<p>The analysis of the acceleration behavior of gradient-based eigensolvers with preconditioning presents a substantial theoretical challenge. In this work, we present a novel framework for preconditioning on Riemannian manifolds and introduce a metric, the leading angle, to evaluate preconditioners for symmetric eigenvalue problems. We extend the locally optimal Riemannian accelerated gradient method for Riemannian convex optimization to develop the Riemannian acceleration with preconditioning (RAP) for symmetric eigenvalue problems, thereby providing theoretical evidence to support its acceleration. Our analysis of the Schwarz preconditioner for elliptic eigenvalue problems demonstrates that RAP achieves a convergence rate of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="211_2025_1451_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$1-C\kappa ^{-1/2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>-</mo> <mi>C</mi> <msup> <mi>κ</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, which is an improvement over the preconditioned steepest descent method’s rate of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="211_2025_1451_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$1-C\kappa ^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>-</mo> <mi>C</mi> <msup> <mi>κ</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. The exponent in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="211_2025_1451_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$\kappa ^{-1/2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>κ</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> is sharp, and numerical experiments confirm our theoretical findings.</p>

Riemannian acceleration with preconditioning for symmetric eigenvalue problems

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2025年1月20日

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Nian Shao,
Wenbin Chen

摘要

The analysis of the acceleration behavior of gradient-based eigensolvers with preconditioning presents a substantial theoretical challenge. In this work, we present a novel framework for preconditioning on Riemannian manifolds and introduce a metric, the leading angle, to evaluate preconditioners for symmetric eigenvalue problems. We extend the locally optimal Riemannian accelerated gradient method for Riemannian convex optimization to develop the Riemannian acceleration with preconditioning (RAP) for symmetric eigenvalue problems, thereby providing theoretical evidence to support its acceleration. Our analysis of the Schwarz preconditioner for elliptic eigenvalue problems demonstrates that RAP achieves a convergence rate of $1-C\kappa ^{-1/2}$ $1 - C κ^{- 1 / 2}$ , which is an improvement over the preconditioned steepest descent method’s rate of $1-C\kappa ^{-1}$ $1 - C κ^{- 1}$ . The exponent in $\kappa ^{-1/2}$ $κ^{- 1 / 2}$ is sharp, and numerical experiments confirm our theoretical findings.