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We consider the design of a passive optical telecommunication access network, where clients have to be connected to an intermediate level of distribution points (DPs) and further on to some central offices (COs) in a tree-like fashion. Each client demands a given number of fiber connections to its CO. Passive optical splitters installed at the DPs allow k connections to share a single common fiber between the DP and the CO. We consider fixed charge costs for the use of an edge of the underlying street network, of a DP, and of a CO and variable costs for installing fibers along the street edges and for installing splitters at the DPs. We present two Lagrangian decomposition approaches that decompose the problem based on the network structure and on the cost structure, respectively. The subproblems are solved using MIP techniques. We report computational results for realistic instances and compare the efficiency of the Lagrangian approaches to the solutions of an integrated MIP model.
Motivation. Modelling, parameter identification, and simulation play an important role in systems biology. Usually, the goal is to determine parameter values that minimise the difference between experimental measurement values and model predictions in a least-squares sense. Large-scale biological networks, however, often suffer from missing data for parameter identification. Thus, the least-squares problems are rank-deficient and solutions are not unique. Many common optimisation methods ignore this detail because they do not take into account the structure of the underlying inverse problem. These algorithms simply return a “solution” without additional information on identifiability or uniqueness. This can yield misleading results, especially if parameters are co-regulated and data are noisy.
Results. The Gauss-Newton method presented in this paper monitors the numerical rank of the Jacobian and converges locally, for the class of adequate problems, to a solution that is unique within the subspace of identifiable parameters. This method has been implemented in BioPARKIN, a software package that combines state-of-the-art numerical algorithms with compliance to system biology standards, most importantly SBML, and an accessible interface.
Availability. The software package BioPARKIN is available for download at http://bioparkin.zib.de .
A basic task in signal analysis is to character-
ize data in a meaningful way for analysis and classification
purposes. Time-frequency transforms are powerful strategies
for signal decomposition, and important recent generalizations
have been achieved in the setting of frame theory. In parallel
recent developments, tools from algebraic topology, traditionally
developed in purely abstract settings, have provided new insights
in applications to data analysis. In this report, we investigate some
interactions of these tools, both theoretically and with numerical
experiments, in order to characterize signals and their frame
transforms. We explain basic concepts in persistent homology
as an important new subfield of computational topology, as well
as formulations of time-frequency analysis in frame theory. Our
objective is to use persistent homology for constructing topo-
logical signatures of signals in the context of frame theory. The
motivation is to design new classification and analysis methods by
combining the strength of frame theory as a fundamental signal
processing methodology, with persistent homology as a new tool
in data analysis.
We derive a formula for the backward error of a complex number $\lambda$ when considered as an approximate eigenvalue
of a Hermitian matrix pencil or polynomial with respect to Hermitian perturbations. The same are also obtained for approximate
eigenvalues of matrix pencils and polynomials with related structures like skew-Hermitian, $*$-even and $*$-odd.
Numerical experiments suggest that in many cases there is a significant difference between the backward
errors with respect to perturbations that preserve structure and those with respect to arbitrary perturbations.
An existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial
conditions. This system is meant to model two-species phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently
introduced and analyzed in [3]. Both systems conform to the general theory developed in [5]: two parabolic PDEs, interpreted as balances of microforces and microenergy,
are to be solved for the order parameter rho and the chemical potential mu. In the system studied in this note, a phase-field equation in rho fairly more general than in [3] is coupled with a highly nonlinear diffusion equation for mu, in which the conductivity coefficient is allowed to depend nonlinearly on both variables.
Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity
(2013)
Existence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic lattice [19]; in the balance equations of microforces and microenergy, the two
unknowns are the order parameter rho and the chemical potential mu. A simpler version of the same system has recently been discussed in [8]. In this paper, a fairly more general phase-field equation for rho is coupled with a genuinely nonlinear diffusion equation for mu. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of constant atom mobility, a new and rather unusual uniqueness
proof is given, based on a suitable combination of variables.
We are concerned with a nonstandard phase field model of
Cahn-Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated
by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011,
and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE
is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient sigma in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution.
The present note deals with a nonstandard systems of differential equations describing a two-species phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors,
as the diffusion coefficient in the equation governing
the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove
a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential.
In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of
nonlinearities. Moreover, for the difference of discrete
and continuous solutions we prove an error estimate of
order one with respect to the time step.
We study the incremental facility location problem, wherein we are given an instance of the uncapacitated facility location problem. We seek an incremental sequence of opening facilities and an incremental sequence of serving customers along with their fixed assignments to facilities open in the partial sequence. Our aim is to have the solution obtained for serving the first l customers in the sequence be competitive with the optimal solution to serve any l customers. We provide an incremental framework that provides an overall competitive factor of 8 and a worst case instance that provides the lower bound of 3. The problem has applications in multi-stage network planning.
