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A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely
representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure is presented that uses linear programming to find a good approximation to the sparse solution on a given refinement level. Then only those parts of the mesh are refined that belong to nonzero expansion coefficients. Error estimates for this procedure are refined and the behavior of the procedure is demonstrated via some simple elliptic model problems.
Interior Point Methods in Function Space for State Constraints - Inexact Newton and Adaptivity
(2009)
We consider an interior point method in function space for PDE constrained optimal control problems with state constraints. Our emphasis is on the construction and analysis of an algorithm that integrates a Newton path-following method with adaptive grid refinement. This is done in the framework of inexact Newton methods in function space, where the discretization error of each Newton step is controlled by adaptive grid refinement in the innermost loop. This allows to perform most of the required Newton steps on coarse grids, such that the overall computational time is dominated by the last few steps. For this purpose we propose an a-posteriori error estimator for a problem suited norm.
A continuity result for Nemyckii Operators and some applications in PDE constrained optimal control
(2008)
This work explores two applications of a classical result on the continuity of Nemyckii operators to optimal control with PDEs. First, we present an alternative approach to the analysis of Newton's method for function space problems involving semi-smooth Nemyckii operators. A concise proof for superlinear convergence is presented, and sharpened bounds on the rate of convergence are derived. Second, we derive second order sufficient conditions for problems, where the underlying PDE has poor regularity properties. We point out that the analytical structure in both topics is essentially the same.
We propose and analyse an interior point path-following method in function space
for state constrained optimal control. Our emphasis is on proving convergence in
function space and on constructing a practical path-following algorithm. In particular, the introduction of a pointwise damping step leads to a very efficient method, as verified by numerical experiments.
We consider hybrid systems of differential-algebraic equations and present
a general framework for general nonlinear over- and underdetermined hybrid
systems that allows the
analysis of existence and uniqueness and the application of index reduction
methods for hybrid differential-algebraic systems.
A particular difficulty in the numerical simulation of hybrid systems is
(numerical) chattering, i.e., fast oscillations between modes of operations.
A regularization technique using sliding modes allows to regularize the
system behavior in the case of chattering.
Further, we show how chattering behavior during the numerical solution can
be prevented using sliding mode simulation. The advantage of the sliding mode
simulation is illustrated by numerical examples.
A generalization of the method of Chu, Liu and Mehrmann
for the computation of the Hamiltonian real Schur form is presented.
The new method avoids some of the difficulties that may arise when
a Hamiltonian matrix has tightly clustered groups of eigenvalues.
A detailed analysis of the method is presented and several numerical examples demonstrate the superior behavior of the method.
We present and analyze novel hierarchical a posteriori error estimates
for self-adjoint elliptic obstacle problems.
Our approach differs from straightforward, but non-reliable estimators~\cite{RHWHoppe_RKornhuber_1994a}
by an additional extra term accounting for the deviation
of the discrete free boundary in the localization step.
We prove efficiency and reliability
on a saturation assumption and a regularity condition on the underlying grid.
Heuristic arguments suggest
that the extra term is of higher order and preserves full locality.
Numerical computations confirm our theoretical findings.
The main focus of this paper is on an a-posteriori analysis for the method of proper orthogonal decomposition (POD) applied to optimal control problems governed by
parabolic and elliptic PDEs. Based on a perturbation method it is deduced how far the suboptimal
control, computed on the basis of the POD model, is from the (unknown)
exact one. Numerical examples illustrate the realization of the proposed approach for linear-quadratic problems governed by parabolic and elliptic partial differential equations.
The complexity of molecular kinetics can be reduced significantly by a restriction to metastable conformations which are almost invariant sets of molecular dynamical systems. With the Robust Perron Cl uster Analysis PCCA+, developed by Weber and Deuflhard, we have a tool available which can be used to identify these conformations from a transition probability matrix. This method can also be applied to the corresponding transition rate matrix which provides important information concerning transition pathways of single molecules. In the present paper, we explain the relationship between these tw o concepts and the extraction of conformation kinetics from transition rates. Moreover, we show how transition rates can be approximated and conclude with numerical examples.
Wigner functions are functions on classical phase space, which are in one-to-one correspondence to square integrable functions on configuration space. For molecular quantum systems, classical transport of Wigner functions provides the basis of asymptotic approximation methods in the high energy regime. The article addresses the sampling of Wigner functions by Monte Carlo techniques. The approximation step is realized by an adaption of the Metropolis algorithm for real-valued functions with disconnected support. The quadrature, which computes values of the Wigner function, uses importance sampling with a Gaussian weight function. The numerical experiments combine the sampling with a surface hopping algorithm for non-adiabatic quantum dynamics. In agreement with theoretical considerations, the obtained results show an accuracy of two to four percent.