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In this review, we intend to clarify the underlying ideas and the relations
between various multigrid methods ranging from subset decomposition,
to projected subspace decomposition and truncated multigrid.
In addition, we present a novel globally convergent inexact active set method
which is closely related to truncated multigrid. The numerical properties
of algorithms are carefully assessed by means of a degenerate problem and
a problem with a complicated coincidence set.
We present a new inexact nonsmooth Newton method for the solution
of convex minimization problems with piecewise smooth, pointwise
nonlinearities. The algorithm consists of a nonlinear smoothing
step on the fine level and a linear coarse correction.
Suitable postprocessing guarantees global convergence
even in the case of a single multigrid step for each linear subproblem.
Numerical examples show that the overall efficiency
is comparable to multigrid for similar linear problems.
In this paper, we propose and investigate numerical methods based on QR factorization for computing all or some Lyapunov or Sacker-Sell spectral intervals for
linear differential-algebraic equations.
Furthermore, a perturbation and error analysis for these methods is presented. We
investigate how errors in the data and in the numerical integration affect the
accuracy of the approximate spectral intervals. Although we need to integrate
numerically some differential-algebraic systems on usually very long
time-intervals, under certain assumptions, it is shown that the error of the
computed spectral intervals can be controlled by the local error of numerical
integration and the error in solving the algebraic constraint.
Some numerical examples are presented to illustrate the theoretical results.
Motivated by the analysis of passive control systems, we undertake a detailed perturbation analysis of Hamiltonian matrices that have eigenvalues on the imaginary axis. We construct minimal Hamiltonian perturbations that move and coalesce eigenvalues of opposite sign characteristic to form multiple eigenvalues with mixed sign characteristics, which are then moved from the imaginary axis to specific locations in the complex plane by small Hamiltonian perturbations. We also present a numerical method to compute upper bounds for the minimal perturbations that move all eigenvalues of a given Hamiltonian matrix outside a vertical strip along the imaginary axis.
The PSurface Library
(2010)
We describe psurface, a C++ library that allows to store and access piecewise linear mappings between simplicial surfaces in $\R^2$ and $\R^3$. These mappings are stored in a graph data structure and can be constructed explicitly, by projection, or by surface simplification. Piecewise linear maps can be used, e.g., to construct boundary
approximations for finite element grids, and grid intersections for domain decomposition methods. In computer graphics the mappings allow to build level-of-detail representations as well as texture- and bump maps. We document the data structures and algorithms used and show how \psurface is used in the numerical analysis framework Dune
and the visualization software Amira.
This paper is devoted to the numerical approximation of Lyapunov and Sacker-Sell spectral intervals for linear differential-algebraic equations (DAEs). The spectral analysis for DAEs is improved and the concepts of leading directions and solution subspaces associated with spectral intervals are extended to DAEs. Numerical methods
based on smooth singular value decompositions are introduced for computing all or only some spectral intervals and their associated leading directions. The numerical algorithms as well as implementation issues are discussed in detail and numerical examples are presented to illustrate the theoretical results.
This paper presents three different adaptive algorithms for eigenvalue problems
associated with non-selfadjoint partial differential
operators. The basis for the developed algorithms is a homotopy method.
The homotopy method starts from a well-understood selfadjoint problem,
for which well-established adaptive methods are available.
Apart from the adaptive grid refinement, the progress of the homotopy as
well as the solution of the iterative method are adapted to balance the contributions
of the different error sources.
The first algorithm balances the homotopy, discretization and approximation errors with respect
to a fixed step-size $\tau$ in the homotopy.
The second algorithm combines the adaptive step-size control for the homotopy
with an adaptation in space that ensures an error below a fixed tolerance $\varepsilon$.
The third algorithm allows the complete adaptivity in space,
homotopy step-size as well as the iterative algebraic eigenvalue solver.
All three algorithms are compared in numerical examples.
In this work we propose a general framework for the structured perturbation
analysis of several classes of structured matrix polynomials in homogeneous
form, including complex symmetric, skew-symmetric, even and odd matrix polynomials. We introduce structured backward errors for approximate eigenvalues and eigenvectors and we construct minimal structured perturbations such that an approximate eigenpair is an exact eigenpair of an appropriately perturbed matrix polynomial. This work extends previous work for the non-homogeneous case (we include infinite eigenvalues) and we show that the structured backward errors improve the known unstructured backward errors.
The paper proposes goal-oriented error estimation and mesh refinement
for optimal control problems with elliptic PDE constraints using the value
of the reduced cost functional as quantity of interest. Error representation,
hierarchical error estimators, and greedy-style error indicators are derived and
compared to their counterparts when using the all-at-once cost functional as
quantity of interest. Finally, the efficiency of the error estimator and generated
meshes are demonstrated on numerical examples.
We investigate geodesic finite elements for functions with values in a space of zero curvature, like a torus or the M\"obius strip. Unlike in the general case, a closed-form expression for geodesic finite element functions is then available. This simplifies computations, and allows us to prove optimal estimates for the interpolation error in 1d and 2d. We also show the somewhat surprising result that the discretization by Kirchhoff transformation of the Richards equation proposed by Berninger et al. is a discretization by geodesic finite elements in the manifold $\mathbb{R}$ with a special metric.
We propose a generalization of the Structured Doubling Algorithm (SDA) to compute invariant subspaces
of structured matrix pencils
that arise in the context of solving linear quadratic optimal control problems.
