arXiv subjects

Christian Lubich

Publications and source records attributed to Christian Lubich.

At least 37 records · Page 2Linked to original sources

arXiv2022

Rank-adaptive time integration of tree tensor networks

A rank-adaptive integrator for the approximate solution of high-order tensor differential equations by tree tensor networks is proposed and analyzed. In a recursion from the leaves to the root, the integrator updates bases and then evolves connection tensors by a Galerkin method in the augmented subspace spanned by the new and old bases. This is followed by rank truncation within a specified error tolerance. The memory requirements are linear in the order of the tensor and linear in the maximal mode dimension. The integrator is robust to small singular values of matricizations of the connection tensors. Up to the rank truncation error, which is controlled by the given error tolerance, the integrator preserves norm and energy for Schrodinger equations, and it dissipates the energy in gradient systems. Numerical experiments with a basic quantum spin system illustrate the behavior of the proposed algorithm.

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arXiv2022

Polarized high-frequency wave propagation beyond the nonlinear Schrödinger approximation

This paper studies highly oscillatory solutions to a class of systems of semilinear hyperbolic equations with a small parameter, in a setting that includes Klein--Gordon equations and the Maxwell--Lorentz system. The interest here is in solutions that are polarized in the sense that up to a small error, the oscillations in the solution depend on only one of the frequencies that satisfy the dispersion relation with a given wave vector appearing in the initial wave packet. The construction and analysis of such polarized solutions is done using modulated Fourier expansions. This approach includes higher harmonics and yields approximations to polarized solutions that are of arbitrary order in the small parameter, going well beyond the known first-order approximation via a nonlinear Schrödinger equation. The given construction of polarized solutions is explicit, uses in addition a linear Schrödinger equation for each further order of approximation, and is accessible to direct numerical approximation.

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arXiv2022

Rank-$1$ matrix differential equations for structured eigenvalue optimization

A new approach to solving eigenvalue optimization problems for large structured matrices is proposed and studied. The class of optimization problems considered is related to computing structured pseudospectra and their extremal points, and to structured matrix nearness problems such as computing the structured distance to instability or to singularity. The structure can be a general linear structure and includes, for example, large matrices with a given sparsity pattern, matrices with given range and co-range, and Hamiltonian matrices. Remarkably, the eigenvalue optimization can be performed on the manifold of complex (or real) rank-1 matrices, which yields a significant reduction of storage and in some cases of the computational cost. The method relies on a constrained gradient system and the projection of the gradient onto the tangent space of the manifold of complex rank-$1$ matrices. It is shown that near a local minimizer this projection is very close to the identity map, and so the computationally favorable rank-1 projected system behaves locally like the %computationally expensive gradient system.

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arXiv2022

On a large-stepsize integrator for charged-particle dynamics

Xiao and Qin [Computer Physics Comm., 265:107981, 2021] recently proposed a remarkably simple modification of the Boris algorithm to compute the guiding centre of the highly oscillatory motion of a charged particle with step sizes that are much larger than the period of gyrorotations. They gave strong numerical evidence but no error analysis. This paper provides an analysis of the large-stepsize modified Boris method in a setting that has a strong non-uniform magnetic field and moderately bounded velocities, considered over a fixed finite time interval. The error analysis is based on comparing the modulated Fourier expansions of the exact and numerical solutions, for which the differential equations of the dominant terms are derived explicitly. Numerical experiments illustrate and complement the theoretical results.

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arXiv2022

Time-dependent electromagnetic scattering from thin layers

The scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a familiar second step, the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell's equations. The time-dependent boundary integral equationis discretized with Runge--Kutta based convolution quadrature in time and Raviart--Thomas boundary elements in space. Using the frequency-explicit bounds from the well-posedness analysis given here together with known approximation properties of the numerical methods, the full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. Taking the same Runge--Kutta based convolution quadrature for discretizing the time-dependent representation formulas, the optimal order of convergence is obtained away from the scattering boundary, whereas an order reduction occurs close to the boundary. The theoretical results are illustrated by numerical experiments.

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arXiv2021

A rank-adaptive robust integrator for dynamical low-rank approximation

A rank-adaptive integrator for the dynamical low-rank approximation of matrix and tensor differential equations is presented. The fixed-rank integrator recently proposed by two of the authors is extended to allow for an adaptive choice of the rank, using subspaces that are generated by the integrator itself. The integrator first updates the evolving bases and then does a Galerkin step in the subspace generated by both the new and old bases, which is followed by rank truncation to a given tolerance. It is shown that the adaptive low-rank integrator retains the exactness, robustness and symmetry-preserving properties of the previously proposed fixed-rank integrator. Beyond that, up to the truncation tolerance, the rank-adaptive integrator preserves the norm when the differential equation does, it preserves the energy for Schrödinger equations and Hamiltonian systems, and it preserves the monotonic decrease of the functional in gradient flows. Numerical experiments illustrate the behaviour of the rank-adaptive integrator.

