相关论文: A low-rank Schwarz method for radiative transport …
We propose a computationally efficient Schwarz method for elliptic equations with rough media. A random sampling strategy is used to find low-rank approximations of all local solution maps in the offline stage; these maps are used to…
We derive radiative transport equations for solutions of a Schr\"odinger equation in a periodic structure with small random inhomogeneities. We use systematically the Wigner transform and the Bloch wave expansion. The streaming part of the…
Low-rank approximation is a technique to approximate a tensor or a matrix with a reduced rank to reduce the memory required and computational cost for simulation. Its broad applications include dimension reduction, signal processing,…
The low-rank approximation is a complexity reduction technique to approximate a tensor or a matrix with a reduced rank, which has been applied to the simulation of high dimensional problems to reduce the memory required and computational…
This paper presents a novel backtracking strategy for additive Schwarz methods for general convex optimization problems as an acceleration scheme. The proposed backtracking strategy is independent of local solvers, so that it can be applied…
The radiative transfer equation models various physical processes ranging from plasma simulations to radiation therapy. In practice, these phenomena are often subject to uncertainties. Modeling and propagating these uncertainties requires…
In this paper, we propose a domain decomposition dynamical low-rank method to solve high-dimensional radiative transfer problems and similar kinetic equations. The algorithm uses a separate low-rank approximation on each spatial subdomain,…
We introduce a dynamical low-rank method to reduce the computational complexity for solving the multi-scale multi-dimensional linear transport equation. The method is based on a macro-micro decomposition of the equation. The proposed…
In the fields of control theory and machine learning, the dynamic low-rank approximation for large-scale matrices has received substantial attention. Considering large-scale semilinear stiff matrix differential equations, we propose…
We present here nonoverlapping optimized Schwarz methods applied to heat transfer problems with heterogeneous diffusion coefficients. After a Laplace transform in time, we derive the error equation and obtain the convergence factor. The…
Historically, analysis for multiscale PDEs is largely unified while numerical schemes tend to be equation-specific. In this paper, we propose a unified framework for computing multiscale problems through random sampling. This is achieved by…
The dynamical low-rank (DLR) approximation is an efficient technique to approximate the solution to matrix differential equations. Recently, the DLR method was applied to radiation transport calculations to reduce memory requirements and…
We describe a numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition. In contrast to existing techniques, our method does not rely on…
We use the Wigner transformation and asymptotic analysis to systematically derive the semi-classical model for the Schr\"{o}dinger equation in arbitrary spatial dimensions, with any periodic structure. Our particular emphasis lies in…
The dynamic iteration method with a restricted additive Schwarz splitting is investigated to co-simulate linear differential algebraic equations system coming from RLC electrical circuit with linear components. We show the pure linear…
Deterministically solving charged particle transport problems at a sufficient spatial and angular resolution is often prohibitively expensive, especially due to their highly forward peaked scattering. We propose a model order reduction…
Atomic and molecular breakup reactions, such as multiple-ionisation, are described by a driven Schr\"odinger equation. This equation is equivalent to a high-dimensional Helmholtz equation and it has solutions that are outgoing waves,…
Optimized Schwarz Methods (OSMs) are based on optimized transmission conditions along the interfaces between the subdomains. Optimized transmission conditions are derived at the theoretical level, using techniques developed in the last…
Reduced basis methods build low-rank approximation spaces for the solution sets of parameterized PDEs by computing solutions of the given PDE for appropriately selected snapshot parameters. Localized reduced basis methods reduce the offline…
In this paper, we develop an efficient numerical solver for unsteady diffusion-type partial differential equations with random coefficients. A major computational challenge in such problems lies in repeatedly handling large-scale linear…