The new algorithm is
designed to attain better accuracy when the classical Riccati equation approach for the solution of the optimal control problem is not well suited because
the stable and unstable invariant subspaces are not well separated (due to eigenvalues near or on the imaginary
axis) or in the case when the Riccati solution does not exist at all. We analyze the convergence
of the method and compare the new method with the classical SDA algorithm as well as some recent structured QR-methods.
This paper presents concepts and implementation of the finite element toolbox Kaskade 7, a flexible C++ code for solving elliptic and parabolic PDE systems. Issues such as problem formulation, assembly and adaptivity are discussed at the example of optimal control problems. Trajectory compression for parabolic optimization problems is considered as a case study.
Deflated and augmented Krylov subspace methods: Basic Facts and a Breakdown-free deflated MINRES
(2011)
In this paper we consider deflation and augmentation techniques for accelerating
the convergence of Krylov subspace methods for the solution of nonsingular linear
algebraic systems. The two techniques are conceptually different from
preconditioning. Deflation "removes" certain parts from the operator, while
augmentation adds a subspace to the Krylov subspace. Both approaches have been
used in a variety of methods and settings. For Krylov subspace methods that
satisfy a (Petrov-) Galerkin condition we show that augmentation can in general
be achieved implicitly by projecting the residuals appropriately and correcting
the approximate solutions in a final step. In this context, we analyze known
methods to deflate CG, GMRes and MinRes. Our analysis reveals that the recently
proposed RMinRes method can break down. We show how such breakdowns can be
avoided by choosing a special initial guess, and we derive a breakdown-free
deflated MinRes method. In numerical experiments we study the properties of
different variants of MinRes analyzed in this paper.
One of the most challenging problems in dynamic concurrent multiscale
simulations is the reflectionless transfer of physical quantities between the
different scales. In particular, when coupling molecular dynamics and finite
element discretizations in solid body mechanics, often spurious wave reflections
are introduced by the applied coupling technique. The reflected waves are
typically of high frequency and are arguably of little importance in the domain
where the finite element discretization drives the simulation.
In this work, we provide an analysis of this phenomenon.
Based on the gained
insight, we derive a new coupling approach, which neatly separates high and low
frequency waves. Whereas low frequency waves are permitted to
bridge the scales, high frequency waves can be removed by applying damping techniques without affecting the coupled share of the solution. As a consequence, our new method almost completely eliminates unphysical wave reflections and deals in a consistent way with waves of arbitrary frequencies. The separation of
wavelengths is achieved by employing a discrete $L^2$-projection, which acts as a
low pass filter. Our coupling constraints enforce matching in the range of this projection. With respect to the numerical realization this approach
has the advantage of a small number of constraints, which is computationally
efficient. Numerical results in one and two dimensions confirm our theoretical
findings and illustrate the performance of our new weak coupling approach.
We investigate optimal elliptic
regularity (within the scale of Sobolev spaces) of anisotropic
div--grad operators in three dimensions at a multi-material vertex on
the Neumann boundary part of a polyhedral spatial domain. The
gradient of a solution to the corresponding elliptic PDE (in a
neighbourhood of the vertex) is integrable to an index greater than
three.
We consider Large Deformation Diffeomorphic Metric Mapping of general $m$-currents. After stating an optimization algorithm in the function space of admissable morph generating velocity fields, two innovative aspects in this framework are presented and numerically investigated: First, we spatially discretize the velocity field with conforming adaptive finite elements and discuss advantages of this new approach. Second, we directly compute the temporal evolution of discrete $m$-current attributes.
We consider the mechanical coupling of a geometrically exact Cosserat rod to a linear elastic continuum. The coupling conditions are formulated in the nonlinear rod configuration space. We describe a Dirichlet--Neumann algorithm for the coupled system, and use it to simulate the static stresses in a human knee joint, where the Cosserat rods are models for the ligaments.
In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and Newton type methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy compression
maintaining a certain error bound turns out to be sufficient.
We introduce geodesic finite elements as a conforming way to discretize partial differential equations for functions $v : \Omega \to M$, where $\Omega$ is an open subset of $\R^d$ and $M$ is a Riemannian manifold. These geodesic finite elements naturally generalize standard first-order
finite elements for Euclidean spaces. They also generalize the geodesic finite elements proposed for $d=1$ by the author. Our formulation is equivariant under isometries of $M$, and hence preserves objectivity of continuous problem formulations. We concentrate on partial differential equations that can be formulated as minimization problems. Discretization leads to algebraic minimization problems on product manifolds $M^n$. These can be solved efficiently
using a Riemannian trust-region method. We propose a monotone multigrid method to solve the constrained inner problems with linear multigrid speed. As an example we numerically compute harmonic maps from a domain in $\R^3$
to $S^2$.
Logical modeling of biological regulatory networks gives rise to a representation of the system's dynamics as a so-called state transition graph. Analysis of such a graph in its entirety allows for a comprehensive understanding of the functionalities and behavior of the modeled system. However, the size of the vertex set of the graph is exponential in the number of the network components making analysis costly, motivating development of reduction methods. In this paper, we present results allowing for a complete description of an asynchronous state transition graph of a Thomas network solely based on the analysis of the subgraph induced by certain extremal states. Utilizing this notion, we compare the behavior of a simple multi-valued network and a corresponding Boolean network and analyze the conservation of dynamical properties between them. Understanding the relation between such coarser and finer models is a necessary step towards meaningful network reduction as well as model refinement methods.