Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning
We analyze the conjugate gradient (CG) method with variable preconditioning for solving a linear system with a real symmetric positive definite (SPD) matrix of coefficients $A$. We assume that the preconditioner is SPD on each step, and that the condition number of the preconditioned system matrix i...
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| Published in: | SIAM journal on matrix analysis and applications Vol. 29; no. 4; pp. 1267 - 1280 |
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| Format: | Journal Article |
| Language: | English |
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Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2007
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| ISSN: | 0895-4798, 1095-7162 |
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| Abstract | We analyze the conjugate gradient (CG) method with variable preconditioning for solving a linear system with a real symmetric positive definite (SPD) matrix of coefficients $A$. We assume that the preconditioner is SPD on each step, and that the condition number of the preconditioned system matrix is bounded above by a constant independent of the step number. We show that the CG method with variable preconditioning under this assumption may not give improvement, compared to the steepest descent (SD) method. We describe the basic theory of CG methods with variable preconditioning with the emphasis on "worst case" scenarios, and provide complete proofs of all facts not available in the literature. We give a new elegant geometric proof of the SD convergence rate bound. Our numerical experiments, comparing the preconditioned SD and CG methods, not only support and illustrate our theoretical findings, but also reveal two surprising and potentially practically important effects. First, we analyze variable preconditioning in the form of inner-outer iterations. In previous such tests, the unpreconditioned CG inner iterations are applied to an artificial system with some fixed preconditioner as a matrix of coefficients. We test a different scenario, where the unpreconditioned CG inner iterations solve linear systems with the original system matrix $A$. We demonstrate that the CG-SD inner-outer iterations perform as well as the CG-CG inner-outer iterations in these tests. Second, we compare the CG methods using a two-grid preconditioning with fixed and randomly chosen coarse grids, and observe that the fixed preconditioner method is twice as slow as the method with random preconditioning. |
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| AbstractList | We analyze the conjugate gradient (CG) method with variable preconditioning for solving a linear system with a real symmetric positive definite (SPD) matrix of coefficients $A$. We assume that the preconditioner is SPD on each step, and that the condition number of the preconditioned system matrix is bounded above by a constant independent of the step number. We show that the CG method with variable preconditioning under this assumption may not give improvement, compared to the steepest descent (SD) method. We describe the basic theory of CG methods with variable preconditioning with the emphasis on "worst case" scenarios, and provide complete proofs of all facts not available in the literature. We give a new elegant geometric proof of the SD convergence rate bound. Our numerical experiments, comparing the preconditioned SD and CG methods, not only support and illustrate our theoretical findings, but also reveal two surprising and potentially practically important effects. First, we analyze variable preconditioning in the form of inner-outer iterations. In previous such tests, the unpreconditioned CG inner iterations are applied to an artificial system with some fixed preconditioner as a matrix of coefficients. We test a different scenario, where the unpreconditioned CG inner iterations solve linear systems with the original system matrix $A$. We demonstrate that the CG-SD inner-outer iterations perform as well as the CG-CG inner-outer iterations in these tests. Second, we compare the CG methods using a two-grid preconditioning with fixed and randomly chosen coarse grids, and observe that the fixed preconditioner method is twice as slow as the method with random preconditioning. |
| Author | Lashuk, Ilya Knyazev, Andrew V. |
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| Keywords | random preconditioning Support conjugate gradient Steepest descent method circular cone Iterative method Iteration variable preconditioning Multigrid Convergence rate Steepest descent Real matrix Conjugate gradient method Cone Grid pattern Solving convergence rate bound Reflection linear systems multigrid; 65F10 Numerical analysis Linear system Symmetric matrix Symmetric system Linear algebra Positive definite matrix Condition number Householder reflection inner-outer iterations preconditioner Preconditioning |
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| SubjectTerms | Acceleration of convergence Algebra Algebraic geometry Algorithms Applied mathematics Exact sciences and technology Iterative methods Linear and multilinear algebra, matrix theory Mathematics Methods Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Sciences and techniques of general use |
| Title | Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning |
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