Existence of dynamical low-rank approximations to parabolic problems

The existence and uniqueness of weak solutions to dynamical low-rank evolution problems for parabolic partial differential equations in two spatial dimensions is shown, covering also non-diagonal diffusion in the elliptic part. The proof is based on a variational time-stepping scheme on the low-rank manifold. Moreover, this scheme is shown to be closely related to practical methods for computing such low-rank evolutions

Проектирование электрической части подстанции 110/10 кВ "Мясокомбинат" Волковысских электрических сетей

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Accuracy aspects of the reaction-diffusion master equation on unstructured meshes

The reaction-diffusion master equation (RDME) is a stochastic model for spatially heterogeneous chemical systems. Stochastic models have proved to be useful for problems from molecular biology since copy numbers of participating chemical species often are small, which gives a stochastic behaviour. The RDME is a discrete space model, in contrast to spatially continuous models based on Brownian motion. In this thesis two accuracy issues of the RDME on unstructured meshes are studied. The first concerns the rates of diffusion events. Errors due to previously used rates are evaluated, and a second order accurate finite volume method, not previously used in this context, is implemented. The new discretisation improves the accuracy considerably, but unfortunately it puts constraints on the mesh, limiting its current usability. The second issue concerns the rates of bimolecular reactions. Using the macroscopic reaction coefficients these rates become too low when the spatial resolution is high. Recently, two methods to overcome this problem by calculating mesoscopic reaction rates for Cartesian meshes have been proposed. The methods are compared and evaluated, and are found to work remarkably well. Their possible extension to unstructured meshes is discussed

Stiff convergence of force-gradient operator splitting methods

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Numerical Methods for Wave Propagation : Analysis and Applications in Quantum Dynamics

We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrödinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable. The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.eSSENC