# Norm bound computation for inverses of linear operators in Hilbert spaces

@article{Watanabe2016NormBC, title={Norm bound computation for inverses of linear operators in Hilbert spaces}, author={Yoshitaka Watanabe and Kaori Nagatou and Michael Plum and Mitsuhiro T. Nakao}, journal={Journal of Differential Equations}, year={2016}, volume={260}, pages={6363-6374} }

## 10 Citations

### An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces

- MathematicsJournal of Differential Equations
- 2019

### An improved method for verifying the existence and bounds of the inverse of second-order linear elliptic operators mapping to dual space

- MathematicsJapan Journal of Industrial and Applied Mathematics
- 2019

This paper presents an improved method for determining the invertibility of second-order linear elliptic operators with a bound on the norm of their inverses by computers in a mathematically rigorous…

### Some improvements of invertibility verifications for second-order linear elliptic operators

- Mathematics
- 2020

### Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems

- MathematicsComput. Math. Appl.
- 2022

### Infinite-Dimensional Newton-Type Method

- MathematicsSpringer Series in Computational Mathematics
- 2019

This chapter presents two numerical verification methods which are based on some infinite-dimensional fixed-point theorems. The first approach is a technique using sequential iteration. Although this…

### Numerical verification methods for a system of elliptic PDEs, and their software library

- Computer Science, MathematicsNonlinear Theory and Its Applications, IEICE
- 2021

Existing verification methods are reformulated using a convergence theorem for simplified Newton-like methods in the direct product space of a computable finite-dimensional space Vh and its orthogonal complement space V⊥ to provide verification methods of solutions to PDEs.

### A numerical verification method for nonlinear functional equations based on infinite-dimensional Newton-like iteration

- MathematicsAppl. Math. Comput.
- 2016

### A new formulation for the numerical proof of the existence of solutions to elliptic problems

- MathematicsArXiv
- 2019

This paper represents the inverse operator ${\mathcal L}^{-1} as an infinite-dimensional operator matrix that can be decomposed into two parts, one finite dimensional and one infinite dimensional, enabling a more efficient verification procedure compared with existing methods for the solution of elliptic PDEs.

### A new formulation using the Schur complement for the numerical existence proof of solutions to elliptic problems: without direct estimation for an inverse of the linearized operator

- Computer Science, MathematicsNumerische Mathematik
- 2020

<jats:p>Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of…

### Equilibrium validation in models for pattern formation based on Sobolev embeddings

- MathematicsDiscrete & Continuous Dynamical Systems - B
- 2021

In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual…

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