TimeEvolver: A program for time evolution with improved error bound

We present TimeEvolver, a program for computing time evolution in a generic quantum system. It relies on well-known Krylov subspace techniques to tackle the problem of multiplying the exponential of a large sparse matrix iH, where His the Hamiltonian, with an initial vector v. The fact that His Hermitian makes it possible to provide an easily computable bound on the accuracy of the Krylov approximation. Apart from effects of numerical roundoff, the resulting a posteriori error bound is rigorous, which represents a crucial novelty as compared to existing software packages such as Expokit[1]. On a standard notebook, TimeEvolverallows to compute time evolution with adjustable precision in Hilbert spaces of dimension greater than 10(6) Program summary Program Title: TimeEvolver CPC Library link to program files: https://doi.org/10.17632/vvwvng9w36.1 Code Ocean capsule: https://codeocean.com/capsule/8431379 Developer's repository link: https://github.com/marco-michel/TimeEvolver Licensing provisions: MIT Programming language: C++ Supplementary material: An example which demonstrates the computation of time evolution in a concrete physical system. Nature of problem: Computing time evolution in a generic physical quantum system can be reduced to the numerical task of calculating exp(-iHt)v. Here His the Hamiltonian matrix, which is large and sparse, icorresponds to the imaginary unit, tdenotes time and the vector vrepresents the initial state. A program is needed to perform this computation efficiently. Since the use of approximation methods is unavoidable, it is important to quantify as rigorously as possible the resulting error. Moreover, in order to facilitate the application to various problems in physics, additional functionalities are needed, in particular for forming the Hamiltonian matrix from a more abstract representation of the Hamiltonian operator. Solution method: The program employs known Krylov subspace methods for calculation the exponential of the large sparse matrix (- iHt) times the vector v. The Arnoldi algorithm is used to form the Krylov subspace and exponentiation of the resulting small matrix is achieved by diagonalization. The fact that (-iHt) is anti-Hermitian makes it possible to calculate the error of the Krylov approximation in terms of an easily-computable integral formula. This allows to choose a maximal size of the time step, after which the method is restarted and a new Krylov subspace is formed, while respecting an adjustable error bound. It is rigorous up to inaccuracies of a one-dimensional numerical integral and effects of finite machine precision, for which we also give an estimate. All linear algebra operations are performed with the Intel (R) Math Kernel Library and Boost is used for numerical integration. The methods for deriving the Hamiltonian matrix rely on a hashtable representation of Hilbert space. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).. Additionally, we provide routines for deriving the matrix Hfrom a more abstract representation of the Hamiltonian operator.

