A dynamical Tikhonov regularization for solving ill-posed linear algebraic systems

DOI10.1007/s10440-012-9766-3zbMath1259.65068OpenAlexW2065119749MaRDI QIDQ1938001

Chein-Shan Liu

Publication date: 1 February 2013

Published in: Acta Applicandae Mathematicae (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10440-012-9766-3

zbMATH Keywords

numerical examplesconjugate gradient methodTikhonov regularizationsteepest descent methodill-posed linear systemadaptive Tikhonov methodBarzilai-Borwein method (BBM)dynamical Tikhonov regularizationoptimal vector method

Mathematics Subject Classification ID

Ill-posedness and regularization problems in numerical linear algebra (65F22) Iterative numerical methods for linear systems (65F10)

Related Items

Метод расширенных нормальных уравнений для задач регуляризации Тихонова с дифференцирующим оператором ⋮ A vector regularization method to solve linear inverse problems ⋮ A double optimal descent algorithm for iteratively solving ill-posed linear inverse problems ⋮ An iterative algorithm for identifying heat source by using a DQ and a Lie-group method ⋮ Optimal algorithms in a Krylov subspace for solving linear inverse problems by MFS ⋮ Adaptive choice of the Tikhonov regularization parameter to solve ill-posed linear algebraic equations via Liapunov optimizing control ⋮ A globally optimal tri-vector method to solve an ill-posed linear system ⋮ A doubly optimized solution of linear equations system expressed in an affine Krylov subspace ⋮ To solve the inverse Cauchy problem in linear elasticity by a novel Lie-group integrator

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Optimally scaled vector regularization method to solve ill-posed linear problems
• Convergence of numerical solution of the Fredholm integral equation of the first kind with degenerate kernel
• Dynamical systems method for solving operator equations
• A study of type I intermittency of a circular differential equation under a discontinuous right-hand side
• An efficient dynamical systems method for solving singularly perturbed integral equations with noise
• Dynamical systems gradient method for solving ill-conditioned linear algebraic systems
• Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates
• On the choice of the regularization parameter for iterated Tikhonov regularization of ill-posed problems
• Numerical solution of linear Fredholm integral equation by using hybrid Taylor and Block-Pulse functions.
• Intermittent transition to quasiperiodicity demonstrated via a circular differential equation
• Modified two-point stepsize gradient methods for unconstrained optimization
• Optimization and dynamical systems
• A new mechanical algorithm for solving the second kind of Fredholm integral equation
• Differential equations and solution of linear systems
• Choosing Regularization Parameters in Iterative Methods for Ill-Posed Problems
• Fast realization algorithms for determining regularization parameters in linear inverse problems
• R-linear convergence of the Barzilai and Borwein gradient method
• A Dynamical Tikhonov Regularization Method for Solving Nonlinear Ill-Posed Problems
• A Revision of Relaxed Steepest Descent Method from the Dynamics on an Invariant Manifold
• The Concept of Best Vector Used to Solve Ill-Posed Linear Inverse Problems
• A Fictitious Time Integration Method for the Numerical Solution of the Fredholm Integral Equation and for Numerical Differentiation of Noisy Data, and Its Relation to the Filter Theory
• On Solving the Ill-Conditioned System Ax=b: General-Purpose Conditioners Obtained From the Boundary-Collocation Solution of the Laplace Equation, Using Trefftz Expansions With Multiple Length Scales
• A Scalar Homotopy Method for Solving an Over/Under-Determined System of Non-Linear Algebraic Equations
• Novel Algorithms Based on the Conjugate Gradient Method for Inverting Ill-Conditioned Matrices, and a New Regularization Method to Solve Ill-Posed Linear Systems
• An Iterative Algorithm for Solving a System of Nonlinear Algebraic Equations, F ( x )= 0 , Using the System of ODEs with an Optimum a in d x = l [ a F +(1 - a ) B T F ; B ij =dF i /dx j]
• A Highly Accurate Technique for Interpolations Using Very High-Order Polynomials, and Its Applications to Some Ill-Posed Linear Problems
• The Barzilai and Borwein Gradient Method for the Large Scale Unconstrained Minimization Problem
• Dynamical systems method for solving linear ill-posed problems
• An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates
• Two-Point Step Size Gradient Methods
• Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter
• Analysis of Discrete Ill-Posed Problems by Means of the L-Curve
• Iterative choices of regularization parameters in linear inverse problems
• Gradient Method with Retards and Generalizations
• The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems
• Comparisons of parameter choice methods for regularization with discrete noisy data
• Alternate minimization gradient method
• An improved model function method for choosing regularization parameters in linear inverse problems
• Solving ill-conditioned linear algebraic systems by the dynamical systems method
• On the Barzilai and Borwein choice of steplength for the gradient method
• Control Perspectives on Numerical Algorithms and Matrix Problems
• Regularization of ill-posed problems in Banach spaces: convergence rates
• An Iteration Formula for Fredholm Integral Equations of the First Kind
• Relaxed steepest descent and Cauchy-Barzilai-Borwein method