arXiv subjects

Daniel Peterseim

Publications and source records attributed to Daniel Peterseim.

At least 55 records · Page 3Linked to original sources

arXiv2020

Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency

We propose a new normalized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself. The gradient flow is well-defined and converges to an eigenfunction. For ground states we can quantify the convergence speed as exponentially fast where the rate depends on spectral gaps of a linearized operator. The forward Euler time discretization of the flow yields a numerical method which generalizes the inverse iteration for the nonlinear eigenvalue problem. For sufficiently small time steps, the method reduces the energy in every step and converges globally in $H^1$ to an eigenfunction. In particular, for any nonnegative starting value, the ground state is obtained. A series of numerical experiments demonstrates the computational efficiency of the method and its competitiveness with established discretizations arising from other gradient flows for this problem.

math.NA↗

arXiv2020

Sparse Compression of Expected Solution Operators

We show that the expected solution operator of prototypical linear elliptic partial differential operators with random coefficients is well approximated by a computable sparse matrix. This result is based on a random localized orthogonal multiresolution decomposition of the solution space that allows both the sparse approximate inversion of the random operator represented in this basis as well as its stochastic averaging. The approximate expected solution operator can be interpreted in terms of classical Haar wavelets. When combined with a suitable sampling approach for the expectation, this construction leads to an efficient method for computing a sparse representation of the expected solution operator.

math.NA↗

arXiv2020

Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials

This paper concerns spectral properties of linear Schrödinger operators under oscillatory high-amplitude potentials on bounded domains. Depending on the degree of disorder, we prove the existence of spectral gaps amongst the lowermost eigenvalues and the emergence of exponentially localized states. We quantify the rate of decay in terms of geometric parameters that characterize the potential. The proofs are based on the convergence theory of iterative solvers for eigenvalue problems and their optimal local preconditioning by domain decomposition.

math.NA↗

arXiv2020

Computational high frequency scattering from high contrast heterogeneous media

This article considers the computational (acoustic) wave propagation in strongly heterogeneous structures beyond the assumption of periodicity. A high contrast between the constituents of microstructured multiphase materials can lead to unusual wave scattering and absorption, which are interesting and relevant from a physical viewpoint, for instance, in the case of crystals with defects. We present a computational multiscale method in the spirit of the Localized Orthogonal Decomposition and provide its rigorous a priori error analysis for two-phase diffusion coefficients that vary between $1$ and very small values. Special attention is paid to the extreme regimes of high frequency, high contrast, and their previously unexplored coexistence. A series of numerical experiments confirms the theoretical results and demonstrates the ability of the multiscale approach to efficiently capture relevant physical phenomena.

math.NA↗

arXiv2019

Localized computation of eigenstates of random Schrödinger operators

This paper concerns the numerical approximation of low-energy eigenstates of the linear random Schrödinger operator. Under oscillatory high-amplitude potentials with a sufficient degree of disorder it is known that these eigenstates localize in the sense of an exponential decay of their moduli. We propose a reliable numerical scheme which provides localized approximations of such localized states. The method is based on a preconditioned inverse iteration including an optimal multigrid solver which spreads information only locally. The practical performance of the approach is illustrated in various numerical experiments in two and three space dimensions and also for a non-linear random Schrödinger operator.

math.NA↗

arXiv2019

Efficient implementation of the Localized Orthogonal Decomposition method

In this paper we present algorithms for an efficient implementation of the Localized Orthogonal Decomposition method (LOD). The LOD is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales. We show how the method can be implemented in a fairly standard Finite Element framework and discuss its realization for different types of problems, such as linear elliptic problems with rough coefficients and linear eigenvalue problems.

math.NA↗

arXiv2019

Numerical stochastic homogenization by quasilocal effective diffusion tensors

This paper proposes a numerical upscaling procedure for elliptic boundary value problems with diffusion tensors that vary randomly on small scales. The resulting effective deterministic model is given through a quasilocal discrete integral operator, which can be further compressed to an effective partial differential operator. Error estimates consisting of a priori and a posteriori terms are provided that allow one to quantify the impact of uncertainty in the diffusion coefficient on the expected effective response of the process.

