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Automatic, or algorithmic, differentiation addresses the need for the accurate
and efficient calculation of derivative values in scientific computing. To this
end procedural programs for the evaluation of problem-specific functions are
transformed into programs that also compute the required derivative values
at the same numerical arguments in floating point arithmetic. Disregarding
many important implementation issues, we examine in this article complexity
bounds and other more mathematical aspects of the program transformation
task sketched above.
Quasi-Newton methods based on least change secant updating
formulas that solve linear equations $Ax=b$ in $n=\dim(x)=\dim(b)$ steps
can be expected to solve corresponding smooth nonlinear
systems $n$-step quadratically, i.e. with an $r$-order
of $\rho = 2^{1/n} = 1 + 1/n +O(1/n^2)$. The best rate one can
possibly expect on general problems is given by the positive root
$\rho_n$ of $\rho^n(\rho -1)=1$, for which
$\rho_n-1 = \ln(n)/n + O(1/n^2)$. To show that this upper bound is
actually achieved one usually has to impose a priori some kind of
linear independence condition on the sequence of steps taken by the
quasi-Newton iteration in question. Without any such assumptions we
establish in this paper the convergence order $\rho_n$ for the
two-sided rank one formula proposed by Schlenkrich et al in \cite{SGW06}.
It requires the evaluation of adjoint vectors, is invariant with respect
to linear transformations on the variable domain and combines the
properties of bounded deterioration and heredity.
We present a new extension of the well-known
Perron-Frobenius theorem to regular matrix pairs $(E,A)$.
The new extension is based on projector chains and is motivated from
the solution of positive differential-algebraic systems or descriptor
systems. We present several examples where the new condition holds, whereas conditions
in previous literature are not satisfied.
We present globally convergent multigrid methods for the nonsymmetric
obstacle problems as arising from the discretization of Black–Scholes models of
American options with local volatilities and discrete data. No tuning or regularization
parameters occur. Our approach relies on symmetrization by transformation
and data recovery by superconvergence.
We present a domain decomposition approach for the computation of the
electromagnetic field within periodic structures. We use a
Schwarz method with transparent boundary conditions at the interfaces of
the domains. Transparent boundary conditions are approximated by the
perfectly matched layer method (PML). To cope with Wood anomalies
appearing in periodic structures an adaptive strategy to determine
optimal PML parameters is developed. \\ We focus on the application to
typical EUV lithography line masks. Light propagation within the
multi-layer stack of the EUV mask is treated analytically. This results
in a drastic reduction of the computational costs and allows for the
simulation of next generation lithography masks
on a standard personal computer.
We present a new solver for large-scale two-body contact problems in nonlinear elasticity. It is based on an SQP-trust-region approach.
This guarantees global convergence to a first-order critical point of
the energy functional. The linearized contact conditions are
discretized using mortar elements. A
special basis transformation known from linear contact problems
allows to use a monotone multigrid solver for the inner quadratic programs.
They can thus be solved with multigrid complexity. Our algorithm
does not contain any regularization or penalization parameters,
and can be used for all hyperelastic material models.
The purpose of the paper is to apply monotone multigrid methods
to static and dynamic biomechanical contact problems.
In space, a finite element method involving a mortar
discretization of the contact conditions is used.
In time, a new contact--stabilized Newmark scheme is presented.
Numerical experiments for a two body Hertzian contact problem
and a biomechanical knee problem are reported.
For the solution of nonlinear equation systems
quasi-Newton methods based on low-rank updates are of particular interest. We analyze a class
of TR1 update formulas to approximate the system Jacobian. The local q-superlinear convergence for nonlinear problems is proved for a particular subclass of updates. Moreover, we give an estimate of the r-order of convergence. Numerical results comparing the TR1 method to Newton's and other quasi-Newton methods atr presented.
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.