The Use of Defect Correction to Refine the Eigenelements of Compact Integral Operators

Ahués, Mario; Chatelin, Françoise

doi:10.1137/0720077

Cited by 13 publications

(13 citation statements)

References 8 publications

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“…Spectral refinement is a procedure that allows one to approximate eigenelements of a very large discrete system by successively improving upon the eigenelements obtained from a coarse model through direct methods. Several refinement methods such as the Rayleigh-Schrodinger method ( [2,7,8,14,15]), the fixed slope Newton scheme and its variants ( [3,4,8,11,15]) and the defect correction method ( [1,8,16]) have been studied for approximating a simple eigenvalue. These methods avoid solving large matrix eigenvalue problems, thus saving time and memory.…”

Section: Introductionmentioning

confidence: 99%

Accelerated spectral refinement Part I: simple eigenvalue

Alam

Kulkarni

Limaye

2000

J. Aust. Math. Soc. Series B, Appl. Math

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Section: Introductionmentioning

confidence: 99%

Accelerated spectral refinement Part I: simple eigenvalue

Alam

Kulkarni

Limaye

2000

J. Aust. Math. Soc. Series B, Appl. Math

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“…Subtracting the exact solution, A −1 r (0) , of the IR residual equation from both sides and noting that e ( 1 2 ) = Ge (0) yields e (1) = e ( 1 2 ) − P B c (P t AP ) −1 P t AGe (0) − P B c (P t AP ) −1 δ = (I − P B c (P t AP ) −1 P t A)Ge (0) − P B c (P t AP ) −1 δ.…”

Section: Convergence Theory: Two-gridmentioning

confidence: 99%

“…Using superscripts as in the previous proof, we first estimate how the T G cycle approximates the solution A −1 r (0) of Ay = r (0) , where r (0) = r + δ 0 , δ 0 ≤ ε r , is r rounded to ε-precision. Specifically, this proof is mostly devoted to showing that the final computed approximation y (1) satisfies (7.10)…”

Section: Taking Energy Norms Of Both Sides Yieldsmentioning

confidence: 99%

“…To this end, for each j ∈ {1, 2}, consider a monotonically energy-convergent stationary linear iteration x ← x − M (j) (Ax − b), where M (j) ∈ R n×n , and let α j be a constant such that computing M (j) z for a vector z ∈ R n in ε-precision yields the result M (j) z + δ, δ ≤ α j ε z . Then the key point here is that the error propagation matrix for relaxation with preconditioner M (1) followed by relaxation with preconditioner M (2) can be written as (I − M (2) A)(I − M (1) A) = I − M (2) A, where M (2) = M (1) + M (2) − M (2) AM (1) . We can therefore think of, and implement, two relaxation sweeps as just the one sweep y ← y − M (2) (Ay − r), which means that we can analyze a V(2, 0)-cycle as just a V(1, 0)-cycle with this M (2) .…”

Section: Supplementary Materials: Algebraic Error Analysismentioning

confidence: 99%

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Algebraic error analysis for mixed-precision multigrid solvers

McCormick¹,

Benzaken²,

Tamstorf³

2020

Preprint

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This paper establishes the first theoretical framework for analyzing the rounding-error effects on multigrid methods using mixed-precision iterative-refinement solvers. While motivated by the sparse symmetric positive definite (SPD) matrix equations that arise from discretizing linear elliptic PDEs, the framework is purely algebraic such that it applies to matrices that do not necessarily come from the continuum. Based on the so-called energy or A norm, which is the natural norm for many problems involving SPD matrices, we provide a normwise forward error analysis, and introduce the notion of progressive precision for multigrid solvers. Each level of the multigrid hierarchy uses three different precisions that each increase with the fineness of the level, but at different rates, thereby ensuring that the bulk of the computation uses the lowest possible precision. The theoretical results developed here in the energy norm differ notably from previous theory based on the Euclidean norm in important ways. In particular, we show that simply rounding an exact result to finite precision causes an error in the energy norm that is proportional to the square root of κ, the associated matrix condition number. (By contrast, this error is of order 1 when measured in the Euclidean norm.) Given this observation, we show that the limiting accuracy for both V-cycles and full multigrid is optimal in the sense that it is also proportional to κ 1/2 in energy. Additionally, we show that the loss of convergence rate due to rounding grows in proportion to κ 1/2 , but argue that this loss is insignificant in practice. The theory presented here is the first forward error analysis in the energy norm of iterative refinement and the first rounding error analysis of multigrid in general.

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“…Several schemes such as the Rayleigh᎐ w x w x Schrodinger scheme 8, 9, 14, 15 , the defect correction scheme 1,8,16 , w x w x the fixed slope Newton scheme 4,8,12,15 , and some of its variants 8,4 , to name only a few, have been developed for this purpose. These schemes successively refine a crude approximation obtained from a coarse model through direct methods to achieve a desired accuracy.…”

Section: Introductionmentioning

confidence: 99%