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Register a new userFast low-rank solution of the Poisson equation with application to the Stokes problem
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We consider the problem of computing approximate solution of Poisson equation in the low-parametric tensor formats. We propose a new algorithm to compute the solution based on the cross approximation algorithm in the frequency space, and it has better complexity with respect to ranks in comparison with standard algorithms, which are based on the exponential sums approximation. To illustrate the effectiveness of our solver, we incorporate into a Uzawa solver for the Stokes problem on semi-staggered grid as a subsolver. The resulting solver outperforms the standard method for $n geq 256$.
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Many problems encountered in plasma physics require a description by kinetic equations, which are posed in an up to six-dimensional phase space. A direct discretization of this phase space, often called the Eulerian approach, has many advantages but is extremely expensive from a computational point of view. In the present paper we propose a dynamical low-rank approximation to the Vlasov--Poisson equation, with time integration by a particular splitting method. This approximation is derived by constraining the dynamics to a manifold of low-rank functions via a tangent space projection and by splitting this projection into the subprojections from which it is built. This reduces a time step for the six- (or four-) dimensional Vlasov--Poisson equation to solving two systems of three- (or two-) dimensional advection equations over the time step, once in the position variables and once in the velocity variables, where the size of each system of advection equations is equal to the chosen rank. By a hierarchical dynamical low-rank approximation, a time step for the Vlasov--Poisson equation can be further reduced to a set of six (or four) systems of one-dimensional advection equations, where the size of each system of advection equations is still equal to the rank. The resulting systems of advection equations can then be solved by standard techniques such as semi-Lagrangian or spectral methods. Numerical simulations in two and four dimensions for linear Landau damping, for a two-stream instability and for a plasma echo problem highlight the favorable behavior of this numerical method and show that the proposed algorithm is able to drastically reduce the required computational effort.
Transport maps have become a popular mechanic to express complicated probability densities using sample propagation through an optimized push-forward. Beside their broad applicability and well-known success, transport maps suffer from several drawbacks such as numerical inaccuracies induced by the optimization process and the fact that sampling schemes have to be employed when quantities of interest, e.g. moments are to compute. This paper presents a novel method for the accurate functional approximation of probability density functions (PDF) that copes with those issues. By interpreting the pull-back result of a target PDF through an inexact transport map as a perturbed reference density, a subsequent functional representation in a more accessible format allows for efficient and more accurate computation of the desired quantities. We introduce a layer-based approximation of the perturbed reference density in an appropriate coordinate system to split the high-dimensional representation problem into a set of independent approximations for which separately chosen orthonormal basis functions are available. This effectively motivates the notion of h- and p-refinement (i.e. ``mesh size and polynomial degree) for the approximation of high-dimensional PDFs. To circumvent the curse of dimensionality and enable sampling-free access to certain quantities of interest, a low-rank reconstruction in the tensor train format is employed via the Variational Monte Carlo method. An a priori convergence analysis of the developed approach is derived in terms of Hellinger distance and the Kullback-Leibler divergence. Applications comprising Bayesian inverse problems and several degrees of concentrated densities illuminate the (superior) convergence in comparison to Monte Carlo and Markov-Chain Monte Carlo methods.
With already demonstrated in previous work the equations that describe the space dependence of the electric potential are determined by the solution of the equation of Poisson-Boltzmann. In this work we consider these solutions for the membrane of the human neuron, using a model simplified for this structure considering the distribution of electrolytes in each side of the membrane, as well as the effect of glycocalyx and the lipidic bilayer. It was assumed that on both sides of the membrane the charges are homogeneously distributed and that the potential depends only on coordinate z.
This paper proposes a computer-assisted solution existence verification method for the stationary Navier-Stokes equation over general 3D domains. The proposed method verifies that the exact solution as the fixed point of the Newton iteration exists around the approximate solution through rigorous computation and error estimation. The explicit values of quantities required by applying the fixed point theorem are obtained by utilizing newly developed quantitative error estimation for finite element solutions to boundary value problems and eigenvalue problems of the Stokes equation.
We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the $L^2$-norm of the curl and the {it det-grad} measure of the fluid. We impose a Tikhonov regularization in the form of a perimeter functional and a volume constraint to address the possibility of topological change. Having been able to establish the existence of an optimal shape, the first order necessary condition was formulated by utilizing the so-called rearrangement method. Finally, numerical examples are presented by utilizing a finite element method on the governing states, and a gradient descent method for the deformation of the domain. On the said gradient descent method, we use two approaches to address the volume constraint: one is by utilizing the augmented Lagrangian method; and the other one is by utilizing a class of divergence-free deformation fields.
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