We consider a generalized version of the rooted connected facility location problem which occurs in planning of telecommunication networks with both survivability and hop-length constraints. Given a set of client nodes, a set of potential facility nodes including one predetermined root facility, a set of optional Steiner nodes, and the set of the potential connections among these nodes, that task is to decide which facilities to open, how to assign the clients to the open facilities, and how to interconnect the open facilities in such a way, that the resulting network contains at least edge-disjoint paths, each containing at most H edges, between the root and each open facility and that the total cost for opening facilities and installing connections is minimal. We study two IP models for this problem and present a branch-and-cut algorithm based on Benders decomposition for nding its solution. Finally, we report computational results.
In the connected facility location problem with buy-at-bulk edge costs we are given a set of clients with positive demands and a set of potential facilities with opening costs in an undirected graph with edge lengths obeying the triangle inequality. Moreover, we are given a set of access cable types, each with a cost per unit length and a capacity such that the cost per capacity decreases from small to large cables, and a core cable type of innite capacity. The task is to open some facilities and to connect them by a Steiner tree using core cables, and to build a forest network using access cables such that the edge capacities suce to simultaneously route all client demands unsplit to the open facilities. The objective is to minimize the total cost of opening facilities, building the core Steiner tree, and installing the access cables. In this paper, we devise a constant-factor approximation algorithm for this problem based on a random sampling technique.
We consider a generalization of the connected facility location problem where the clients must be connected to the open facilities via shared capacitated (tree) networks instead of independent shortest paths. This problem arises in the planning of ber optic telecommunication access networks, for example. Given a set of clients with positive demands, a set of potential facilities with opening costs, a set of capacitated access cable types, and a core cable type of innite capacity, one has to decide which facilities to open, how to interconnect them using a Steiner tree of innite capacity core cables, and which access cable types to install on which potential edges such that these edges form a forest and the installed capacities suce to simultaneously route the client demands to the open facilities via single paths. The objective is to minimize the total cost of opening facilities, building the core Steiner tree among them, and installing the access cables. In this paper, we devise a constant-factor approximation algorithm for problem instances where the access cable types obey economies of scale. In the special case where only multiples of a single cable type can be installed on the access edges, a variant of our algorithm achieves a performance guarantee of 6.72.
This paper presents efficient computational techniques for solving an optimization problem in cardiac defibrillation governed by the monodomain equations. Time-dependent electrical currents injected at different spatial positions act as the control. Inexact Newton-CG methods are used, with reduced gradient computation by adjoint solves. In order to reduce the computational complexity, adaptive mesh refinement for state and adjoint equations is performed. To reduce the high storage and bandwidth demand imposed by adjoint gradient and Hessian-vector evaluations, a lossy compression technique for storing trajectory data is applied. An adaptive choice of quantization tolerance based on error estimates is developed in order to ensure convergence. The efficiency of the proposed approach is demonstrated on numerical examples.
In this paper a condition number for linear-quadratic two-stage stochastic optimization problems is introduced as the Lipschitz modulus of the multifunction assigning to a (discrete) probability distribution the solution set of the problem. Being the outer norm of the Mordukhovich coderivative of this multifunction, the condition number can be estimated from above explicitly in terms of the problem data by applying appropriate calculus rules. Here, a chain rule for the extended partial second-order subdifferential recently proved by Mordukhovich and Rockafellar plays a crucial role. The obtained results are illustrated for the example of two-stage stochastic optimization problems with simple recourse.
The chemical master equation is a fundamental equation in chemical kinetics. It is an adequate substitute for the classical reaction-rate equations whenever stochastic effects become relevant. In the present paper we give a simple argument showing that the solutions of a large class of chemical master equations, including all those in which elementary reactions between two and more molecules do not generate a larger number of molecules than existed before,
are bounded in weighted $\ell_1$-spaces. As an illustration for the implications of this kind of regularity we analyze the effect of truncating the state space. This leads to an error analysis of the finite state projection of the chemical master equation, an approximation that underlies many numerical methods.
A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints
(2013)
We illustrate the solution of a mixed-integer stochastic nonlinear optimization problem in an application of power management. In this application, a coupled system consisting of a hydro power station and a wind farm is considered. The objective is to satisfy the local energy demand and sell any surplus energy on a spot market for a short time horizon. Generation of wind energy is assumed to be random, so that demand satisfaction is modeled by a joint probabilistic constraint taking into account the multivariate distribution. The turbine is forced to either operate between given positive limits or to be shut down. This introduces additional binary decisions. The numerical solution procedure is presented and results are illustrated.
We present new residual estimates based on Kato's square root theorem for spectral approximations of diagonalizable non-self-adjoint differential operators of convection-diffusion-reaction type. These estimates are incorporated as part of an hp-adaptive finite element algorithm for practical spectral computations, where it is shown that the
resulting a posteriori error estimates are reliable. Provided experiments demonstrate the efficiency and reliability of our approach.
