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Reconstructing a space-dependent source term via the quasi-reversibility method

This paper proposes a quasi-reversibility method that truncates the Fourier series of the solution to a governing equation, transforming the inverse source problem into a system of linear elliptic equations to directly reconstruct the space-dependent source function.

Original authors: Loc H. Nguyen, Huong T. Vu

Published 2026-09-16

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CC0 1.0

Original authors: Loc H. Nguyen, Huong T. Vu

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to see inside a sealed box without opening it, using only the ripples that bounce off its surface. This is the fundamental challenge of a field known as inverse scattering, a branch of physics that seeks to reveal the hidden interior of an object by analyzing how waves, such as light or sound, scatter when they hit it. This technique is vital for many modern technologies, from medical imaging that looks for tumors inside the body to security scanners that detect buried explosives or nanostructures too small to see with the naked eye. The difficulty lies in the fact that the relationship between the scattered waves and the hidden object is incredibly complex and unstable; tiny errors in measurement can lead to wildly incorrect pictures of what is inside. To make this problem manageable, scientists often simplify the situation by assuming the hidden object is a small disturbance in a uniform background, turning a tangled, non-linear puzzle into a more straightforward, linear one. This simplification allows researchers to focus on finding the shape and strength of that specific disturbance, known as a source term, which represents the unknown object.

In a recent study, researchers Loc H. Nguyen and Huong T.T. Vu tackled this simplified version of the problem with a new approach designed to be robust against the noise that inevitably plagues real-world measurements. Their goal was to reconstruct a space-dependent source term, which essentially means figuring out exactly where an unknown object is located within a region and how strong it is, based solely on wave data collected at the boundary of that region. The authors propose a method that breaks the complex wave function down into a series of simpler components, much like separating a chord into individual musical notes, but using a specific set of mathematical building blocks that are particularly well-suited for this task. By truncating this series to include only the most significant components, they transform the difficult inverse problem into a system of linear equations. This system describes how these components interact within the region, allowing the researchers to solve for the hidden source without needing to process high-frequency data that is notoriously sensitive to noise.

The core of their strategy involves a technique called the quasi-reversibility method, which acts as a stabilizer for the calculation. Because the data collected from the boundary is often incomplete or slightly corrupted by errors, trying to solve the equations directly can lead to chaotic results. The quasi-reversibility method introduces a small amount of mathematical smoothing that forces the solution to remain stable and physically plausible, even when the input data is imperfect. The researchers tested this algorithm on a series of computer simulations where they knew the true shape of the hidden source beforehand. They created scenarios where the hidden object was a simple rectangle, a rotated square, a ring with a hollow center, and even the complex shape of the letter Y. In each case, they introduced varying levels of random noise into the data, simulating the imperfections found in real experiments, with noise levels ranging from five percent up to a very challenging fifty percent.

The results of these simulations were striking. The algorithm successfully reconstructed the shapes of all the hidden objects, including the intricate letter Y, with high fidelity. Even when the data was corrupted by fifty percent noise, the method managed to recover the general form of the objects and accurately estimate their maximum intensity. For instance, in the test involving the letter Y, the error in the maximum intensity value was less than two percent even with such heavy noise. The researchers found that the method remained stable and effective across all tested scenarios, suggesting that it does not require the high-frequency data that often makes other methods fail. This stability is crucial because it means the technique could potentially work with data that is noisier or less precise than what is typically required by other reconstruction methods.

The paper does not claim to have solved the full, original inverse scattering problem, which remains a highly complex and non-linear challenge. Instead, it demonstrates a successful and robust way to solve the linearized version, which serves as a critical first step in understanding more complicated scenarios. The authors explicitly note that while their mathematical approximation relies on a specific type of series expansion that is difficult to prove rigorously for all cases, the numerical evidence strongly supports its validity. They emphasize that their approach is particularly effective because it avoids the need for high-frequency data, which is often the source of instability in other techniques. By combining a clever truncation of the wave function with a stabilizing mathematical method, the researchers have provided a tool that can reliably uncover hidden structures from noisy boundary measurements, offering a promising path forward for applications in imaging and non-destructive testing where clarity is often obscured by noise.

Technical Summary: Reconstructing a Space-Dependent Source Term via the Quasi-Reversibility Method

Problem Statement
The paper addresses an inverse source problem (Problem 1.1) arising from the linearization of the nonlinear inverse scattering problem. The objective is to determine a space-dependent source function $p(x)$ , which is compactly supported within a bounded domain $\Omega \subset \mathbb{R}^d$ , using boundary measurements. The governing physics is described by the Helmholtz equation with a background dielectric constant $c_0(x)$ and an incident plane wave $u_0(x, \theta)$ at various angles $\theta \in [\theta, \theta]$ . The available data consists of the wave function $u(x, \theta)$ and its normal derivative $\partial_\nu u(x, \theta)$ measured on the boundary $\partial\Omega$ for all incident angles.

