(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods

We consider ultraweak variational formulations for (parametrized) linear first order transport equations in time and/or space. Computationally feasible pairs of optimally stable trial and test spaces are presented, starting with a suitable test space and defining an optimal trial space by the application of the adjoint operator. As a result, the inf-sup constant is one in the continuous as well as in the discrete case and the computational realization is therefore easy. In particular, regarding the latter, we avoid a stabilization loop within the greedy algorithm when constructing reduced models within the framework of reduced basis methods. Several numerical experiments demonstrate the good performance of the new method

High performance computing on smartphones

Nowadays there is a strong demand to simulate even real-world engineering problems on small computing devices with very limited capacity, such as a smartphone. We explain, using a concrete example, how we can obtain a reduction in complexity – to enable such computations – using mathematical methods

A Space-Time Variational Method for Optimal Control Problems

We consider a space-time variational formulation of a PDE-constrained optimal control problem with box constraints on the control and a parabolic PDE with Robin boundary conditions. In this setting, the optimal control problem reduces to an optimization problem for which we derive necessary and sufficient optimality conditions. Next, we introduce a space-time (tensorproduct) discretization using finite elements in space and piecewise linear functions in time. This setting is known to be equivalent to a Crank-Nicolson time stepping scheme for parabolic problems. The optimization problem is solved by a projected gradient method. We show numerical comparisons for problems in 1d, 2d and 3d in space. It is shown that the classical semi-discrete primal-dual setting is more efficient for small problem sizes and moderate accuracy. However, the space-time discretization shows good stability properties and even outperforms the classical approach as the dimension in space and/or the desired accuracy increases.Comment: 20 page

The Wavelet Element Method

The Wavelet Element Method (WEM) provides a construction of multiresolution systems and biorthogonal wavelets on fairly general domains. These are split into subdomains that are mapped to a single reference hypercube. Tensor products of scaling functions and wavelets defined on the unit interval are used on the reference domain. By introducing appropriate matching conditions across the interelement boundaries, a globally continuous biorthogonal wavelet basis on the general domain is obtained. This construction does not uniquely define the basis functions but rather leaves some freedom for fulfilling additional features. In this paper we detail the general construction principle of the WEM to the 1D, 2D and 3D cases. We address additional features such as symmetry, vanishing moments and minimal support of the wavelet functions in each particular dimension. The construction is illustrated by using biorthogonal spline wavelets on the interval