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- geodesic finite elements (2)
- Adaptive Finite Elements (1)
- Atomistic-Continuum Coupling (1)
- Bridging Domain (1)
- CG (1)
- Cosserat rod (1)
- Currents (1)
- Diffeomorphic Registration (1)
- Dirichlet-Neumann (1)
- Elliptic div-grad operators (1)
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.
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.
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 $S^2$.