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Multi-Level Monte-Carlo Finite Element Methods for stochastic elliptic variational inequalities
(2013)
Multi-Level Monte-Carlo Finite Element (MLMC--FE) methods
for the solution of stochastic elliptic variational inequalities
are introduced, analyzed, and numerically investigated.
Under suitable assumptions on the random diffusion coefficient,
the random forcing function, and the deterministic obstacle,
we prove existence and uniqueness of solutions of ``mean-square''
and ``pathwise'' formulations.
Suitable regularity results for deterministic,
elliptic obstacle problems lead
to uniform pathwise error bounds, providing
optimal-order error estimates of the statistical error
and upper bounds for the
corresponding computational cost for
classical Monte--Carlo and novel MLMC--FE methods.
Utilizing suitable multigrid solvers for the occurring sample problems,
in two space dimensions
MLMC--FE methods then provide numerical
approximations of the expectation of the random solution
with the same order of efficiency as for a corresponding
deterministic problem, up to logarithmic terms.
Our theoretical findings are illustrated by numerical experiments.
We introduce and analyze nonsmooth Schur-Newton methods for a class of nonsmooth saddle point problems. The method is able to solve problems where the primal energy decomposes into a convex smooth part and a convex separable but nonsmooth part. The method is based on nonsmooth Newton techniques for an equivalent unconstrained dual problem. Using this we show that it is globally convergent even for inexact evaluation of the linear subproblems.
We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad-div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad-div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator's matrices are identical to those of grad-div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad-div stabilization, and in all cases, solutions are found to be very similar.
The computation of guided modes in photonic crystal wave-guides is a key issue in the process of designing devices in photonic communications. Existing methods, such as the super-cell method, provide an efficient computation of well-confined modes. However, if the modes are not well-confined, the modelling error of the super-cell method becomes prohibitive and advanced methods applying transparent boundary conditions for periodic media are needed. In this work we demonstrate the numerical realization of a recently proposed Dirichlet-to-Neumann approach and compare the results with those of the super-cell method. For the resulting non-linear eigenvalue problem we propose an iterative solution based on Newton's method and a direct solution using Chebyshev interpolation of the non-linear operator. Based on the Dirichlet-to-Neumann approach, we present a formula for the group velocity of guided modes that can serve as an objective function in the optimization of photonic crystal wave-guides.
We study the perturbation theory of structured matrices under structured
rank one perturbations, with emphasis on matrices that are unitary, orthogonal, or symplectic
with respect to an indefinite inner product. The rank one perturbations are not necessarily of
arbitrary small size (in the sense of norm).
In the case of sesquilinear forms, results on selfadjoint matrices can be applied to
unitary matrices by using the Cayley transformation, but
in the case of real or complex symmetric or skew-symmetric bilinear forms
additional considerations are necessary. For complex symplectic matrices, it turns out that
generically (with respect to the perturbations) the behavior of the Jordan form of the
perturbed matrix follows the pattern established earlier for unstructured matrices and their unstructured perturbations, provided the specific properties of the Jordan
form of complex symplectic matrices are accounted for. For instance,
the number of Jordan blocks of fixed odd size corresponding to the eigenvalue $1$ or $-1$ have to be even.
For complex orthogonal matrices, it is shown that the behavior of
the Jordan structures corresponding to the original eigenvalues that are not moved by
perturbations follows again the pattern established earlier for unstructured matrices,
taking into account the specifics of Jordan forms of complex orthogonal
matrices.
The proofs are based on general results developed in the paper concerning Jordan forms of
structured matrices (which include in particular the classes of orthogonal and symplectic matrices)
under structured rank one perturbations. These results are presented and proved in the framework of
real as well as of complex matrices.
Periodic Solutions to Dissipative Hyperbolic Systems. II: Hopf Bifurcation for Semilinear Problems
(2013)
We consider boundary value problems for semilinear hyperbolic systems of the type
$$
\partial_tu_j + a_j(x,\la)\partial_xu_j + b_j(x,\la,u) = 0, \; x\in(0,1), \;j=1,\dots,n
$$
with smooth coefficient functions $a_j$
and $b_j$
such that
$b_j(x,\la,0) = 0$ for all $x \in [0,1]$, $\la \in \R$, and $j=1,\ldots,n$.
We state conditions for Hopf bifurcation, i.e.,
for existence, local uniqueness (up to phase shifts), smoothness and smooth dependence
on $\la$
of time-periodic solutions bifurcating from the zero stationary solution. Furthermore,
we derive a formula which determines the bifurcation direction.