math.NA↗

arXiv2018

Computational multiscale methods for linear heterogeneous poroelasticity

We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling. This poroelasticity problem suffers from rapidly oscillating material parameters, which calls for a thorough numerical treatment. In this paper, we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity. Therein, local corrector problems are constructed in line with the static equations, whereas we propose to consider the full system. This allows to benefit from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure. We prove the optimal first-order convergence of this method and verify the result by numerical experiments.

math.NA↗

arXiv2018

Computational Multiscale Methods for Linear Poroelasticity with High Contrast

In this work, we employ the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) to solve the problem of linear heterogeneous poroelasticity with coefficients of high contrast. The proposed method makes use of the idea of energy minimization with suitable constraints in order to generate efficient basis functions for the displacement and the pressure. These basis functions are constructed by solving a class of local auxiliary optimization problems based on eigenfunctions containing local information on the heterogeneity. Techniques of oversampling are adapted to enhance the computational performance. Convergence of first order is shown and illustrated by a number of numerical tests.

math.NA↗

arXiv2018

From Domain Decomposition to Homogenization Theory

This paper rediscovers a classical homogenization result for a prototypical linear elliptic boundary value problem with periodically oscillating diffusion coefficient. Unlike classical analytical approaches such as asymptotic analysis, oscillating test functions, or two-scale convergence, the result is purely based on the theory of domain decomposition methods and standard finite elements techniques. The arguments naturally generalize to problems far beyond periodicity and scale separation and we provide a brief overview on such applications.

math.NA↗

arXiv2018

Explicit Computational Wave Propagation in Micro-Heterogeneous Media

Explicit time stepping schemes are popular for linear acoustic and elastic wave propagation due to their simple nature which does not require sophisticated solvers for the inversion of the stiffness matrices. However, explicit schemes are only stable if the time step size is bounded by the mesh size in space subject to the so-called CFL condition. In micro-heterogeneous media, this condition is typically prohibitively restrictive because spatial oscillations of the medium need to be resolved by the discretization in space. This paper presents a way to reduce the spatial complexity in such a setting and, hence, to enable a relaxation of the CFL condition. This is done using the Localized Orthogonal Decomposition method as a tool for numerical homogenization. A complete convergence analysis is presented with appropriate, weak regularity assumptions on the initial data.

math.NA↗

arXiv2018

An analysis of a class of variational multiscale methods based on subspace decomposition

Numerical homogenization tries to approximate the solutions of elliptic partial differential equations with strongly oscillating coefficients by functions from modified finite element spaces. We present in this paper a class of such methods that are very closely related to the method of Målqvist and Peterseim [Math. Comp. 83, 2014]. Like the method of Målqvist and Peterseim, these methods do not make explicit or implicit use of a scale separation. Their compared to that in the work of Målqvist and Peterseim strongly simplified analysis is based on a reformulation of their method in terms of variational multiscale methods and on the theory of iterative methods, more precisely, of additive Schwarz or subspace decomposition methods.

math.NA↗

arXiv2017

Computation of quasilocal effective diffusion tensors and connections to the mathematical theory of homogenization

This paper aims at bridging existing theories in numerical and analytical homogenization. For this purpose the multiscale method of Målqvist and Peterseim [Math. Comp. 2014], which is based on orthogonal subspace decomposition, is reinterpreted by means of a discrete integral operator acting on standard finite element spaces. The exponential decay of the involved integral kernel motivates the use of a diagonal approximation and, hence, a localized piecewise constant coefficient. In a periodic setting, the computed localized coefficient is proved to coincide with the classical homogenization limit. An a priori error analysis shows that the local numerical model is appropriate beyond the periodic setting when the localized coefficient satisfies a certain homogenization criterion, which can be verified a posteriori. The results are illustrated in numerical experiments.

