Related papers: Dynamic Inverse Wave Problems - Part II: Operator …
In this paper we consider new regularization methods for linear inverse problems of dynamic type. These methods are based on dynamic programming techniques for linear quadratic optimal control problems. Two different approaches are…
We consider ill-posed inverse problems where the forward operator $T$ is unknown, and instead we have access to training data consisting of functions $f_i$ and their noisy images $Tf_i$. This is a practically relevant and challenging…
Consider an exterior problem of the three-dimensional elastic wave equation, which models the scattering of a time-harmonic plane wave by a rigid obstacle. The scattering problem is reformulated into a boundary value problem by introducing…
We introduce a framework for the reconstruction and representation of functions in a setting where these objects cannot be directly observed, but only indirect and noisy measurements are available, namely an inverse problem setting. The…
We consider the inverse problem for the wave equation which consists of determining an unknown space-dependent force function acting on a vibrating structure from Cauchy boundary data. Since only boundary data are used as measurements, the…
Partial differential equations (PDEs) play a foundational role in modeling physical phenomena. This study addresses the challenging task of determining variable coefficients within PDEs from measurement data. We introduce a novel neural…
Inverse problems arise anywhere we have indirect measurement. As, in general they are ill-posed, to obtain satisfactory solutions for them needs prior knowledge. Classically, different regularization methods and Bayesian inference based…
Several novel imaging and non-destructive testing technologies are based on reconstructing the spatially dependent coefficient in an elliptic partial differential equation from measurements of its solution(s). In practical applications, the…
We consider, in a Hilbert space $H$, the convolution integro-differential equation $u''(t)-h*Au(t)=f(t)$, $0\le t\le T$, $h*v(t)=\int_0^t h(t-s)v(s) ds$, where $A$ is a linear closed densely defined (possibly selfadjoint and/or positive…
Inverse spectral problems are studied for the second order integro-differential operators on a finite interval. Properties of spectral characteristic are established, and the uniqueness theorem is proved for this class of inverse problems.
Learning underlying dynamics from data is important and challenging in many real-world scenarios. Incorporating differential equations (DEs) to design continuous networks has drawn much attention recently, however, most prior works make…
There is a tendency to write the equations of general relativity as a first order symmetric system of time dependent partial differential equations. However, for numerical reasons, it might be advantageous to use a second order formulation…
We provide the first proof of local well-posedness for the two-dimensional gravity water wave equations with spatially quasi-periodic initial conditions. We represent the solution using holomorphic coordinates, which are equivalent to a…
We consider the identification of a nonlinear friction law in a one-dimensional damped wave equation from additional boundary measurements. Well-posedness of the governing semilinear hyperbolic system is established via semigroup theory and…
There are two main approaches to solve inverse coefficient determination problems for wave equations: the Boundary Control method and an approach based on geometric optics. These notes focus on the Boundary Control method, but we will have…
In the theory and practice of inverse problems for partial differential equations (PDEs) much attention is paid to the problem of the identification of coefficients from some additional information. This work deals with the problem of…
To quantify uncertainties in inverse problems of partial differential equations (PDEs), we formulate them into statistical inference problems using Bayes' formula. Recently, well-justified infinite-dimensional Bayesian analysis methods have…
Deterministic neural operators perform well on many PDEs but can struggle with the approximation of high-frequency wave phenomena, where strong input-to-output sensitivity makes operator learning challenging, and spectral bias blurs…
Here, Darboux's classical results about transformations with differential substitutions for hyperbolic equations are extended to the case of parabolic equations of the form $L u = \big(D^2_{x} + a(x,y) D_x + b(x,y) D_y + c(x,y)\big)u=0$. We…
In recent years, deep learning for modeling physical phenomena which can be described by partial differential equations (PDEs) have received significant attention. For example, for learning Hamiltonian mechanics, methods based on deep…