The proof is done by means of a Liapunov-Schmidt reduction procedure.
For this purpose, Fredholm properties of the linearized
system and implicit function
theorem techniques are used.
There are at least two distinguishing features of Hopf bifurcation theorems for hyperbolic PDEs in comparison with those for parabolic PDEs or for ODEs:
First, the question if a non-degenerate time-periodic solution depends smoothly on the system parameters
is much more delicate. And second,
a sufficient amount of dissipativity is needed in the system, and a priori
it is not clear how to verify this in terms of the data of the PDEs and of the boundary conditions.
Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems
(2013)
This paper concerns linear first-order hyperbolic systems in one space dimension of the type
$$
\partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; x \in (0,1),\; j=1,\ldots,n,
$$
with periodicity conditions in time and reflection boundary conditions in space. We state a kind of dissipativity condition (depending on the coefficients $a_j$ and $b_{jj}$ and the boundary reflection coefficients), which implies Fredholm solvability of the problem, i.e., either there is a nontrivial solution to the homogeneous problem (in this case the space of such solutions has finite dimension) or the nonhomogeneous problem is uniquely solvable for any right-hand side (in this case the solution depends continuously on the right-hand side). In particular, under those conditions no small denominator effects occur.
Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity.
Finally, in the case that all coefficients $a_j$ are $t$-independent, we show that the solutions are $C^\infty$-smooth if the data are $C^\infty$-smooth.
We examine robustness of exponential dichotomies of boundary value problems for general linear first-order one-dimensional hyperbolic systems. The boundary conditions are supposed to be of types ensuring smoothing solutions in finite time, which includes reflection boundary conditions. We show that the dichotomy survives in the space of continuous functions under small perturbations of all coefficients in the differential equations.
We give an exposition of recent results on regularity and Fredholm properties for first-order one-dimensional hyperbolic PDEs. We show that large classes of boundary operators cause an effect that smoothness increases with time. This property is the key in finding regularizers
(parametrices) for hyperbolic problems. We construct regularizers for periodic problems for dissipative first-order linear hyperbolic PDEs and show that these problems are modeled by Fredholm operators of index zero.
We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic $\lambda$-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory.
Grad-div stabilization has been proved to be a very useful tool in discretizations
of incompressible flow problems. Standard error analysis for inf-sup stable conforming pairs of
finite element spaces predicts that the stabilization parameter should be optimally chosen
to be $\mathcal O(1)$. This paper revisits this choice for the Stokes equations on the basis
of minimizing the $H^1(\Omega)$ error of the velocity and the $L^2(\Omega)$ error of the pressure.
It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle
than known so far in the literature. It depends on the used norm,
the solution, the family of finite
element spaces, and the type of mesh. Depending on the situation, the
optimal
stabilization parameter might range from being very small to very large.
The analytic results
are supported by numerical examples.
Complete damage in linear elastic materials — Modeling, weak formulation and existence results
(2013)
In this work, we introduce a degenerating PDE system with a time-depending
domain for complete damage processes under time-varying
Dirichlet boundary conditions. The evolution of the system is
described by a doubly nonlinear differential inclusion for the damage
process and a degenerating quasi-static balance equation for the displacement field
which are strongly nonlinearly coupled.
In our proposed model, the material
may completely disintegrate which is indispensable for a realistic modeling of
damage processes in elastic materials. Complete damage theories
lead to several mathematical problems since, for instance, coercivity properties
of the free energy are lost and, therefore, several difficulties arise.
For the introduced complete damage model, we propose a classical
formulation and a corresponding suitable weak formulation in an
$SBV$-framework. The main aim is to prove existence of weak solutions
for the introduced degenerating model. In addition, we show that the classical
differential inclusion can be regained from the notion of weak solutions under
certain regularity assumptions which is a novelty in the theory of complete damage
models of this type.
For the existence results, we had to handle the following problem:
During the damage process it might occur that not completely damaged material regions are isolated
from the Dirichlet boundary. In this case, the
deformation field cannot be controlled in the transition from incomplete
to complete damage. To tackle this problem, we consider the evolution
process on a time-depending domain. In this context, two major challenges
arise:
Firstly, the time-dependent domain approach leads to jumps in the energy
which have to be accounted for in the energy inequality of the notion of
weak solutions. To handle this problem, several energy estimates are established
by $\Gamma$-convergence techniques. Secondly, the time-depending domain
might have bad smoothness properties such that Korn's inequality cannot be
applied. To this end, a covering result for such sets with smooth
compactly embedded domains has been shown.
