# Christian Lubich's research while affiliated with University of Tuebingen and other places

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## Publications (182)

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 mildly non-uniform strong magnetic field, taking step sizes that are much larger than the period of the Larmor rotations. For th...

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 singular...

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...

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 app...

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 fo...

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 evolvin...

A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional sur- faces, 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 s...

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 m...

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 evolvin...

We correct a sign error in the paper

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 app...

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...

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 materi...

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'), 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...

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 m...

As an indicator of the stability of spectral clustering of an undirected weighted graph into k clusters, the kth spectral gap of the graph Laplacian is often considered. The kth 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 v...

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 sec...

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 mater...

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...

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 differential equations of motion of a charged particle in a strong non-uniform magnetic field have the magnetic moment as an adiabatic invariant. This quantity is nearly conserved over long time scales covering arbitrary negative powers of the small parameter, which is inversely proportional to the strength of the magnetic field. The numerical...

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...

The semiclassically scaled time-dependent multi-particle Schr\"odinger 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...

A numerical integrator is presented that computes a symmetric or skew-symmetric low-rank approximation to large symmetric or skew-symmetric time-dependent matrices that are either given explicitly or are the unknown solution to a matrix differential equation. A related algorithm is given for the approximation of symmetric or anti-symmetric time-dep...

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 mea...

A proof of convergence is given for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the discrete surface like in Dziuk’s method, and linearly implicit backward difference formulae for time integration. Th...

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...

A numerical integrator is presented that computes a symmetric or skew-symmetric low-rank approximation to large symmetric or skew-symmetric time-dependent matrices that are either given explicitly or are the unknown solution to a matrix differential equation. A related algorithm is given for the approximation of symmetric or anti-symmetric time-dep...

As an indicator of the stability of spectral clustering of an undirected weighted graph into k clusters, the kth spectral gap of the graph Laplacian is often considered. The kth spectral gap is characterized here as an unstructured distance to ambiguity, namely as the minimal distance of the Laplacian to arbitrary symmetric matrices with vanishing...

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...

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 here as an unstructured distance to ambiguity, namely as the minimal distance of the Laplacian to arbitrary symmetric matrices with vani...

A novel algorithmic approach is presented for the problem of partitioning a connected undirected weighted graph under constraints such as cardinality or membership requirements or must-link and cannot-link constraints. Such constrained problems can be NP-hard combinatorial optimization problems. They are restated as matrix nearness problems for the...

For a large class of fully nonlinear parabolic equations, which include gradient flows for energy functionals that depend on the solution gradient, the semidiscretization in time by implicit Runge-Kutta methods such as the Radau IIA methods of arbitrary order is studied. Error bounds are obtained in the $W^{1,\infty}$ norm uniformly on bounded time...

Stability and convergence of full discretizations of various surface evolution equations are studied in this paper. The proposed discretization combines a higher-order evolving-surface finite element method (ESFEM) for space discretization with higher-order linearly implicit backward difference formulae (BDF) for time discretization. The stability...

Numerical methods that approximate the solution of the Vlasov-Poisson equation by a low-rank representation have been considered recently. These methods can be extremely effective from a computational point of view, but contrary to most Eulerian Vlasov solvers, they do not conserve mass and momentum, neither globally nor in respecting the correspon...

The Boris algorithm is a widely used numerical integrator for the motion of particles in a magnetic field. This article proves near-conservation of energy over very long times in the special cases where the magnetic field is constant or the electric potential is quadratic. When none of these assumptions is satisfied, it is illustrated by numerical...

A proof of convergence is given for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the discrete surface like in {Dziuk's} method, and linearly implicit backward difference formulae for time integration....

Semidiscretization in time is studied for a class of quasi-linear evolution equations in a framework due to Kato, which applies to symmetric first-order hyperbolic systems and to a variety of fluid and wave equations. In the regime where the solution is suffciently regular, we show stability and optimal-order convergence of the linearly implicit an...

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 dynamic...

Given a connected undirected weighted graph, we are concerned with problems related to partitioning the graph. First of all we look for the closest disconnected graph (the minimum cut problem), here with respect to the Euclidean norm. We are interested in the case of constrained minimum cut problems, where constraints include cardinality or members...

For a parabolic surface partial differential equation coupled to surface evolution, convergence of the spatial semidiscretization is studied in this paper. The velocity of the evolving surface is not given explicitly, but depends on the solution of the parabolic equation on the surface. Various velocity laws are considered: elliptic regularization...

Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic integrators for Hamiltonian ordinary and partial differential equations, of dynamical low-rank approximation of...

Dynamical low-rank approximation in the Tucker tensor format of given large time-dependent tensors and of tensor differential equations is the subject of this paper. In particular, a discrete time integration method for rank-constrained Tucker tensors is presented and analyzed. It extends the known projector-splitting integrator for dynamical low-r...

Maxwell's equations are considered with transparent boundary conditions, with initial conditions and inhomogeneity having support in a bounded, not necessarily convex three-dimensional domain or in a collection of such domains. The proposed computational scheme only involves the interior domain and its boundary. The transparent boundary conditions...

The variational equations of motion of the multi-configuration time-dependent Hartree (MCTDH) approach contain the inverse of reduced density matrices which are typically ill-conditioned and therefore lead to small stepsizes for numerical time integration. This problem is usually dealt with via regularization of the density matrices, which works we...

For a parabolic surface partial differential equation coupled to surface evolution, convergence of the spatial semidiscretization is studied in this paper. The velocity of the evolving surface is not given explicitly, but depends on the solution of the parabolic equation on the surface. Various velocity laws are considered: elliptic regularization...

We analyze fully implicit and linearly implicit backward difference formula (BDF) methods for quasilinear parabolic equations, without making any assumptions on the growth or decay of the coefficient functions. We combine maximal parabolic regularity and energy estimates to derive optimal-order error bounds for the time-discrete approximation to th...

A coercivity property of temporal convolution operators is an essential tool in the analysis of time-dependent boundary integral equations and their space and time discretisations. It is known that this coercivity property is inherited by convolution quadrature time discretisation based on A-stable multistep methods, which are of order at most two....

Given a regular matrix pencil A + µE, we consider the problem of determining the nearest singular matrix pencil with respect to the Frobenius norm. We present new approaches based on the solution of matrix differential equations for determining the nearest singular pencil A + ∆A + µ(E + ∆E): one approach for general singular pencils and another one...

We consider the problem of stabilizing a matrix by a correction of minimal norm: Given a square matrix that has some eigenvalues with positive real part, find the nearest matrix having no eigenvalue with positive real part. It can be further required that the correction have a prescribed structure, e.g., be real, have a prescribed sparsity pattern,...

It is shown that for a parabolic problem with maximal $L^p$-regularity (for
$1<p<\infty$), the time discretization by a linear multistep method or
Runge--Kutta method has maximal $\ell^p$-regularity uniformly in the stepsize
if the method is A-stable (and satisfies minor additional conditions). In
particular, the implicit Euler method, the Crank-Ni...

We show that the time-dependent variational principle provides a unifying
framework for time-evolution methods and optimisation methods in the context of
matrix product states. In particular, we introduce a new integration scheme for
studying time-evolution, which can cope with arbitrary Hamiltonians, including
those with long-range interactions. R...

A semilinear wave equation with slowly varying wave speed is considered in one to three space dimensions on a bounded interval, a rectangle or a box, respectively. It is shown that the action, which is the harmonic energy divided by the wave speed and multiplied with the diameter of the spatial domain, is an adiabatic invariant: it remains nearly c...

Low-rank approximations to large time-dependent matrices and tensors are the subject of this paper. These matrices and tensors either are given explicitly or are the unknown solutions of matrix and tensor differential equations. Based on splitting the orthogonal projection onto the tangent space of the low-rank manifold, novel time integrators for...

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential
equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods.

The long-time behaviour of the Störmer–Verlet–leapfrog method is studied when this method is applied to highly oscillatory Hamiltonian systems with a slowly varying, solution-dependent high frequency. Using the technique of modulated Fourier expansions with state-dependent frequencies, which is newly developed here, the following results are proved...

A numerical integrator is proposed for solving the multiconfiguration time-dependent Hartree (MCTDH) equations of motion,
which are widely used in computations of molecular quantum dynamics. In contrast to existing integrators, the proposed algorithm
does not require inverses of ill-conditioned density matrices and obviates the need for their regul...

A linear wave equation on a moving surface is derived from Hamilton's principle of stationary action. The variational principle is discretized with functions that are piecewise linear in space and time. This yields a discretization of the wave equation in space by evolving surface finite elements and in time by a variational integrator, a version o...

For a Hamiltonian matrix with purely imaginary eigenvalues, we aim to determine the nearest Hamiltonian matrix such that some or all eigenvalues leave the imaginary axis. Conversely, for a Hamiltonian matrix with all eigenvalues lying off the imaginary axis, we look for a nearest Hamiltonian matrix that has a pair of imaginary eigenvalues. The Hami...

