Reconstruction of the Initial Condition for a Non-Homogeneous Heat Equation from Finite Measurements
Abstract
We address the inverse problem of reconstructing the initial temperature distribution in a one-dimensional non-homogeneous heat equation with Dirichlet boundary conditions from a finite number of pointwise-in-time measurements at a single spatial location. We propose an explicit approximation scheme that incorporates the source term and utilizes a refined sequence of measurement times. Unlike previous approaches that often rely on exponentially growing observation times, our chosen time sequence allows for efficient recovery within a fixed time horizon. We establish algebraic convergence rates under suitable regularity assumptions on the initial data and the forcing, and we validate the theoretical sharpness with numerical experiments for varying numbers of measurements.
Related
- An adaptive weighted extrapolation neural network for dispersive shock waves generated by square-root initial condition of the KdV equation
- Mathematical and Numerical Analysis of Tsunami Generation and Propagation: The Initial Tsunami Condition
- The Two-Dimensional Narrow Escape Problem with Deterministic Initial Condition: Mean Escape Time and Escape location probabilities
- Initial condition recovery in nonlinear damped viscous photoacoustic tomography using a convolutional neural network-guided gradient-free optimization framework
- Backward Homogeneous Heat Equation with Ψ-Caputo Time Derivative and Spectral Fractional Laplacian
BibTeX
@misc{dollente2026reconstruction,
title = {{Reconstruction of the Initial Condition for a Non-Homogeneous Heat Equation from Finite Measurements}},
author = {Randy Dollente and Samuel Frerichs and Hee Seok Nam and Jianru Shen and Grace Walters and Young Hwan You},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30181},
eprint = {2609.30181},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30181},
}
RIS
TY - UNPB
TI - Reconstruction of the Initial Condition for a Non-Homogeneous Heat Equation from Finite Measurements
AU - Randy Dollente
AU - Samuel Frerichs
AU - Hee Seok Nam
AU - Jianru Shen
AU - Grace Walters
AU - Young Hwan You
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30181
UR - https://arxiv.org/abs/2609.30181
AB - We address the inverse problem of reconstructing the initial temperature distribution in a one-dimensional non-homogeneous heat equation with Dirichlet boundary conditions from a finite number of pointwise-in-time measurements at a single spatial location. We propose an explicit approximation scheme that incorporates the source term and utilizes a refined sequence of measurement times. Unlike previous approaches that often rely on exponentially growing observation times, our chosen time sequence allows for efficient recovery within a fixed time horizon. We establish algebraic convergence rates under suitable regularity assumptions on the initial data and the forcing, and we validate the theoretical sharpness with numerical experiments for varying numbers of measurements.
ER -