This paper discusses adaptive finite element methods (AFEMs) for the solution of elliptic eigenvalue problems associated with partial differential operators. An adaptive method based on nodal-patch refinement leads to an asymptotic error reduction property for the computed sequence of simple eigenvalues and eigenfunctions. This justifies the use of the proven saturation property for a class of reliable and efficient hierarchical a posteriori error estimators. Numerical experiments confirm that the saturation property is present even for very coarse meshes for many examples; in other cases the smallness assumption on the initial mesh may be severe.
Flux variability analysis (FVA) is an important tool to further analyze the results obtained by flux balance analysis (FBA) on genome-scale metabolic networks. Standard FVA may predict unbounded fluxes through some reactions in the network even if the nutrient uptake rate is bounded. These fluxes violate the second law of thermodynamics. They may be eliminated by extending flux variability analysis with thermodynamic constraints.
We present a new algorithm for efficient flux variability (and flux balance) analysis with thermodynamic constraints, suitable for analyzing genome-scale metabolic networks. We first show that flux balance analysis with thermodynamic constraints is NP-hard. Then we derive a theoretical tractability result, which can be applied to metabolic networks in practice. We use this result to develop a new constraint programming algorithm Fast-tFVA for fast flux variability analysis with thermodynamic constraints (tFVA). Computational comparisons with previous methods demonstrate the efficiency of the new method. For tFVA, a speed-up of factor 30-300 is achieved.
In an analysis of genome-scale metabolic networks in the BioModels database, we found that in 485 out of 716 networks additional irreversible or fixed reactions could be detected.
We discuss the possibility of computing eigenpairs of some prototypical linear second-order self-adjoint elliptic partial differential operator (or its high-resolution finite element discretization) by numerical upscaling techniques. We compute a low-dimensional generalized finite element space that preserves small eigenvalues in a superconvergent way. The approximate eigenpairs are then obtained by solving the corresponding low-dimensional algebraic eigenvalue problem. The rigorous error bounds are based on two-scale decompositions of H1 by means of a certain Clement-type quasi-interpolation operator.
We present a discretization for dynamic large deformation contact problems without friction. Our model is based on Hamilton’s principle, which avoids the explicit appearance of the contact forces. The resulting differential inclusion is discretized in time using a modified midpoint rule. This modification, which concerns the evaluation of the generalized gradient, allows to achieve energy dissipativity. For the space discretization we use a dual-basis mortar method. The resulting spatial algebraic
problems are nonconvex minimization problems with nonconvex inequality constraints. These can be solved efficiently using a trust-region SQP framework with a monotone multigrid inner solver.
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coeffcient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear finescale computations in small patches that have a diameter of order H |log(H)| where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coeffcients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh
size. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.
We discuss shape optimization problems for cylindrical tubes that are loaded by time-dependent applied force. This is a problem of shape optimization that leads to optimal control in linear elasticity theory. We determine the optimal thickness of a cylindrical tube minimizing the deformation of the tube under the influence of the external force. The main difficulty is that the state equation is a hyperbolic partial differential equation of 4th order. First order necessary conditions for the optimal solution are derived. Based on them, a numerical method is set up and numerical examples are presented.
Optimal Thickness of a Cylindrical Shell -- An Optimal Control Problem in Linear Elasticity Theory
(2012)
In this paper we discuss optimization problems for cylindrical tubes which are loaded by an applied force. This is a problem of optimal control in linear elasticity theory (shape optimization). We are looking for an optimal thickness minimizing the deflection (deformation) of the tube under the influence of an external force.
From basic equations of mechanics, we derive the equation of deformation. We apply the displacement approach from shell theory and make use of the hypotheses of
Mindlin and Reissner. A corresponding optimal control problem is formulated and first order necessary conditions for the optimal solution (optimal thickness) are derived.
We present numerical examples which were solved by the finite element method.
In incompressible flows with vanishing
normal velocities at the boundary, irrotational forces in the momentum
equations should be balanced
completely by the pressure gradient.
Unfortunately, nearly all available discretization methods for incompressible flows violate this property.
The origin of the problem is that discrete velocity approximations
of incompressible flows are usually not
divergence-free. Hence, the use of divergence-free velocity reconstructions is
proposed wherever an $L^2$ scalar product appears in the discrete
variational formulation.
The approach is illustrated and applied to a nonconforming MAC-like discretization for unstructured Delaunay grids.
It is numerically demonstrated that a divergence-free velocity reconstruction based on the lowest-order Raviart-Thomas element
increases the robustness and accuracy of an existing convergent discretization, when irrotational forces appear in the momentum equations.