Continuous Adaptive Cross Approximation for Ill-posed Problems with Chebfun

TL;DR

This paper investigates using Chebfun to solve ill-posed problems directly in function space, offering a more analytical approach compared to traditional matrix-based methods.

Contribution

It introduces a novel application of Chebfun for ill-posed problems, bridging functional analysis and numerical linear algebra.

Findings

Chebfun enables working with functions instead of matrices

The approach aligns numerical solutions more closely with theoretical analysis

Potential for improved handling of ill-posed problems

Abstract

The analysis of linear ill-posed problems often is carried out in function spaces using tools from functional analysis. However, the numerical solution of these problems typically is computed by first discretizing the problem and then applying tools from (finite-dimensional) linear algebra. The present paper explores the feasibility of applying the Chebfun package to solve ill-posed problems. This approach allows a user to work with functions instead of matrices. The solution process therefore is much closer to the analysis of ill-posed problems than standard linear algebra-based solution methods.