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In this paper, we propose a novel adaptive finite element method for an elliptic equation with line Dirac delta functions as a source term. We first study the well-posedness and global regularity of the solution in the whole domain. Instead…
In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at…
Diffusion is a fundamental graph procedure and has been a basic building block in a wide range of theoretical and empirical applications such as graph partitioning and semi-supervised learning on graphs. In this paper, we study…
We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the…
We propose and analyze a numerical method to solve an elliptic transmission problem in full space. The method consists of a variational formulation involving standard boundary integral operators on the coupling interface and an ultra-weak…
Physical systems with complex unsteady dynamics, such as fluid flows, are often poorly represented by a single mean solution. For many practical applications, it is crucial to access the full distribution of possible states, from which…
In computational system biology, the mesoscopic model of reaction-diffusion kinetics is described by a continuous time, discrete space Markov process. To simulate diffusion stochastically, the jump coefficients are obtained by a…
Diffusion models have become the go-to method for many generative tasks, particularly for image-to-image generation tasks such as super-resolution and inpainting. Current diffusion-based methods do not provide statistical guarantees…
An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary…
In this paper we perform the rigorous derivation of the topological derivative for optimization problems constrained by a class of quasi-linear elliptic transmission problems. In the case of quasi-linear constraints, techniques using…
Diffusion models are a class of probabilistic generative models that have been widely used as a prior for image processing tasks like text conditional generation and inpainting. We demonstrate that these models can be adapted to make…
Explicit analytical expressions for the drag and diffusion coefficients of a spherical particle attached to the interface between two immiscible fluids are constructed for the case of a small viscosity ratio between the fluid phases. The…
We consider the inverse obstacle scattering problem of determining both the shape and the "equivalent impedance" from far field measurements at a fixed frequency. In this work, the surface impedance is represented by a second order surface…
A free boundary diffusive logistic model finds application in many different fields from biological invasion to wildfire propagation. However, many of these processes show a random nature and contain uncertainties in the parameters. In this…
We analyze the matter wave transmission above a step potential within the framework of the cubic-nonlinear Schr\"odinger equation. We present a comprehensive analysis of the corresponding stationary problem based on an exact second-order…
The present article proposes a diffuse interface model for compressible multicomponent flows with transport phenomena of mass, momentum and energy (i.e., mass diffusion, viscous dissipation and heat conduction). The model is reduced from…
In this paper we present the complete derivation of the effective contour model for electrical discharges which appears as the asymptotic limit of the minimal streamer model for the propagation of electric discharges, when the electron…
We study computing geometric problems on uncertain points. An uncertain point is a point that does not have a fixed location, but rather is described by a probability distribution. When these probability distributions are restricted to a…
Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups and discontinuities.…
An adaptive finite difference scheme for variable-order fractional-time subdiffusion equations in the Caputo form is studied. The fractional time derivative is discretized by the L1 procedure but using nonhomogeneous timesteps. The size of…