Lagrange–Schwarz Waveform Relaxation domain decomposition methods for linear and nonlinear quantum wave problems

International audienceA Schwarz Waveform Relaxation (SWR) algorithm is proposed to solve by Domain Decomposition Method (DDM) linear and nonlinear Schrödinger equations. The symbols of the transparent fractional transmission operators involved in Optimized Schwarz Waveform Relaxation (OSWR) algorithms are approximated by low order Lagrange polynomials to derive Lagrange-Schwarz Waveform Relaxation (LSWR) algorithms based on local transmission operators. The LSWR methods are numerically shown to be computationally efficient, leading to convergence rates almost similar to OSWR techniques

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

International audienceThis paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for variable coefficients linear and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [12] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis

Multilevel preconditioning techniques for Schwarz waveform relaxation domain decomposition methods for real-and imaginary-time nonlinear Schrödinger equations

International audienceThis paper is dedicated to the derivation of a multilevel Schwarz Waveform Relaxation (SWR) Domain Decomposition Method (DDM) in real-and imaginary-time for the NonLinear Schrödinger Equation (NLSE). In imaginary-time, it is shown that the use of the multilevel SWR-DDM accelerates the convergence compared to the one-level SWR-DDM, resulting in an important reduction of the computational time and memory storage. In real-time, the method requires in addition the storage of the solution in overlapping zones at any time, but on coarser discretization levels. The method is numerically validated on the Classical SWR and Robin-based SWR methods but can however be applied to any SWR approach

On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation

International audienceThis paper is dedicated to the analysis of the rate of convergence of the classical and quasi-optimal Schwarz waveform relaxation (SWR) method for solving the linear Schrödinger equation with space-dependent potential. The strategy is based on i) the rewriting of the SWR algorithm as a fixed point algorithm in frequency space, and ii) the explicit construction of contraction factors thanks to pseudo-differential calculus. Some numerical experiments illustrating the analysis are also provided

An analysis of Schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations

International audienceThe aim of this paper is to derive and numerically validate some asymptotic estimates of the convergence rate of Classical and Optimized Schwarz Waveform Relaxation (SWR) domain decomposition methods applied to the computation of the stationary states of the one-dimensional linear and nonlinear Schrödinger equations with a potential. Although SWR methods are currently used for efficiently solving high dimensional partial differential equations, their convergence analysis and most particularly obtaining expressions of their convergence rate remains largely open even in one dimension, except in simple cases. In this work, we tacke this problem for linear and nonlinear one-dimensional Schrödinger equations by developing techniques which can be extended to higher dimensional problems and other types of PDEs. The approach combines the method developed in [24] for the linear advection reaction diffusion equation and the theory of inhomogeneous pseu-dodifferential operators in conjunction with the associated symbolical asymptotic expansions. For computing the stationary states, we consider the imaginary-time formulation of the Schrödinger equation based on the Continuous Normalized Gradient Flow (CNGF) method and use a semi-implicit Euler scheme for the discretization. Some numerical results in the one-dimensional case illustrate the analysis for both the linear Schrödinger and Gross-Pitaevskii equations

Optimal Control and Synchronization of Dynamic Ensemble Systems

Ensemble control involves the manipulation of an uncountably infinite collection of structurally identical or similar dynamical systems, which are indexed by a parameter set, by applying a common control without using feedback. This subject is motivated by compelling problems in quantum control, sensorless robotic manipulation, and neural engineering, which involve ensembles of linear, bilinear, or nonlinear oscillating systems, for which analytical control laws are infeasible or absent. The focus of this dissertation is on novel analytical paradigms and constructive control design methods for practical ensemble control problems. The first result is a computational method %based on the singular value decomposition (SVD) for the synthesis of minimum-norm ensemble controls for time-varying linear systems. This method is extended to iterative techniques to accommodate bounds on the control amplitude, and to synthesize ensemble controls for bilinear systems. Example ensemble systems include harmonic oscillators, quantum transport, and quantum spin transfers on the Bloch system. To move towards the control of complex ensembles of nonlinear oscillators, which occur in neuroscience, circadian biology, electrochemistry, and many other fields, ideas from synchronization engineering are incorporated. The focus is placed on the phenomenon of entrainment, which refers to the dynamic synchronization of an oscillating system to a periodic input. Phase coordinate transformation, formal averaging, and the calculus of variations are used to derive minimum energy and minimum mean time controls that entrain ensembles of non-interacting oscillators to a harmonic or subharmonic target frequency. In addition, a novel technique for taking advantage of nonlinearity and heterogeneity to establish desired dynamical structures in collections of inhomogeneous rhythmic systems is derived

Contribution to the study of efficient iterative methods for the numerical solution of partial differential equations

Multigrid and domain decomposition methods provide efficient algorithms for the numerical solution of partial differential equations arising in the modelling of many applications in Computational Science and Engineering. This manuscript covers certain aspects of modern iterative solution methods for the solution of large-scale problems issued from the discretization of partial differential equations. More specifically, we focus on geometric multigrid methods, non-overlapping substructuring methods and flexible Krylov subspace methods with a particular emphasis on their combination. Firstly, the combination of multigrid and Krylov subspace methods is investigated on a linear partial differential equation modelling wave propagation in heterogeneous media. Secondly, we focus on non-overlapping domain decomposition methods for a specific finite element discretization known as the hp finite element, where unrefinement/refinement is allowed both by decreasing/increasing the step size h or by decreasing/increasing the polynomial degree p of the approximation on each element. Results on condition number bounds for the domain decomposition preconditioned operators are given and illustrated by numerical results on academic problems in two and three dimensions. Thirdly, we review recent advances related to a class of Krylov subspace methods allowing variable preconditioning. We examine in detail flexible Krylov subspace methods including augmentation and/or spectral deflation, where deflation aims at capturing approximate invariant subspace information. We also present flexible Krylov subspace methods for the solution of linear systems with multiple right-hand sides given simultaneously. The efficiency of the numerical methods is demonstrated on challenging applications in seismics requiring the solution of huge linear systems of equations with multiple right-hand sides on parallel distributed memory computers. Finally, we expose current and future prospectives towards the design of efficient algorithms on extreme scale machines for the solution of problems coming from the discretization of partial differential equations