Philipp Bader

Philipp Bader, Sergio Blanes, Enrique Ponsoda

We present structure preserving integrators for solving linear quadratic optimal control problems. This problem requires the numerical integration of matrix Riccati differential equations whose exact solution is a symmetric positive definite time-dependent matrix which controls the stability of the equation for the state. This property is not preserved, in general, by the numerical methods. We propose second order exponential methods based on the Magnus series expansion which unconditionally preserve positivity for this problem and analyze higher order Magnus integrators. This method can also be used for the integration of nonlinear problems if they are previously linearized. The performance of the algorithms is illustrated with the stabilization of a quadrotor which is an unmanned aerial vehicle.

Philipp Bader, Sergio Blanes

We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since the system can be split into the kinetic and remaining part, and each part can be solved efficiently using Fast Fourier Transforms. To split the system into the quantum harmonic oscillator problem and the remaining part allows to get higher accuracies in many cases, but it requires to change between Hermite basis functions and the coordinate space, and this is not efficient for time-dependent frequencies or strong nonlinearities. We show how to build new methods which combine the advantages of using Fourier methods while solving the timedependent harmonic oscillator exactly (or with a high accuracy by using a Magnus integrator and an appropriate decomposition).

Philipp Bader, Sergio Blanes, Nikita Kopylov

We consider the numerical integration of the Schrödinger equation with a time-dependent Hamiltonian given as the sum of the kinetic energy and a time-dependent potential. Commutator-free (CF) propagators are exponential propagators that have shown to be highly efficient for general time-dependent Hamiltonians. We propose new CF propagators that are tailored for Hamiltonians of said structure, showing a considerably improved performance. We obtain new fourth- and sixth-order CF propagators as well as a novel sixth-order propagator that incorporates a double commutator that only depends on coordinates, so this term can be considered as cost-free. The algorithms require the computation of the action of exponentials on a vector similarly to the well known exponential midpoint propagator, and this is carried out using the Lanczos method. We illustrate the performance of the new methods on several numerical examples.

Philipp Bader, Sergio Blanes, Muaz Seydaoğlu

We propose splitting methods for the computation of the exponential of perturbed matrices which can be written as the sum $A=D+\varepsilon B$ of a sparse and efficiently exponentiable matrix $D$ with sparse exponential $e^D$ and a dense matrix $\varepsilon B$ which is of small norm in comparison with $D$. The predominant algorithm is based on scaling the large matrix $A$ by a small number $2^{-s}$, which is then exponentiated by efficient Padé or Taylor methods and finally squared in order to obtain an approximation for the full exponential. In this setting, the main portion of the computational cost arises from dense-matrix multiplications and we present a modified squaring which takes advantage of the smallness of the perturbed matrix $B$ in order to reduce the number of squarings necessary. Theoretical results on local error and error propagation for splitting methods are complemented with numerical experiments and show a clear improvement over existing methods when medium precision is sought.

Philipp Bader, Sergio Blanes, Fernando Casas et al.

Two families of symplectic methods specially designed for second-order time-dependent linear systems are presented. Both are obtained from the Magnus expansion of the corresponding first-order equation, but otherwise they differ in significant aspects. The first family is addressed to problems with low to moderate dimension, whereas the second is more appropriate when the dimension is large, in particular when the system corresponds to a linear wave equation previously discretised in space. Several numerical experiments illustrate the main features of the new schemes.

Philipp Bader, Sergio Blanes, Fernando Casas

In this work, we present a new way to compute the Taylor polynomial of the matrix exponential which reduces the number of matrix multiplications in comparison with the de-facto standard Patterson-Stockmeyer method. This reduction is sufficient to make the method superior in performance to Padé approximants by 10-30% over a range of values for the matrix norms and thus we propose its replacement in standard software kits. Numerical experiments show the performance of the method and illustrate its stability.

Philipp Bader, Arieh Iserles, Karolina Kropielnicka et al.

We build an efficient and unitary (hence stable) method for the solution of the semi-classical Schrödinger equation subject with explicitly time-dependent potentials. The method is based on a combination of the Zassenhaus decomposition (Bader, Iserles, Kropielnicka & Singh 2014) with the Magnus expansion of the time-dependent Hamiltonian. We conclude with numerical experiments.

Philipp Bader, Sergio Blanes, Enrique Ponsoda et al.

We consider the numerical integration of the matrix Hill's equation. Parametric resonances can appear and this property is of great interest in many different physical applications. Usually, the Hill's equations originate from a Hamiltonian function and the fundamental matrix solution is a symplectic matrix. This is a very important property to be preserved by the numerical integrators. In this work we present new sixth-and eighth-order symplectic exponential integrators that are tailored to the Hill's equation. The methods are based on an efficient symplectic approximation to the exponential of high dimensional coupled autonomous harmonic oscillators and yield accurate results for oscillatory problems at a low computational cost. Several numerical examples illustrate the performance of the new methods.

Philipp Bader

We present a new method to propagate rotating Bose-Einstein condensates subject to explicitly time-dependent trapping potentials. Using algebraic techniques, we combine Magnus expansions and splitting methods to yield any order methods for the multivariate and nonautonomous quadratic part of the Hamiltonian that can be computed using only Fourier transforms at the cost of solving a small system of polynomial equations. The resulting scheme solves the challenging component of the (nonlinear) Hamiltonian and can be combined with optimized splitting methods to yield efficient algorithms for rotating Bose-Einstein condensates. The method is particularly efficient for potentials that can be regarded as perturbed rotating and trapped condensates, e.g., for small nonlinearities, since it retains the near-integrable structure of the problem. For large nonlinearities, the method remains highly efficient if higher order p > 2 is sought. Furthermore, we show how it can adapted to the presence of dissipation terms. Numerical examples illustrate the performance of the scheme.

Philipp Bader, David I McLaren, G. R. W. Quispel et al.

It is a classical theorem of Liouville that Hamiltonian systems preserve volume in phase space. Any symplectic Runge-Kutta method will respect this property for such systems, but it has been shown that no B-Series method can be volume preserving for all volume preserving vector fields (BIT 47 (2007) 351-378 and IMA J. Numer. Anal. 27 (2007) 381-405). In this paper we show that despite this result, symplectic Runge-Kutta methods can be volume preserving for a much larger class of vector fields than Hamiltonian systems, and discuss how some Runge-Kutta methods can preserve a modified measure exactly.