Adaptive Finite Element Methods for Multiscale Partial Differential Equations

Achim Nonnenmacher

Engineers rely on efficient simulations that provide them with reliable data in order to make proper engineering design decisions. The purpose of this thesis is to design adaptive numerical methods for multiscale problems in this spirit. We consider elliptic homogenization problems discretized by the finite element heterogeneous multiscale method (FE-HMM). Unlike standard (single-scale) finite element methods, our multiscale discretization scheme relies on coupled macro and micro finite elements. The framework of the HMM allows to design an algorithm that follows the classical finite element structure on the macro level. The fine scales of the multiscale problems are taken into account by replacing the element-wise numerical integration over unknown macroscopic data by a numerical integration over suitably averaged micro solutions. These micro solutions are obtained from micro FE problems on sampling domains within the macro elements. This thesis is divided into two parts. In the first part, we discuss a short and versatile FE implementation of the multiscale algorithm. The implementation is flexible, easy to use and to modify and can handle simplicial or quadrilateral FE and various macro-micro coupling conditions for the constrained micro problems. The implementation of time-dependent problems is also discussed. Numerical examples including three dimensional problems are presented and demonstrate the efficiency and the versatility of the computational strategy. In the second part (the main part of this thesis), we present an a posteriori error analysis for the FE-HMM. The a posteriori analysis enables us to estimate the accuracy of a numerical solution (and therefore its reliability) and further it allows for the design of adaptive numerical methods, which are the most efficient. The crucial component for the design of an adaptive multiscale method is the introduction of appropriate error indicators. As the error indicators depend on macroscopic data (such as the macroscopic diffusion tensor) that are not readily available, we construct error indicators that only depend on the available macro and micro FE solutions, available from previous computations. We provide a posteriori estimates for the upper and lower bound in the energy norm. The corresponding macroscopic mesh refinement strategy is therefore both reliable and efficient. The microscopic mesh is refined simultaneously and – under appropriate assumptions – optimally with the macroscopic mesh. This means that the strategy reduces the macro and micro error at the same rate. In the case of a uniformly oscillating tensor and exact micro computations, the standard a posteriori error estimates for the FEM applied to the homogenized problem are recovered. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method and demonstrate the optimality of the chosen macro-micro coupling. We extend the adaptive FE-HMM to higher order FE. We further derive a posteriori estimates for the error in quantities of interest that are needed to make certain design decisions; the quantity of interest is represented by a linear functional. We derive and analyze a multiscale counterpart to the classical dual-weighted residual method and design a corresponding goal-oriented adaptive multiscale method. The efficiency of the method is shown in numerical experiments.

EPFL2011

Numerical approximation of the unsteady Navier-Stokes equations with an application to the flow past a square cylinder

Francesco Casalegno

The goal of this project is to numerically solve the Navier-Stokes equations by using different numerical methods with particular emphasis on solving the problem of the flow past a square cylinder. In particular, we use the finite element method based on P2/P1 elements for the velocity and pressure fields for the spatial approximation, while the backward Euler method (with semi-implicit treatment of the nonlinear term) for the time discretization. We firstly test the numerical schemes on a benchmark problem with known exact solution. Then, we discuss in detail the advantages, in terms of computational costs, in using the algebraic Chorin-Temam method with additional implementation improvements. We finally investigate the problem of the two-dimensional flow past a square cylinder, focusing our attention on the range 0.1-300 for the Reynolds number (Re). We describe the two different regimes associated to the steady and the unsteady flows and we remark as the latter is in fact due to a Hopf bifurcation of the system. We also discuss the relation between the Strouhal and Reynolds numbers concluding that the Strouhal number attains its maximum value in the range 169-170 for the Reynolds number. In particular, a cubic model is proposed, showing very good matching with observed data and a better fitting than other models available in literature.

2013

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

Claudia Maria Colciago, Simone Deparis

This paper deals with fast simulations of the hemodynamics in large arteries by considering a reduced model of the associated fluid-structure interaction problem, which in turn allows an additional reduction in terms of the numerical discretisation. The resulting method is both accurate and computationally cheap. This goal is achieved by means of two levels of reduction: first, we describe the model equations with a reduced mathematical formulation which allows to write the fluid-structure interaction problem as a Navier-Stokes system with non-standard boundary conditions; second, we employ numerical reduction techniques to further and drastically lower the computational costs. The non standard boundary condition is of a generalized Robin type, with a boundary mass and boundary stiffness terms accounting for the arterial wall compliance. The numerical reduction is obtained coupling two well-known techniques: the proper orthogonal decomposition and the reduced basis method, in particular the greedy algorithm. We start by reducing the numerical dimension of the problem at hand with a proper orthogonal decomposition and we measure the system energy with specific norms; this allows to take into account the different orders of magnitude of the state variables, the velocity and the pressure. Then, we introduce a strategy based on a greedy procedure which aims at enriching the reduced discretization space with low offline computational costs. As application, we consider a realistic hemodynamics problem with a perturbation in the boundary conditions and we show the good performances of the reduction techniques presented in the paper. The results obtained with the numerical reduction algorithm are compared with the one obtained by a standard finite element method. The gains obtained in term of CPU time are of three orders of magnitude.