Numerical Reconstruction of Inverse Coefficient Problems for Parabolic Equations with Rough Potentials

Professor Introduction

J. F | Ph.D. in Applied Mathematics

Home Institute：Chongqing University of Technology

[ Research Interests ] Observability and controllability of partial differential equations (PDEs), inverse problems and control of PDEs, harmonic analysis
[ Teaching Experience ] Lecturer in the Department of Information and Computational Science at Chongqing University of Technology
[ Additional Experience ] Member of the Chinese Society of Industrial and Applied Mathematics, member of the Western Society of Partial Differential Equations, participated in many research projects at home and abroad

Project Description

This project focuses on the numerical reconstruction of inverse coefficient problems for parabolic equations with rough, spatiotemporally bounded potentials, specifically addressing the Calderón problem. The primary objective is to develop and analyze numerical methods for solving these problems in high-dimensional spaces. By employing various finite difference techniques on Dirichlet boundary conditions, the project aims to prove the existence and stability of numerical solutions.

Project Keywords

Project Outline

Part 1 :  Introduction to Parabolic Equations and Inverse Problems
• Fundamentals of parabolic equations
• Overview of inverse coefficient problems
• Introduction to the Calderón problem

Part 2 : Rough Potentials in Parabolic Equations
• Definition and properties of rough, spatiotemporally bounded potentials
• Challenges in dealing with rough potentials
• Examples and case studies

Part 3 : Numerical Methods for High-Dimensional Spaces
• Finite difference techniques for Dirichlet boundary conditions
• Numerical methods for high-dimensional PDEs
• Stability and convergence analysis of numerical solutions

Part 4 : Existence and Stability of Numerical Solutions
• Proof of existence of numerical solutions
• Stability analysis of numerical methods
• Error estimation and control

Part 5 : Practical Applications and Advanced Topics
• Application of numerical reconstruction methods to real-world problems
• Advanced topics in numerical analysis of PDEs
• Future research directions and open problems

Suitable for

• High School Students interested in applied mathematics and computational science