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This paper is concerned with the inverse time-harmonic elastic scattering problem of recovering unbounded rough surfaces in two dimensions. We assume that elastic plane waves with different directions are incident onto a rigid rough surface…
Based on the matrix expression of general nonlinear numerical analogues presented by the present author, this paper proposes a novel philosophy of nonlinear computation and analysis. The nonlinear problems are considered an ill-posed linear…
We give an alternative method to obtain normal forms of reversible equivariant vector fields. We adapt the classical method using tools from invariant theory to establish formulae that take symmetries into account as a starting point.…
Waveform inversion is theoretically a powerful tool to reconstruct subsurface structures, but a usually encountered problem is that accurate sources are very rare, causing the computation unstable and divergent. This challenging problem,…
The inverse scattering problem, whose goal is to reconstruct an unknown scattering object from its scattered wave, is essential in fundamental wave physics and its wide applications in imaging sciences. However, it remains challenging to…
Wavefront-free behavior and the resulting quantification difficulties are intrinsic limitations of vibrothermography due to the diffusive nature of thermal fields. This work proposes a generalized virtual wave reconstruction framework to…
Inverse Problems in medical imaging and computer vision are traditionally solved using purely model-based methods. Among those variational regularization models are one of the most popular approaches. We propose a new framework for applying…
Deep neural networks have shown great potential in image reconstruction problems in Euclidean space. However, many reconstruction problems involve imaging physics that are dependent on the underlying non-Euclidean geometry. In this paper,…
The problem of inverting a system in presence of a series-defined output is analyzed. Inverse models are derived that consist of a set of algebraic equations. The inversion is performed explicitly for an output trajectory functional, which…
The article suggests a new approach what is called a consistency method for the inversion of the spherical Radon transform in 2D with detectors on a line. It is known that there is not an exact inversion formula in 2D. By means of the…
We consider the behavior of generalized Laplacian vector fields on a Jordan domain of $\mathbb{R}^{3}$ with fractal boundary. Our approach is based on properties of the Teodorescu transform and suitable extension of the vector fields.…
In this article we study the problem of recovering the initial data of the two-dimensional wave equation from Neumann measurements on a convex domain with smooth boundary in the plane. We derive an explicit inversion formula of a so-called…
We reduce the broken ray transform on some Riemannian manifolds (with corners) to the geodesic ray transform on another manifold, which is obtained from the original one by reflection. We give examples of this idea and present injectivity…
For the reconstruction problem, the universal representation of inverse Radon transforms implies the needed complexity of the direct Radon transforms which leads to the additional contributions. In the standard theory of generalized…
We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three…
A general method for analytic inversion of geometric integral transforms is proposed
In this paper, we consider the conical Radon transform on all cones with horizontal central axis whose vertices are on a straight line. We derive an explicit inversion formula for such transform. The inversion makes use of the vertical…
The linear inverse problem is fundamental to the development of various scientific areas. Innumerable attempts have been carried out to solve different variants of the linear inverse problem in different applications. Nowadays, the rapid…
This paper discusses basic results and recent developments on variational regularization methods, as developed for inverse problems. In a typical setup we review basic properties needed to obtain a convergent regularization scheme and…
This paper proposes a neural network approach for solving two classical problems in the two-dimensional inverse wave scattering: far field pattern problem and seismic imaging. The mathematical problem of inverse wave scattering is to…