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Continuum Vlasov simulations can be utilized for highly accurate modelling of fully kinetic plasmas. Great progress has been made recently regarding the applicability of the method in realistic plasma configurations. However, a reduction of the high computational cost that is inherent to fully kinetic simulations would be desirable, especially at high velocity space resolutions. For this purpose, low-rank approximations can be employed. The so far available low-rank solvers are restricted to either electrostatic systems or low dimensionality and can therefore not be applied to most space, astrophysical and fusion plasmas. In this paper we present a new parallel low-rank solver for the full six-dimensional electromagnetic Vlasov-Maxwell equations that can utilize distributed memory architectures. Special care is taken to ensure the conservation of mass and a good representation of Gauss's law. The low-rank Vlasov solver is applied to standard benchmark problems of plasma turbulence and magnetic reconnection and compared to the full grid method. It yields accurate results at significantly reduced computational cost.

The time-dependent Schrödinger equation describes quantum dynamical phenomena. Solving it numerically, the small-scale interactions that are modeled require very fine spatial resolution. At the same time, the solutions are localized and confined to small regions in space. Using the required resolution over the entire high-dimensional domain often makes the model problems intractable due to the prohibitively large grids that result from such a discretization. In this paper, we present a block-structured adaptive mesh refinement scheme, aiming at efficient adaptive discretization of high-dimensional partial differential equations such as the time-dependent Schrödinger equation. Our framework allows for anisotropic grid refinement in order to avoid unnecessary refinement. For spatial discretization, we use standard finite difference stencils together with summation-by-parts operators and simultaneous-approximation-term interface treatment. We propagate in time using exponential integration with the Lanczos method. Our theoretical and numerical results show that our adaptive scheme is stable for long time integrations. We also show that the discretizations meet the expected convergence rates.

In this paper, we present a discretization of the time-dependent Schrödinger equation based on a Magnus-Lanczos time integrator and high-order Gauss-Lobatto finite elements in space. A truncated Galerkin orthogonality is used to obtain duality-based a posteriori error estimates that address the temporal and the spatial error separately. Based on this theory, a space-time adaptive solver for the Schrödinger equation is devised. An efficient matrix-free implementation of the differential operator, suited for spectral elements, is used to enable computations for realistic configurations. We demonstrate the performance of the algorithm for the example of matter-field interaction.

The time-dependent Schrödinger equation (TDSE) models the quantum nature of molecular processes.  Numerical simulations based on the TDSE help in understanding and predicting the outcome of chemical reactions. This thesis is dedicated to the derivation and analysis of efficient and reliable simulation tools for the TDSE, with a particular focus on models for the interaction of molecules with time-dependent electromagnetic fields.

Various time propagators are compared for this setting and an efficient fourth-order commutator-free Magnus-Lanczos propagator is derived. For the Lanczos method, several communication-reducing variants are studied for an implementation on clusters of multi-core processors. Global error estimation for the Magnus propagator is devised using a posteriori error estimation theory. In doing so, the self-adjointness of the linear Schrödinger equation is exploited to avoid solving an adjoint equation. Efficiency and effectiveness of the estimate are demonstrated for both bounded and unbounded states. The temporal approximation is combined with adaptive spectral elements in space. Lagrange elements based on Gauss-Lobatto nodes are employed to avoid nondiagonal mass matrices and ill-conditioning at high order. A matrix-free implementation for the evaluation of the spectral element operators is presented. The framework uses hybrid parallelism and enables significant computational speed-up as well as the solution of larger problems compared to traditional implementations relying on sparse matrices.

As an alternative to grid-based methods, radial basis functions in a Galerkin setting are proposed and analyzed. It is found that considerably higher accuracy can be obtained with the same number of basis functions compared to the Fourier method. Another direction of research presented in this thesis is a new algorithm for quantum optimal control: The field is optimized in the frequency domain where the dimensionality of the optimization problem can drastically be reduced. In this way, it becomes feasible to use a quasi-Newton method to solve the problem.

The time-dependent Schrödinger equation models the quantum nature of molecular processes. Numerical simulations of these models help in understanding and predicting the outcome of chemical reactions.

In this thesis, several numerical algorithms for evolving the Schrödinger equation with an explicitly time-dependent Hamiltonian are studied and their performance is compared for the example of a pump-probe and an interference experiment for the rubidium diatom. For the important application of interaction dynamics between a molecule and a time-dependent field, an efficient fourth order Magnus-Lanczos propagator is derived. Error growth in the equation is analyzed by means of a posteriori error estimation theory and the self-adjointness of the Hamiltonian is exploited to yield a low-cost global error estimate for numerical time evolution. Based on this theory, an h,p-adaptive Magnus-Lanczos propagator is developed that is capable to control the global error. Numerical experiments for various model systems (including a three dimensional model and a dissociative configuration) show that the error estimate is effective and the number of time steps needed to meet a certain accuracy is reduced due to adaptivity.