math.NA↗

arXiv2017

Thermo-optical interactions in a dye-microcavity photon Bose-Einstein condensate

Superfluidity and Bose-Einstein condensation are usually considered as two closely related phenomena. Indeed, in most macroscopic quantum systems, like liquid helium, ultracold atomic Bose gases, and exciton-polaritons, condensation and superfluidity occur in parallel. In photon Bose-Einstein condensates realized in the dye microcavity system, thermalization does not occur by direct interaction of the condensate particles as in the above described systems, i.e. photon-photon interactions, but by absorption and re-emission processes on the dye molecules, which act as a heat reservoir. Currently, there is no experimental evidence for superfluidity in the dye microcavity system, though effective photon interactions have been observed from thermo-optic effects in the dye medium. In this work, we theoretically investigate the implications of effective thermo-optic photon interactions, a temporally delayed and spatially non-local effect, on the photon condensate, and derive the resulting Bogoliubov excitation spectrum. The calculations suggest a linear photon dispersion at low momenta, fulfilling the Landau's criterion of superfluidity . We envision that the temporally delayed and long-range nature of the thermo-optic photon interaction offer perspectives for novel quantum fluid phenomena.

quant-ph↗

arXiv2017

Numerical Homogenization of Heterogeneous Fractional Laplacians

In this paper, we develop a numerical multiscale method to solve the fractional Laplacian with a heterogeneous diffusion coefficient. When the coefficient is heterogeneous, this adds to the computational costs. Moreover, the fractional Laplacian is a nonlocal operator in its standard form, however the Caffarelli-Silvestre extension allows for a localization of the equations. This adds a complexity of an extra spacial dimension and a singular/degenerate coefficient depending on the fractional order. Using a sub-grid correction method, we correct the basis functions in a natural weighted Sobolev space and show that these corrections are able to be truncated to design a computationally efficient scheme with optimal convergence rates. A key ingredient of this method is the use of quasi-interpolation operators to construct the fine scale spaces. Since the solution of the extended problem on the critical boundary is of main interest, we construct a projective quasi-interpolation that has both $d$ and $d+1$ dimensional averages over subsets in the spirit of the Scott-Zhang operator. We show that this operator satisfies local stability and local approximation properties in weighted Sobolev spaces. We further show that we can obtain a greater rate of convergence for sufficient smooth forces, and utilizing a global $L^2$ projection on the critical boundary. We present some numerical examples, utilizing our projective quasi-interpolation in dimension $2+1$ for analytic and heterogeneous cases to demonstrate the rates and effectiveness of the method.

math.NA↗

arXiv2017

Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials

This paper analyses the numerical solution of a class of non-linear Schrödinger equations by Galerkin finite elements in space and a mass- and energy conserving variant of the Crank-Nicolson method due to Sanz-Serna in time. The novel aspects of the analysis are the incorporation of rough, discontinuous potentials in the context of weak and strong disorder, the consideration of some general class of non-linearities, and the proof of convergence with rates in $L^{\infty}(L^2)$ under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.

math.NA↗

arXiv2017

Relaxing the CFL condition for the wave equation on adaptive meshes

The Courant-Friedrichs-Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. However, it limits the choice of the time step size to be bounded by the minimal mesh size in the spatial finite element mesh. This essentially prohibits any sort of adaptive mesh refinement that would be required to reveal optimal convergence rates on domains with re-entrant corners. This paper shows how a simple subspace projection step inspired by numerical homogenisation can remove the critical time step restriction so that the CFL condition and approximation properties are balanced in an optimal way, even in the presence of spatial singularities.

math.NA↗

arXiv2016

Error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in 2d

We formulate a stabilized quasi-optimal Petrov-Galerkin method for singularly perturbed convection-diffusion problems based on the variational multiscale method. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local patch problems, which depend on the direction of the velocity field and the singular perturbation parameter, is rigorously justified. Under moderate assumptions, this stabilization guarantees stability and quasi-optimal rate of convergence for arbitrary mesh Péclet numbers on fairly coarse meshes at the cost of additional inter-element communication.

math.NA↗