Date of Award

2025

Degree Name

Mathematics

College

College of Science

Type of Degree

M.A.

Document Type

Thesis

First Advisor

Dr. Trung Truong

Second Advisor

Dr. Tom Cuchta

Third Advisor

Dr. Scott A. Sarra

Abstract

This thesis investigates a numerical method for solving the periodic inverse source problem governed by the Helmholtz equation. The problem involves reconstructing an unknown periodic source term from boundary measurements, which is inherently ill-posed. To address this challenge, we employ a quasi-reversibility method (QRM) combined with a basis function expansion to stabilize the inverse reconstruction. The forward problem is solved using the Lippmann-Schwinger equation, discretized via the trapezoidal rule, and the inverse problem is formulated as a constrained least-squares minimization. The discretized system is efficiently solved using sparse matrix techniques and regularization strategies. Numerical experiments demonstrate the robustness of the proposed method under various noise levels and different source configurations, including elliptical, sinusoidal, and bar-shaped sources. The results highlight the effectiveness of the approach in reconstructing periodic sources while mitigating the effects of measurement noise and numerical instability. Future work will focus on refining regularization techniques and extending the method to more complex periodic structures.

Subject(s)

Mathematics.

Discrete mathematics.

Helmholtz equation.

Mathematical models.

Differential equations, Partial.

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