Moreover, the thesis proposes an efficient numerical optimization framework for the design of femtosecond laser pulses with the aim of manipulating chemical reactions. This task can be formulated as an optimal control problem with the electric field of the laser being the control variable. In the algorithm described here, the electric field is Fourier transformed and it is optimized over the Fourier coefficients. Then, the frequency band is narrowed which facilitates the application of a quasi-Newton method. Furthermore, the restrictions on the frequency band make sure that the optimized pulse can be realized by the experimental equipment. A numerical comparison shows that the new method can outperform the Krotov method, which is a standard scheme in this field.

We consider an optimal control problem for the time-dependent Schrödinger equation modeling molecular dynamics. Given a molecule in its ground state, the interaction with a tuned laser pulse can result in an excitation to a state of interest. By these means, one can optimize the yield of chemical reactions. The problem of designing an optimal laser pulse can be posed as an optimal control problem. We reformulate the optimization problem by Fourier-transforming the electric field of the laser and narrow the frequency band. In this way, we reduce the dimensionality of the control variable. This allows for storing an approximate Hessian and, thereby, we can solve the optimization problem with a quasi-Newton method. Such an implementation provides superlinear convergence. We show computational results for a Raman-transition example and give numerical evidence that our algorithm can outperform the standard Krotov-like method which does not employ approximative second derivatives. end{abstract}

We use a posteriori error estimation theory to derive a relation between local and global error in the propagation for the time-dependent Schrödinger equation. Based on this result, we design a class of h,p-adaptive Magnus-Lanczos propagators capable of controlling the global error of the time-stepping scheme by only solving the equation once. We provide results for models of several different small molecules including bounded and dissociative states, illustrating the efficiency and wide applicability of the new methods.

In this article, we consider the discretization of the time-dependent Schrödinger equation using radial basis functions (RBF). We formulate the discretized problem over an unbounded domain without imposing explicit boundary conditions. Since we can show that time-stability of the discretization is not guaranteed for an RBF-collocation method, we propose to employ a Galerkin ansatz instead. For Gaussians, it is shown that exponential convergence is obtained up to a point where a systematic error from the domain where no basis functions are centered takes over. The choice of the shape parameter and of the resolved region is studied numerically. Compared to the Fourier method with periodic boundary conditions, the basis functions can be centered in a smaller domain which gives increased accuracy with the same number of points.

We present a memory-efficient and parallel framework for finite element operator application implemented in the generic open-source library deal.II. Instead of assembling a sparse matrix and using it for matrix-vector products, the operation is applied by cell-wise quadrature. The evaluation of shape functions is implemented with a sum-factorization approach. Our implementation is parallelized on three levels to exploit modern supercomputer architecture in an optimal way: MPI over remote nodes, thread parallelization with dynamic task scheduling within the nodes, and explicit vectorization for utilizing processors' vector units. Special data structures are designed for high performance and to keep the memory requirements to a minimum. The framework handles adaptively refined meshes and systems of partial differential equations. We provide performance tests for both linear and nonlinear PDEs which show that our cell-based implementation is faster than sparse matrix-vector products for polynomial order two and higher on hexahedral elements and yields ten times higher Gflops rates.

This work presents the efficient, matrix-free finite-element library hyper deal for solving partial differential equations in two up to six dimensions with high-order discontinuous Galerkin methods. It builds upon the low-dimensional finite-element library deal. II to create complex low-dimensional meshes and to operate on them individually. These meshes are combined via a tensor product on the fly, and the library provides new special-purpose highly optimized matrix-free functions exploiting domain decomposition as well as shared memory via MPI-3.0 features. Both node-level performance analyses and strong/weak-scaling studies on up to 147,456 CPU cores confirm the efficiency of the implementation. Results obtained with the library hyper . deal are reported for high-dimensional advection problems and for the solution of the Vlasov-Poisson equation in up to six-dimensional phase space.

In this article we apply a discrete action principle for the Vlasov-Maxwell equations in a structure-preserving particle-field discretization framework. In this framework the finite-dimensional electromagnetic potentials and fields are represented in a discrete de Rham sequence involving general finite element spaces, and the particle-field coupling is represented by a set of projection operators that commute with the differential operators. With a minimal number of assumptions which allow for a variety of finite elements and shape functions for the particles, we show that the resulting variational scheme has a general discrete Poisson structure and thus leads to a semi-discrete Hamiltonian system. By introducing discrete interior products we derive a second type of space discretization which is momentum preserving, based on the same finite elements and shape functions. We illustrate our method by applying it to spline finite elements, and to a new spectral discretization where the particle-field coupling relies on discrete Fourier transforms.