Related papers: On parameter identification problems for elliptic …
We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator, a concave-convex powers term together with mixed Dirichlet-Neumann boundary conditions.
An inverse boundary value problem for the Helmholtz equation in a bounded domain is considered. The problem is to extract information about an unknown obstacle embedded in the domain with unknown impedance boundary condition (the Robin…
The tangential condition was introduced in [Hanke et al., 95] as a sufficient condition for convergence of the Landweber iteration for solving ill-posed problems. In this paper we present a series of time dependent benchmark inverse…
We analyze a reaction coefficient identification problem for the spectral fractional powers of a symmetric, coercive, linear, elliptic, second-order operator in a bounded domain $\Omega$. We realize fractional diffusion as the…
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…
Initial-boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so-called unified transform techniques. The main obstruction to applying these methods…
In this paper, we use probabilistic approach to prove that there exists a unique weak solution to the Dirichlet boundary value problem for second order elliptic equations whose coefficients are signed measures, and we will give a…
We develop an interpolation-based modeling framework for parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The solution is discretized in the physical domain using…
Inverse problems in partial differential equations (PDEs) involve estimating the physical parameters of a system from observed spatiotemporal solution fields. Neural networks are well-suited for PDE parameter estimation due to their…
We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form $\m{A}x=\lambda\m{B}x$, where the matrices $\m{A}$ and/or $\m{B}$ may depend on a scalar parameter.…
A new method is introduced for studying boundary value problems for a class of linear PDEs with {\it variable} coefficients. This method is based on ideas recently introduced by the author for the study of boundary value problems for PDEs…
This paper presents new results allowing an unknown non-Gaussian positive matrix-valued random field to be identified through a stochastic elliptic boundary value problem, solving a statistical inverse problem. A new general class of…
In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional…
Nonlinear ultrasound imaging leverages harmonic wave generation to enhance contrast and spatial resolution beyond the capabilities of conventional linear techniques. This behavior is commonly modeled by the Westervelt equation, which…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
We consider an overdetermined fourth order boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to the homogeneous Dirichlet boundary condition. It is known that, in the case where…
Inverse problems involving differential equations often require identifying unknown parameters or functions from data. Existing approaches, such as Physics-Informed Neural Networks (PINNs), Universal Differential Equations (UDEs) and…
In statistical inference, a discrepancy between the parameter-to-observable map that generates the data and the parameter-to-observable map that is used for inference can lead to misspecified likelihoods and thus to incorrect estimates. In…
Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with the Dirichlet boundary condition are considered. The principal eigen map of such elliptic operators is introduced…