ALBERT

Beschreibung: The stability radius of an n x n matrix A (or distance to instability) is a well-known measure of robustness of stability of the linear stable dynamical system $\dot{x} = Ax$ . Such a distance is commonly measured either in the 2-norm or in the Frobenius norm. Even if the matrix A is real, the distance to instability is most often considered with respect to complex-valued matrices (in such case the two norms turn out to be equivalent) and restricting the distance to real matrices makes the problem more complicated, and in the case of Frobenius norm—to our knowledge—unresolved. Here, we present a novel approach to approximate real stability radii, particularly well-suited for large sparse matrices. The method consists of a two-level iteration, the inner one aiming to compute the -pseudospectral abscissa of a low-rank (1 or 2) dynamical system, and the outer one consisting of an exact Newton iteration. Due to its local convergence property, it generally provides upper bounds for the stability radii, but in practice usually computes the correct values. The method requires the computation of the rightmost eigenvalue of a sequence of matrices, each of them given by the sum of the original matrix A and a low-rank 1. This makes it particularly suitable for large sparse problems, for which several existing methods become inefficient, due to the fact that they require to solve full Hamiltonian eigenvalue problems and/or compute multiple singular value decompositions (SVDs).

Beschreibung: Two fully discrete, discontinuous Galerkin schemes with time-dynamic, locally refined meshes in space are developed for a fourth-order Cahn–Hilliard equation with an added nonlinear reaction term, a phenomenological model that can describe cancerous tumour growth. The proposed schemes, which are both second-order in time, are based on a primitive-variable discontinuous Galerkin spatial formulation that is valid in any number of space dimensions. We prove that the schemes are convergent, with optimal-order error bounds, even in the case where the mesh is changing with time, provided that the number of mesh changes is bounded by some constant. The schemes represent flexible, high-order accurate alternatives to the standard mixed C 0 finite element methods and nonconforming (plate) finite element methods for solving fourth-order parabolic partial differential equations. We conclude the paper with tests showing the convergence of the scheme at the predicted rates and the flexibility of the method for approximating complex solution dynamics efficiently.

Beschreibung: We analyse a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete logarithm, discrete exponential maps and discrete parallel transport, and we prove convergence to their continuous counterparts. The presented analysis is based on the direct method in the calculus of variation, on -convergence and on weighted finite element error estimation. The convergence results of the discrete geodesic calculus are experimentally confirmed for a basic model on a two-dimensional Riemannian manifold. This provides a theoretical basis for the application to shape spaces in computer vision, for which we present one specific example.

Beschreibung: This paper analyses the approximate solution of very weakly well-posed hyperbolic Cauchy problems. These problems have very sensitive dependence on initial data. We treat a single family of such problems showing that, in spite of the sensitive dependence, approximate solutions with desired precision can be computed in finite-precision arithmetic with cost growing polynomially in 1/. The sensitive dependence requires high finite precision. The analysis required a new Gevrey stability estimate for the leapfrog scheme. The latter depends on a new discrete Glaeser inequality. The cost of calculating solutions with features on a scale 〈〈1 grows as e C –1/2 .

Beschreibung: We discuss the construction of volume-preserving splitting methods based on a tensor product of single-variable basis functions. The vector field is decomposed as the sum of elementary divergence-free vector fields (EDFVFs), each of them corresponding to a basis function. The theory is a generalization of the monomial basis approach introduced in Xue & Zanna (2013, BIT Numer. Math. , 53 , 265–281) and has the trigonometric splitting of Quispel & McLaren (2003, J. Comp. Phys. , 186 , 308–316) and the splitting in shears of McLachlan & Quispel (2004, BIT , 44 , 515–538) as special cases. We introduce the concept of diagonalizable EDFVFs and identify the solvable ones as those corresponding to the monomial basis and the exponential basis. In addition to giving a unifying view of some types of volume-preserving splitting methods already known in the literature, the present approach allows us to give a closed-form solution also to other types of vector fields that could not be treated before, namely those corresponding to the mixed tensor product of monomial and exponential (including trigonometric) basis functions.

