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Implicit Neural Spatial Representation (INSR) has emerged as an effective representation of spatially-dependent vector fields. This work explores solving time-dependent PDEs with INSR. Classical PDE solvers introduce both temporal and…
Immersed boundary methods have attracted substantial interest in the last decades due to their potential for computations involving complex geometries. Often these cannot be efficiently discretized using boundary-fitted finite elements.…
A robust and fast solver for the fractional differential equation (FDEs) involving the Riesz fractional derivative is developed using an adaptive finite element method on non-uniform meshes. It is based on the utilization of hierarchical…
Radiative transfer is essential in astronomy, both for interpreting observations and simulating various astrophysical phenomena. However, self-consistent line radiative transfer is computationally expensive, especially in 3D. To reduce the…
The radiative transfer equation (RTE) has been established as a fundamental tool for the description of energy transport, absorption and scattering in many relevant societal applications, and requires numerical approximations. However,…
This study concerns numerical methods for efficiently solving the Richards equation where different weak formulations and computational techniques are analyzed. The spatial discretizations are based on standard or mixed finite element…
In this work we develop and analyze an adaptive finite element method for efficiently solving electrical impedance tomography -- a severely ill-posed nonlinear inverse problem for recovering the conductivity from boundary voltage…
We present a framework for solving time-dependent partial differential equations (PDEs) in the spirit of the random feature method. The numerical solution is constructed using a space-time partition of unity and random feature functions.…
In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to…
When evolving in time the solution of a hyperbolic partial differential equation, it is often desirable to use high order strong stability preserving (SSP) time discretizations. These time discretizations preserve the monotonicity…
In this article, we propose novel boundary treatment algorithms to avoid order reduction when implicit-explicit Runge-Kutta time discretization is used for solving convection-diffusion-reaction problems with time-dependent Di\-richlet…
We present high-order, finite element-based Second Moment Methods (SMMs) for solving radiation transport problems in two spatial dimensions. We leverage the close connection between the Variable Eddington Factor (VEF) method and SMM to…
Linear kinetic transport equations play a critical role in optical tomography, radiative transfer and neutron transport. The fundamental difficulty hampering their efficient and accurate numerical resolution lies in the high dimensionality…
We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology.…
We present a new method for the numerical solution of the radiative-transfer equation (RTE) in multidimensional scenarios commonly encountered in computational astrophysics. The method is based on the direct solution of the Boltzmann…
Conventional numerical solvers for the radiative transfer equation (RTE) exhibit severe sensitivity to medium parameters. To address this, we propose an operator learning framework that approximates the RTE solution map as a function of…
In this paper the efficiency of multilevel sparse tensor approximation methods for high-dimensional affine parametric diffusion equations is investigated. Methodologically, the recently presented Sparse Alternating Least Squares (SALS)…
Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial…
The time-dependent Schr\"odinger equation (TDSE) in real space is fundamental to understanding the dynamics of many-electron quantum systems, with applications ranging from quantum chemistry to condensed matter physics and materials…
In this paper a class of higher order finite element methods for the discretization of surface Stokes equations is studied. These methods are based on an unfitted finite element approach in which standard Taylor-Hood spaces on an underlying…