The instability of some gradient methods for ill-posed problems

DOI10.1007/BF01385614zbMath0714.65056MaRDI QIDQ751191

Bertolt Eicke, Alfred K. Louis, Robert Plato

Publication date: 1990

Published in: Numerische Mathematik (Search for Journal in Brave)

Full work available at URL: https://eudml.org/doc/133491

zbMATH Keywords

Hilbert spaceerror boundssteepest descentill-posed problemsconjugate gradientsa-priori parameter choice regularization methodsnonclosed range

Mathematics Subject Classification ID

Numerical solutions to equations with linear operators (65J10) Numerical solutions of ill-posed problems in abstract spaces; regularization (65J20)

Related Items (15)

Regularization properties of Krylov iterative solvers CGME and LSMR for linear discrete ill-posed problems with an application to truncated randomized SVDs ⋮ Generalized cross-validation applied to conjugate gradient for discrete ill-posed problems ⋮ Approximate Inverse Preconditioners for the Conjugate Gradient Method ⋮ A Modified Minimal Error Method for Solving Nonlinear Integral Equations via Multiscale Galerkin Methods ⋮ Iteration methods for convexly constrained ill-posed problems in hilbert space ⋮ A modified steepest descent method for solving non-smooth inverse problems ⋮ A convergence analysis of a method of steepest descent and a two–step algorothm for nonlinear ill–posed problems ⋮ An inner-outer regularizing method for ill-posed problems ⋮ Accelerated Landweber iterations for the solution of ill-posed equations ⋮ Convergence rate results for steepest descent type method for nonlinear ill-posed equations ⋮ A New Gradient Method for Ill-Posed Problems ⋮ Iterative Solution Methods ⋮ A conjugate-gradient-type rational Krylov subspace method for ill-posed problems ⋮ The low rank approximations and Ritz values in LSQR for linear discrete ill-posed problem ⋮ On general convergence behaviours of finite-dimensional approximants for abstract linear inverse problems

Cites Work

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• The rate of convergence of conjugate gradients
• Iterative solution methods for inconsistent linear equations with nonself-adjoint operators
• A uniform approach to gradient methods for linear operator equations
• Optimal algorithms for linear ill-posed problems yield regularization methods
• Some Superlinear Convergence Results for the Conjugate Gradient Method
• Necessary and sufficient conditions for convergence of regularization methods for solving linear operator equations of the first kind
• Regularizing algorithms based on the conjugate-gradient method
• The regularizing properties of the adjoint gradient method in ill-posed problems
• Conditions for Termination of the Method of Steepest Descent after a Finite Number of Iterations
• The Conjugate Gradient Method for Linear and Nonlinear Operator Equations

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