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arXiv2021

Finding the nearest passive or non-passive system via Hamiltonian eigenvalue optimization

We propose and study an algorithm for computing a nearest passive system to a given non-passive linear time-invariant system (with much freedom in the choice of the metric defining `nearest', which may be restricted to structured perturbations), and also a closely related algorithm for computing the structured distance of a given passive system to non-passivity. Both problems are addressed by solving eigenvalue optimization problems for Hamiltonian matrices that are constructed from perturbed system matrices. The proposed algorithms are two-level methods that optimize the Hamiltonian eigenvalue of smallest positive real part over perturbations of a fixed size in the inner iteration, using a constrained gradient flow. They optimize over the perturbation size in the outer iteration, which is shown to converge quadratically in the typical case of a defective coalescence of simple eigenvalues approaching the imaginary axis. For large systems, we propose a variant of the algorithm that takes advantage of the inherent low-rank structure of the problem. Numerical experiments illustrate the behavior of the proposed algorithms.

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arXiv2021

Large-stepsize integrators for charged-particle dynamics over multiple time scales

The Boris algorithm, a closely related variational integrator and a newly proposed filtered variational integrator are studied when they are used to numerically integrate the equations of motion of a charged particle in a non-uniform strong magnetic field, taking step sizes that are much larger than the period of the Larmor rotations. For the Boris algorithm and the standard (unfiltered) variational integrator, satisfactory behaviour is only obtained when the component of the initial velocity orthogonal to the magnetic field is filtered out. The particle motion shows varying behaviour over multiple time scales: fast Larmor rotation, guiding centre motion, slow perpendicular drift, near-conservation of the magnetic moment over very long times and conservation of energy for all times. Using modulated Fourier expansions of the exact and numerical solutions, it is analysed to which extent this behaviour is reproduced by the three numerical integrators used with large step sizes.

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arXiv2020

Stable and convergent fully discrete interior-exterior coupling of Maxwell's equations

Maxwell's equations are considered with transparent boundary conditions, for initial conditions and inhomogeneity having support in a bounded, not necessarily convex three-dimensional domain or in a collection of such domains. The numerical method only involves the interior domain and its boundary. The transparent boundary conditions are imposed via a time-dependent boundary integral operator that is shown to satisfy a coercivity property. The stability of the numerical method relies on this coercivity and on an anti-symmetric structure of the discretized equations that is inherited from a weak first-order formulation of the continuous equations. The method proposed here uses a discontinuous Galerkin method and the leapfrog scheme in the interior and is coupled to boundary elements and convolution quadrature on the boundary. The method is explicit in the interior and implicit on the boundary. Stability and convergence of the spatial semidiscretization are proven, and with a computationally simple stabilization term, this is also shown for the full discretization.

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arXiv2020

An unconventional robust integrator for dynamical low-rank approximation

We propose and analyse a numerical integrator that computes a low-rank approximation to large time-dependent matrices that are either given explicitly via their increments or are the unknown solution to a matrix differential equation. Furthermore, the integrator is extended to the approximation of time-dependent tensors by Tucker tensors of fixed multilinear rank. The proposed low-rank integrator is different from the known projector-splitting integrator for dynamical low-rank approximation, but it retains the important robustness to small singular values that has so far been known only for the projector-splitting integrator. The new integrator also offers some potential advantages over the projector-splitting integrator: It avoids the backward time integration substep of the projector-splitting integrator, which is a potentially unstable substep for dissipative problems. It offers more parallelism, and it preserves symmetry or anti-symmetry of the matrix or tensor when the differential equation does. Numerical experiments illustrate the behaviour of the proposed integrator.

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arXiv2020

Time integration of tree tensor networks

Dynamical low-rank approximation by tree tensor networks is studied for the data-sparse approximation to large time-dependent data tensors and unknown solutions of tensor differential equations. A time integration method for tree tensor networks of prescribed tree rank is presented and analyzed. It extends the known projector-splitting integrators for dynamical low-rank approximation by matrices and Tucker tensors and is shown to inherit their favorable properties. The integrator is based on recursively applying the Tucker tensor integrator. In every time step, the integrator climbs up and down the tree: it uses a recursion that passes from the root to the leaves of the tree for the construction of initial value problems on subtree tensor networks using appropriate restrictions and prolongations, and another recursion that passes from the leaves to the root for the update of the factors in the tree tensor network. The integrator reproduces given time-dependent tree tensor networks of the specified tree rank exactly and is robust to the typical presence of small singular values in matricizations of the connection tensors, in contrast to standard integrators applied to the differential equations for the factors in the dynamical low-rank approximation by tree tensor networks.

