Spectral methods approximate the solutions of variational problems, boundary value problems and partial differential equations with high degree polynomials. Such methods are especially well-suited to problems where the solution is holomorphic. We focus on convex optimization problems such as the p-Laplacian, which has long been considered hard to solve. We solve these problems by the barrier method. Theoretically, the barrier method requires the use of “short t-steps.” Our new spectral barrier (SPB) method uses “long t-steps.” By computing these long steps on progressively higher degree polynomial spaces, we ensure that the overall method converges to a tolerance <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathop {\textrm{tol}}\limits }>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>tol</mtext> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\hat{O}(\log {\mathop {\textrm{tol}}\limits }^{-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>O</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <msup> <mrow> <mtext>tol</mtext> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> Newton iterations, where the hat indicates that we neglect very slow growing functions like <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\log \log $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mo>log</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\log ^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mo>log</mo> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>, and provided the problem is reverse Hölder regular. We confirm this theoretical performance estimate with numerical experiments.

The spectral barrier method to solve analytic convex optimization problems in function spaces

期刊论文
出版模式：

开放获取
发表日期：

2025年11月20日

查看全文

Sébastien Loisel

摘要

Spectral methods approximate the solutions of variational problems, boundary value problems and partial differential equations with high degree polynomials. Such methods are especially well-suited to problems where the solution is holomorphic. We focus on convex optimization problems such as the p-Laplacian, which has long been considered hard to solve. We solve these problems by the barrier method. Theoretically, the barrier method requires the use of “short t-steps.” Our new spectral barrier (SPB) method uses “long t-steps.” By computing these long steps on progressively higher degree polynomial spaces, we ensure that the overall method converges to a tolerance ${\mathop {\textrm{tol}}\limits }>0$ $tol > 0$ in $\hat{O}(\log {\mathop {\textrm{tol}}\limits }^{-1})$ $\hat{O} (log {tol}^{- 1})$ Newton iterations, where the hat indicates that we neglect very slow growing functions like $\log \log $ $log log$ and $\log ^*$ ${log}^{*}$ , and provided the problem is reverse Hölder regular. We confirm this theoretical performance estimate with numerical experiments.