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We propose a direct imaging method based on the reverse time migration to reconstruct extended obstacles in the half space with finite aperture elastic scattering data at a fixed frequency. We prove the resolution of the reconstruction…
We present a new integral equation method for the calculation of two-dimensional scattering from periodic structures involving triple-points (multiple materials meeting at a single point). The combination of a robust and high-order accurate…
An inverse problem of finding an unknown heat source for a class of linear parabolic equations is considered. Such problems can typically be converted to a direct problem with non-local conditions in time instead of an initial value…
To close the moment model deduced from kinetic equations, the canonical approach is to provide an approximation to the flux function not able to be depicted by the moments in the reduced model. In this paper, we propose a brand new closure…
Minimization of the $L_\infty$ norm, which can be viewed as approximately solving the non-convex least median estimation problem, is a powerful method for outlier removal and hence robust regression. However, current techniques for solving…
Partial differential equations are often used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and image processing, and many analytic approaches or traditional…
Models with dominant advection always posed a difficult challenge for projection-based reduced order modelling. Many methodologies that have recently been proposed are based on the pre-processing of the full-order solutions to accelerate…
We solve the anisotropic diffusion equation in 2D, where the dominant direction of diffusion is defined by a vector field which does not conform to a Cartesian grid. Our method uses operator splitting to separate the diffusion perpendicular…
In this article, we derive a new, fast, and robust preconditioned iterative solution strategy for the all-at-once solution of optimal control problems with time-dependent PDEs as constraints, including the heat equation and the non-steady…
Blind super-resolution methods based on stable diffusion showcase formidable generative capabilities in reconstructing clear high-resolution images with intricate details from low-resolution inputs. However, their practical applicability is…
A new method is formulated and analyzed for the approximate solution of a two-dimensional time-fractional diffusion-wave equation. In this method, orthogonal spline collocation is used for the spatial discretization and, for the…
In this paper, we present a numerical algorithm for the accurate and efficient computation of the convolution of the frequency domain layered media Green's function with a given density function. Instead of compressing the convolution…
In this paper we study the application of a simplified method to solve the dynamic radiative transfer problem in expanding envelopes. The method, which requires a computational effort similar to that of the diffusion approximation, is based…
In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schr\"odinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate…
We propose an alternative method for one-dimensional continuum diffusion models with spatially variable (heterogeneous) diffusivity. Our method, which extends recent work on stochastic diffusion, assumes the constant-coefficient homogenized…
We present a comprehensive study for common second order PDE's in two dimensional disk-like systems and show how their solution can be approximated by finding the Green function of an effective one dimensional system. After elaborating on…
A jump-diffusion process along with a particle scheme is devised as an accurate and efficient particle solution to the Boltzmann equation. The proposed process (hereafter Gamma-Boltzmann model) is devised to match the evolution of all…
Learning the Green's function using deep learning models enables to solve different classes of partial differential equations. A practical limitation of using deep learning for the Green's function is the repeated computationally expensive…
Linear singularly perturbed convection-diffusion problems with characteristic layers are considered in three dimensions. We demonstrate the sharpness of our recently obtained upper bounds for the associated Green's function and its…
Models of diffusive processes that occur on evolving domains are frequently employed to describe biological and physical phenomena, such as diffusion within expanding tissues or substrates. Previous investigations into these models either…