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We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of…
Particle diffusion in a two dimensional curved surface embedded in $R_3$ is considered. In addition to the usual diffusion flow, we find a new flow with an explicit curvature dependence. New diffusion equation is obtained in $\epsilon$…
The problem of recovering a diffusion coefficient $a$ in a second-order elliptic partial differential equation from a corresponding solution $u$ for a given right-hand side $f$ is considered, with particular focus on the case where $f$ is…
We propose a double obstacle phase field approach to the recovery of piece-wise constant diffusion coefficients for elliptic partial differential equations. The approach to this inverse problem is that of optimal control in which we have a…
In this paper, we consider inverse shape problems coming from diffuse optical tomography and inverse scattering. In both problems, our goal is to reconstruct small volume interior regions from measured data on the exterior surface of an…
This work concerns the direct and inverse potential problems for the stochastic diffusion equation driven by a multiplicative time-dependent white noise. The direct problem is to examine the well-posedness of the stochastic diffusion…
We study diffusive mixing in the presence of thermal fluctuations under the assumption of large Schmidt number. In this regime we obtain a limiting equation that contains a diffusive thermal drift term with diffusion coefficient obeying a…
This paper is concerned with weighted energy estimates and diffusion phenomena for the initial-boundary problem of the wave equation with space-dependent damping term in an exterior domain. In this analysis, an elliptic problem was…
We propose a method to reconstruct the electrical current density inside a conducting medium from acoustically-modulated boundary measurements of the electric potential. We show that the current can be uniquely reconstructed with Lipschitz…
This letter announces and summarizes results obtained in arXiv:1111.5051 and considers several natural extensions. The aforementioned paper proposes a procedure to reconstruct coefficients in a second-order, scalar, elliptic equation from…
As a simplified model for subsurface flows elliptic equations may be utilized. Insufficient measurements or uncertainty in those are commonly modeled by a random coefficient, which then accounts for the uncertain permeability of a given…
We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The…
Metasurfaces are ultra-thin optical elements composed of engineered sub-wavelength structures that enable precise control of light. Their inverse design - determining a geometry that yields a desired optical response - is challenging due to…
Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these…
The inverse problem of backward diffusion is known to be ill-posed and highly unstable. Backward diffusion processes appear naturally in image enhancement and deblurring applications. It is therefore greatly desirable to establish a…
We address the inverse problem of recovering a degeneracy point within the diffusion coefficient of a one-dimensional complex parabolic equation by observing the normal derivative at one point of the boundary. The strongly degenerate case…
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…
The reconstruction of unsteady flow fields from limited measurements is a challenging and crucial task for many engineering applications. Machine learning models are gaining popularity for solving this problem due to their ability to learn…
This paper explores the reconstruction of a space-dependent parameter in inverse diffusion problems, proposing a shape-optimization-based approach. We consider a Robin boundary condition, physically motivated in diffuse optical tomography…
In this work, we investigate the recovery of a parameter in a diffusion process given by the order of derivation in time for a class of diffusion type equations, including both classical and time-fractional diffusion equations, from the…