The authors note that while inverse source problems for homogeneous media ( $c_0 = \text{constant}$ ) have been studied extensively using least-squares optimization, those methods typically require high wavenumbers ( $k$ ) to achieve quality reconstructions. However, high-frequency data is highly sensitive to noise due to rapid oscillations. This paper proposes a method that does not rely on high-frequency data, aiming to reconstruct the source with reasonable values of $k$ while maintaining stability against noise.

Methodology
The proposed numerical solver operates in three distinct steps:

Derivation of a Linear Elliptic System:
The authors introduce a transformation variable $w(x, \theta) = -u(x, \theta) / (k^2 u_0(x, \theta))$ . By substituting this into the governing Helmholtz equation, they derive a third-order partial differential equation (PDE) for $w$ . To eliminate the unknown source $p(x)$ , they differentiate this equation with respect to the angle $\theta$ . Solving this third-order equation directly is deemed analytically and numerically intractable.

Consequently, the authors employ a "Galerkin" approximation approach. They truncate the Fourier series expansion of $w(x, \theta)$ with respect to a special orthonormal basis $\{\Psi_n\}_{n=1}^\infty$ of $L^2([\theta, \theta])$ . This basis, introduced in prior work [35], is constructed via Gram-Schmidt orthonormalization of polynomials multiplied by an exponential function. By truncating the series at a cut-off number $N$ , the problem reduces to finding the first $N$ Fourier coefficients, $w_n(x)$ . This process yields a system of $N$ coupled linear elliptic equations for the vector $W = (w_1, \dots, w_N)^T$ .
Solution via Quasi-Reversibility Method (QRM):
The resulting system of PDEs is over-determined, as it is subject to both Dirichlet ( $W = F$ ) and Neumann ( $\partial_\nu W = G$ ) boundary conditions derived from the measured data. To solve this ill-posed over-determined system, the authors apply the Quasi-Reversibility Method (QRM), as developed in [49].

The QRM formulates the solution as the minimizer of a functional $J(W)$ , which combines a least-squares term (measuring the residual of the PDE system) and a Tikhonov regularization term ( $\epsilon \|W\|^2_{H^2}$ ). The inclusion of the Tikhonov term ensures the coercivity of the functional and the existence of a unique minimizer. The authors note that while the theoretical convergence relies on specific noise models, the numerical implementation uses standard noisy data generation without explicit smoothing.
Source Reconstruction:
Once the vector $W$ is computed, the function $w(x, \theta)$ is reconstructed via the truncated series. The source function $p(x)$ is then directly computed by substituting the reconstructed $w$ back into the original governing equation (specifically, equation 3.3), effectively integrating out the angular dependence.

Key Contributions and Results

Algorithmic Framework: The paper presents a complete algorithm (Algorithm 1) that couples a specific basis truncation technique with the Quasi-Reversibility Method to solve the inverse source problem without requiring high-frequency data.
Numerical Stability: The numerical study, conducted in 2D with a square domain, demonstrates the method's robustness against noise. The authors tested four distinct source geometries: a rectangle, a rotated square, a ring (annulus), and a complex shape (the letter "Y").
Performance Metrics: The method successfully reconstructed the support and magnitude of the source functions even with high noise levels.
- For a rectangular source, the error in the maximum value was approximately 6.7% with 5% noise and 6.9% with 10% noise.
- For a ring-shaped source, the method handled the "void" in the center, with errors of 6.35% at 5% noise and 8.13% at 50% noise.
- For the complex "Y" shape, the reconstruction was highly accurate, with a maximum value error of only 0.83% at 10% noise and 1.92% at 50% noise.
Cut-off Selection: The authors provide a numerical criterion for selecting the truncation number $N$ (set to 35 in their tests) by monitoring the $L^\infty$ error of the truncated series on the boundary.

Significance and Claims
The paper positions this work as a contribution to the field of inverse problems by solving the linearization of the inverse scattering problem. The authors emphasize that their approach is distinct because it avoids the high-frequency data requirement typical of least-squares optimization methods, thereby mitigating sensitivity to noise.

The authors are modest regarding the theoretical justification of their approximation. They explicitly state that rigorously verifying the convergence of the truncated Fourier series approximation as $N \to \infty$ is "extremely challenging" and is outside the scope of the paper. Instead, they rely on the empirical success of this "Galerkin" approximation in various inverse problems (citing previous works [29, 31, 36, 39, 40, 41, 47, 48, 49, 50, 51]) and the proven convergence of the Quasi-Reversibility Method as noise tends to zero. The primary claim is that the proposed method is effective and stable for reconstructing space-dependent sources from boundary measurements, even in the presence of significant noise (up to 50%).

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