian and skew-Hermitian splitting (SGHSS) method for solving ill-posed inverse problems. Based on an...

Computational Mathematics and Numerical Analysis | 65F10 | Mathematical Applications in Computer Science | Mathematics | Ill-posed problem | Applications of Mathematics | Image restoration | Mathematical Applications in the Physical Sciences | Generalized Hermitian and skew-Hermitian splitting (GHSS) | Tikhonov regularization | 65D18 | 65N20 | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | ITERATIVE METHODS

Computational Mathematics and Numerical Analysis | 65F10 | Mathematical Applications in Computer Science | Mathematics | Ill-posed problem | Applications of Mathematics | Image restoration | Mathematical Applications in the Physical Sciences | Generalized Hermitian and skew-Hermitian splitting (GHSS) | Tikhonov regularization | 65D18 | 65N20 | HERMITIAN SPLITTING METHODS | MATHEMATICS, APPLIED | ITERATIVE METHODS

Journal Article

Journal of Inverse and Ill-posed Problems, ISSN 0928-0219, 10/2016, Volume 24, Issue 5, pp. 625 - 636

The relations between stability and accuracy of the operator marching method (OMM) are usually conflicting in waveguides with strong range dependence. To...

DtN reformulation,Helmholtz equation | 65N12 | Operator marching method | error analysis | local orthogonal transform | 65N20 | Helmholtz equation | DtN reformulation | MATHEMATICS | MATHEMATICS, APPLIED | INVERSE PROBLEMS | Waveguides | Transformations (Mathematics) | Usage | Wave propagation | Error analysis (Mathematics) | Analysis | Error analysis

DtN reformulation,Helmholtz equation | 65N12 | Operator marching method | error analysis | local orthogonal transform | 65N20 | Helmholtz equation | DtN reformulation | MATHEMATICS | MATHEMATICS, APPLIED | INVERSE PROBLEMS | Waveguides | Transformations (Mathematics) | Usage | Wave propagation | Error analysis (Mathematics) | Analysis | Error analysis

Journal Article

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