Beschreibung: Splitting methods constitute a well-established class of numerical schemes for the solution of evolution equations. Their efficient application, however, requires spatial smoothness of the underlying exact solution. If smoothness is lacking, the methods usually react with order reduction. Depending on the data and the type of boundary condition, splitting methods on rectangular domains, in general, suffer from such order reductions. This is mainly due to corner singularities arising in the solution. In this paper, the regularity of the Dirichlet problem on a rectangle is studied. On the one hand, this analysis reveals compatibility conditions that lead to smooth solutions; on the other hand, it motivates a modification of the original scheme to overcome the order reduction. This idea is exemplified for the Lie splitting. Numerical experiments illustrate the efficiency of the approach.

Beschreibung: We prove several discrete Gagliardo–Nirenberg–Sobolev and Poincaré–Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The key point of our approach is to use the continuous embedding of the space BV( ) into L N /( N –1) ( ) for a Lipschitz domain R N , with N ≥2. Finally, we give several applications to discrete duality finite volume schemes which are used for the approximation of nonlinear and nonisotropic elliptic and parabolic problems.

Beschreibung: We consider an optimal control problem subject to the one-dimensional Landau–Lifshitz–Gilbert equation, which describes the evolution of magnetizations in S 2 . The problem is motivated in order to control switching processes of ferromagnets. Existence of an optimum and the first-order necessary optimality system are derived. We show (up to subsequences) convergence of state, adjoint and control variables of a time discretization (semi-implicit Euler method) for vanishing time step size. A main step here is to verify corresponding stability properties for the semidiscrete state, which is nontrivial since the iterates take values which only approximate S 2 . We use a perturbation argument within a variational discretization in order to show error bounds for the semidiscrete state variables, from which we may then infer uniform bounds for the semidiscrete state and also adjoint variables. Numerical studies underline these results and compare this discretization with a further variant, which bases on a projection strategy for the state equation to enhance iterates to better approximate S 2 .

Beschreibung: In the past 10 years, the ‘parareal’ (parallel-in-time) algorithm has attracted lots of attention thanks to its excellent performance in scientific computing. The parareal algorithm is iterative and is characterized by two propagators G and F which are associated with a coarse step size T and a fine step size t , respectively, where T = J t and J ≥2 is an integer. When we apply this algorithm to large-scale systems of ordinary differential equations obtained by semidiscretizing partial differential equations, two questions arise naturally. (I) Is the error between the iterate and the target solution contractive at each iteration for any choice of the discretization parameters T , J and x ? (II) How small can the contraction factor be and can such a contraction factor be independent of the discretization parameters? For linear problems u '= A u + g with symmetric negative-definite matrix A , when the implicit Euler method is used as both the G - and F-propagators, positive answers to these two questions were given by Mathew et al. (2010, SIAM J. Sci. Comput. , 32 , 1180–1200) and the contraction factor can be bounded by 0.298 for any choice of the discretization parameters. In this paper, for the case that the implicit Euler method is used as the G -propagator, we provide a positive answer to (I) for three second-order F -propagators: the trapezoidal method, the TR/BDF2 method and the two-stage diagonally implicit Runge–Kutta (2s-DIRK) method. For (II), we prove that the contraction factors can be bounded by 0.316 and 1/3 for the 2s-DIRK method and the TR/BDF2 method (provided the parameter involved in TR/BDF2 satisfies [0.043, 0.977]), respectively, and both bounds are independent of the discretization parameters. For the trapezoidal method, we show that a uniform bound (less than 1) of the contraction factor does not exist. Numerical results are presented to validate the theoretical prediction.

Beschreibung: In this paper, we develop a spectral method based on generalized Laguerre functions for the Camassa–Holm equation. We first introduce four normed spaces and present their equivalence relations, which enables us to develop and to analyse generalized Laguerre approximations efficiently. We also establish some basic results on generalized Laguerre orthogonal approximations. The spectral scheme for the Camassa–Holm equation is proposed, and the convergence is proved. Numerical results demonstrate the spectral accuracy.