A robust and efficient time integrator for dynamical tensor approximation in
the tensor train or matrix product state format is presented. The method is
based on splitting the projector onto the tangent space of the tensor manifold.
The algorithm can be used for updating time-dependent tensors in the given
data-sparse tensor train / matrix product...

Quasi-linear parabolic equations are discretised in time by fully implicit backward difference formulae (BDF) as well as by implicit–explicit and linearly implicit BDF methods up to order five. Under appropriate stability conditions for the various methods considered, we establish optimal order a priori error bounds by energy estimates, which becom...

Space and time discretizations of parabolic differential equations with dynamic boundary conditions are studied in a weak
formulation that fits into the standard abstract formulation of parabolic problems, just that the usual $L_2(\Omega)$ inner product is replaced by an $L_2(\Omega)\oplus L_2(\partial\Omega)$ inner product. The class of parabolic...

This note reviews differential equations on manifolds of matrices or tensors of low rank. They serve to approximate, in a low-rank format, large timedependentmatrices and tensors that are either given explicitly via their increments or are unknown solutions of differential equations. Furthermore, low-rank differential equations are used in novel al...

For gradient systems in Euclidean space or on a Riemannian manifold the energy decreases monotonically along solutions. Algebraically stable Runge-Kutta methods are shown to also reduce the energy in each step under a mild step-size restriction. In particular, Radau IIA methods can combine energy monotonicity and damping in stiff gradient systems....

This paper deals with the numerical integration of Hamiltonian systems in
which a stiff anharmonic potential causes highly oscillatory solution behavior
with solution-dependent frequencies. The Impulse Method, which uses micro- and
macro-steps for the integration of fast and slow parts, respectively, does not
work satisfactorily on such problems. H...

For trigonometric and modified trigonometric integrators applied to
oscillatory Hamiltonian differential equations with one or several constant
high frequencies, near-conservation of the total and oscillatory energies are
shown over time scales that cover arbitrary negative powers of the step size.
This requires non-resonance conditions between the...

We study differential equations that lead to extremal points in symplectic pseudospectra. In a two-level approach, where on the inner level we compute extremizers of the symplectic epsilon-pseudospectrum for a given epsilon and on the outer level we optimize over epsilon, this is used to solve symplectic matrix nearness problems such as the followi...

A linear parabolic differential equation on a moving surface is first discretized in space by evolving surface finite elements and then in time by an implicit Runge–Kutta method. For al-gebraically stable and stiffly accurate Runge–Kutta methods, unconditional stability of the full discretization is proven and the convergence properties are analyse...

The acoustic wave equation on the whole three-dimensional space is considered with initial data and inhomogeneity having support in a bounded domain, which need not be convex. We propose and study a numerical method that approximates the solution using computations only in the interior domain and on its boundary. The transmission conditions between...

A method of choice for the long-time integration of constrained Hamiltonian systems is the Rattle algorithm. It is symmetric, symplectic, and nearly preserves the Hamiltonian, but it is only of order two and thus not efficient for high accuracy requirements. In this article we prove that certain symmetric linear multistep methods have the same qual...

We consider multiscale Hamiltonian systems in which harmonic oscillators with
several high frequencies are coupled to a slow system. It is shown that the
oscillatory energy is nearly preserved over long times eps^{-N} for arbitrary
N>1, where eps^{-1} is the size of the smallest high frequency. The result is
uniform in the frequencies and does not...

We derive a posteriori estimates for single-step methods, including Runge–Kutta and Galerkin methods for the time discretization of linear parabolic equations. We focus on the estimation of the error at the nodes and derive a posteriori estimates that show the full classical order (superconvergence order) in the interior of the spatial domain witho...

Plane wave solutions to the cubic nonlinear Schr\"odinger equation on a torus
have recently been shown to behave orbitally stable under generic perturbations
of the initial data that are small in a high-order Sobolev norm, over long
times that extend to arbitrary negative powers of the smallness parameter. The
present paper studies the question as...

We extend results on the dynamical low-rank approximation for the treatment of time-dependent matrices and tensors (Koch and Lubich; see [SIAM J. Matrix Anal. Appl., 29 (2007), pp. 434-454], [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2360-2375]) to the recently proposed hierarchical Tucker (HT) tensor format (Hackbusch and Kühn; see [J. Fourier An...