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arXiv2020

A convergent algorithm for forced mean curvature flow driven by diffusion on the surface

The evolution of a closed two-dimensional surface driven by both mean curvature flow and a reaction--diffusion process on the surface is formulated into a system, which couples the velocity law not only to the surface partial differential equation but also to the evolution equations for the geometric quantities, namely the normal vector and the mean curvature on the surface. Two algorithms are considered for the obtained system. Both methods combine surface finite elements as a space discretisation and linearly implicit backward difference formulae for time integration. Based on our recent results for mean curvature flow, one of the algorithms directly admits a convergence proof for its full discretisation in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. Numerical examples are provided to support and complement the theoretical convergence results (demonstrating the convergence properties of the method without error estimate), and demonstrate the effectiveness of the methods in simulating a three-dimensional tumour growth model.

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arXiv2020

A convergent evolving finite element algorithm for Willmore flow of closed surfaces

A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector and mean curvature, reformulated as a system of second-order equations, and uses these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical method admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The error analysis combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix--vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.

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arXiv2020

Measuring the stability of spectral clustering

As an indicator of the stability of spectral clustering of an undirected weighted graph into $k$ clusters, the $k$th spectral gap of the graph Laplacian is often considered. The $k$th spectral gap is characterized in this paper as an unstructured distance to ambiguity, namely as the minimal distance of the Laplacian to arbitrary symmetric matrices with vanishing $k$th spectral gap. As a conceptually more appropriate measure of stability, the structured distance to ambiguity of the $k$-clustering is introduced as the minimal distance of the Laplacian to Laplacians of graphs with the same vertices and edges but with weights that are perturbed such that the $k$th spectral gap vanishes. To compute a solution to this matrix nearness problem, a two-level iterative algorithm is proposed that uses a constrained gradient system of matrix differential equations in the inner iteration and a one-dimensional optimization of the perturbation size in the outer iteration. The structured and unstructured distances to ambiguity are compared on some example graphs. The numerical experiments show, in particular, that selecting the number $k$ of clusters according to the criterion of maximal stability can lead to different results for the structured and unstructured stability indicators.

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arXiv2020

Time-dependent acoustic scattering from generalized impedance boundary conditions via boundary elements and convolution quadrature

Generalized impedance boundary conditions are effective, approximate boundary conditions that describe scattering of waves in situations where the wave interaction with the material involves multiple scales. In particular, this includes materials with a thin coating (with the thickness of the coating as the small scale) and strongly absorbing materials. For the acoustic scattering from generalized impedance boundary conditions, the approach taken here first determines the Dirichlet and Neumann boundary data from a system of time-dependent boundary integral equations with the usual boundary integral operators, and then the scattered wave is obtained from the Kirchhoff representation. The system of time-dependent boundary integral equations is discretized by boundary elements in space and convolution quadrature in time. The well-posedness of the problem and the stability of the numerical discretization rely on the coercivity of the Calderón operator for the Helmholtz equation with frequencies in a complex half-plane. Convergence of optimal order in the natural norms is proved for the full discretization. Numerical experiments illustrate the behaviour of the proposed numerical method.

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arXiv2020

Computing quantum dynamics in the semiclassical regime

The semiclassically scaled time-dependent multi-particle Schrödinger equation describes, inter alia, quantum dynamics of nuclei in a molecule. It poses the combined computational challenges of high oscillations and high dimensions. This paper reviews and studies numerical approaches that are robust to the small semiclassical parameter. We present and analyse variationally evolving Gaussian wave packets, Hagedorn's semiclassical wave packets, continuous superpositions of both thawed and frozen Gaussians, and Wigner function approaches to the direct computation of expectation values of observables. Making good use of classical mechanics is essential for all these approaches. The arising aspects of time integration and high-dimensional quadrature are also discussed.

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arXiv2020

Higher-order linearly implicit full discretization of the Landau--Lifshitz--Gilbert equation

For the Landau--Lifshitz--Gilbert (LLG) equation of micromagnetics we study linearly implicit backward difference formula (BDF) time discretizations up to order $5$ combined with higher-order non-conforming finite element space discretizations, which are based on the weak formulation due to Alouges but use approximate tangent spaces that are defined by $L^2$-averaged instead of nodal orthogonality constraints. We prove stability and optimal-order error bounds in the situation of a sufficiently regular solution. For the BDF methods of orders $3$ to~$5$, this requires %a mild time step restriction $τ\leqslant ch$ and that the damping parameter in the LLG equations be above a positive threshold; this condition is not needed for the A-stable methods of orders $1$ and $2$, for which furthermore a discrete energy inequality irrespective of solution regularity is proved.

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arXiv2019

A filtered Boris algorithm for charged-particle dynamics in a strong magnetic field

A modification of the standard Boris algorithm, called filtered Boris algorithm, is proposed for the numerical integration of the equations of motion of charged particles in a strong non-uniform magnetic field in the asymptotic scaling known as maximal ordering. With an appropriate choice of filters, second-order error bounds in the position and in the parallel velocity, and first-order error bounds in the normal velocity are obtained with respect to the scaling parameter. The proof compares the modulated Fourier expansions of the exact and the numerical solutions. Numerical experiments illustrate the error behaviour of the filtered Boris algorithm.

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