Speaker
Description
Recovering hidden causes from observable effects is a fundamental challenge in many scientific and engineering applications. Examples include inferring subsurface properties from magnetic field measurements in geophysics and reconstructing sharper images from blurred ones in medical imaging. These tasks are commonly formulated as inverse problems. Such problems are often ill-posed and lack closed-form solutions. As a result, reliable numerical methods are essential.
In this work, motivated by my master’s thesis, we study an inverse problem for identifying a space-dependent potential in a linear reaction–diffusion equation. We present a pseudo-spectral method that expands the solution in a suitable basis, transforming the governing partial differential equation into an infinite system of ordinary differential equations. Following Galerkin’s approach, this framework leads to a finite-dimensional inverse problem for recovering the potential coefficients from measurement data.
While recent studies using pseudo-spectral methods have focused on the one-dimensional noise-free case and did not include numerical comparisons with other recovery techniques (Audu et al., 2022), we present an extension of the investigation to noisy observations. A finite difference method is presented for benchmarking. Our results indicate that the proposed pseudo-spectral approach remains stable and robust in the presence of noise, yielding accurate reconstructions of the unknown potential.
These findings suggest that pseudo-spectral methods provide a promising computational framework for inverse problems arising in diffusion-driven models.
| References | Audu, J. D., Boumenir, A., Furati, K. M., and Sarumi, I. O., Identifying the heat sink, Discrete and Continuous Dynamical Systems–S, 15(5), 1045–1059 (2022). |
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| Country | Saudi Arabia |
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