[Paper Review] A Mathematica Program for heat source function of 1D heat equation reconstruction by three types of data

Tomasz M. Łapiński, Sergey Leble|arXiv (Cornell University)|Oct 26, 2014

Numerical methods in inverse problems5 references3 citations

TL;DR

This paper presents a Mathematica-based computational framework for reconstructing the heat source function in a 1D heat equation using three data types: single-point time-series, uniform spatial points at fixed time, and random spatial-temporal data. It employs Tikhonov regularization with two parameter selection strategies—minimizing error to the true source and the discrepancy principle—demonstrating stable and accurate reconstructions despite the ill-posed nature of the inverse problem.

ABSTRACT

We solve an inverse problem for the one-dimensional heat diffusion equation. We reconstruct the heat source function for the three types of data: 1) single position point and different times, 2) constant time and uniformly distributed positions, 3) random position points and different times. First we demonstrate reconstruction using simple inversion of discretized Kernel matrix. Then we apply Tikhonov regularization for two types of the parameter of regularization estimation. The first one, which is in fact exemplary simulation, is based on minimization of the distance in C space of reconstructed function to the initial source function. Second rule is known as Discrepancy principle. We generate the data from the chosen source function. In order to get some measure of accuracy of reconstruction we compare the result with the function from which data was generated. We also deliver corresponding application in symbolic computation environment of Mathematica. The program has a lot of flexibility, it can perform reconstruction for much more general input then one considered in the paper.

Motivation & Objective

• Address the ill-posed inverse problem of reconstructing a heat source function from limited temperature measurements in a 1D heat equation.
• Develop a flexible symbolic computation tool in Mathematica to solve the inverse problem for various data configurations.
• Evaluate the effectiveness of Tikhonov regularization with two parameter selection rules: error minimization and discrepancy principle.
• Demonstrate that regularization stabilizes the solution and enables accurate source reconstruction even with noisy or sparse data.
• Provide a reusable, extensible program capable of handling diverse input parameters and data types beyond the presented cases.

Proposed method

• Formulate the inverse problem as a Fredholm integral equation of the first kind using Fourier series expansion.
• Discretize the kernel matrix to transform the continuous problem into a linear system with high condition number.
• Apply Tikhonov regularization via the functional $ M^{eta}(f) = \|Kf - u\|^2 + \alpha \|f\|^2 $, stabilizing the ill-conditioned system.
• Implement two regularization parameter selection methods: (1) minimizing the L2 error between reconstructed and true source, (2) the discrepancy principle using measurement error $ \delta $.
• Use symbolic computation in Mathematica to numerically solve the regularized system and generate reconstructions for all three data types.
• Apply discretization weights $ h_i $ and $ h_s $ to account for non-uniform sampling in time or space during matrix formulation.

Experimental results

Research questions

• RQ1 Can Tikhonov regularization effectively stabilize the inverse source problem for the 1D heat equation under different data configurations?
• RQ2 How does the choice of regularization parameter—via error minimization or the discrepancy principle—affect reconstruction accuracy?
• RQ3 To what extent does the reconstruction quality depend on the data type: time-series at one point, spatial data at fixed time, or random spatio-temporal samples?
• RQ4 How does the program’s symbolic implementation in Mathematica support flexibility and extensibility for real-world inverse problem applications?
• RQ5 What is the impact of measurement error and numerical precision on the stability and accuracy of source reconstruction?

Key findings

• Without regularization, the inverse problem yields unstable and inaccurate reconstructions due to the ill-posed nature of the problem.
• Tikhonov regularization successfully stabilizes the solution, producing reconstructions that are visually and quantitatively close to the true source function.
• Using the discrepancy principle for regularization parameter selection yields stable and accurate reconstructions without requiring prior knowledge of the true source.
• The method based on minimizing the error to the true source function produces excellent results but requires knowledge of the original source, limiting its real-world use.
• All three data types—single-point time-series, uniform spatial sampling, and random spatio-temporal data—yield acceptable reconstructions when regularization is applied.
• The Mathematica program successfully implements the entire workflow, including data generation, regularization, and reconstruction, with high flexibility for parameter adjustment.