The huge number of elementary flux modes (EFM) in genome-scale metabolic networks makes analysis based on elementary flux modes intrinsically difficult. However, it has been shown that the elementary flux modes with optimal yield often contain highly redundant information. The set of optimal-yield elementary flux modes can be compressed using modules. Up to now, this compression was only possible by first enumerating the whole set of all optimal-yield elementary flux modes.
We present a direct method for computing modules of the thermodynamically constrained optimal flux space of a metabolic network. This method can be used to decompose the set of optimal-yield elementary flux modes in a modular way and to speed up their computation. In addition, it provides a new form of coupling information that is not obtained by classical flux coupling analysis. We illustrate our approach on a set of model organisms.
Constraint-based analysis of metabolic networks has become a widely used approach in computational systems biology.
In the simplest form, a metabolic network is represented by a stoichiometric matrix and thermodynamic information on the irreversibility of certain reactions.
Then one studies the set of all steady-state flux vectors satisfying these stoichiometric and thermodynamic constraints.
We introduce a new lattice-theoretic framework for the computational analysis of metabolic networks, which focuses on the support of the flux vectors,
i.e., we consider only the qualitative information whether or not a certain reaction is active, but not its specific flux rate.
Our lattice-theoretic view includes classical metabolic pathway analysis as a special case, but turns out to be much more flexible and general,
with a wide range of possible applications.
We show how important concepts from metabolic pathway analysis, such as blocked reactions, flux coupling, or elementary modes, can be generalized to arbitrary lattice-based models. We develop corresponding general algorithms and present a number of computational results.
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.
A novel Finite Element Method (FEM) for the computational simulation in particle reinforced composite materials with many inclusions is presented. It is based on an adapted mesh which consists of triangles and parametric quadrilaterals in 2D. The number of elements and, hence, the number of degrees of freedom are proportional to the number of inclusions. The error of the method is independent of the distance of the neighboring inclusions. While being related to network methods, the approach can tackle more general settings. We present an efficient residual a posteriori error estimator which enables to compute reliable upper and lower error bounds. Several numerical examples illustrate the performance of the method and the error estimator. Moreover, it is demonstrated that the assumption of a lattice structure of inclusions can easily lead to incorrect predictions about material properties.
We formulate the static mechanical coupling of a geometrically exact Cosserat rod
to a nonlinearly elastic continuum. In this setting, appropriate coupling conditions have
to connect a one-dimensional model with director variables to a three-dimensional
model without directors.
Two alternative coupling conditions are proposed,
which correspond to two different configuration trace spaces.
For both we show existence of solutions of the coupled problems, using the direct
method of the calculus of variations. From the first-order optimality conditions
we also derive the corresponding conditions for the dual variables. These are
then interpreted in mechanical terms.
Recent research has shown that
in some practically relevant situations like multi-physics flows[11]
divergence-free mixed finite elements may have a significantly
smaller discretization error than standard non-divergence-free
mixed finite elements. In order to judge the overall performance of
divergence-free mixed finite elements, we
investigate linear solvers for the saddle point linear systems arising in $((P_k)^d,P_{k-1}^{disc})$ Scott-Vogelius finite element implementations of the incompressible Navier-Stokes equations. We investigate both direct and iterative solver methods.
Due to discontinuous pressure elements in the case of Scott-Vogelius elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements like
Taylor-Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes, and discuss extensions. For iterative methods, we test augmented Lagrangian and H-LU preconditioners with GMRES, on both full and statically condensed systems.
Several numerical experiments are provided that show these classes of solvers are well suited for use with Scott-Vogelius elements, and could deliver an interesting overall performance in several applications.
We consider discretizations for reaction-diffusion systems with nonlinear
diffusion in two space dimensions. The applied model allows to handle heterogeneous
materials and uses the chemical potentials of the involved species as primary variables.
We propose an implicit Voronoi finite volume discretization on regular Delaunay
meshes that allows to prove uniform, mesh-independent global upper and lower L1
bounds for the chemical potentials. These bounds provide the main step for a convergence
analysis for the full discretized nonlinear evolution problem. The fundamental
ideas are energy estimates, a discrete Moser iteration and the use of discrete
Gagliardo-Nirenberg inequalities. For the proof of the Gagliardo-Nirenberg inequalities
we exploit that the discrete Voronoi finite volume gradient norm in 2d coincides
with the gradient norm of continuous piecewise linear finite elements.
The authors propose a recycling MINRES scheme for a solution of subsequent self-adjoint linear systems as appearing, for example, in the Newton process for solving nonlinear equations. Ritz vectors are automatically extracted from one MINRES run and then used for self-adjoint deflation in the next. The method is designed to work with a preconditioner and arbitrary inner products. Numerical experiments with nonlinear Schrödinger equations indicate a substantial decrease in computation time when recycling is used.
Mathematical modeling often helps to provide a systems perspective on gene regulatory networks. In particular, qualitative approaches are useful when detailed kinetic information is lacking. Multiple methods have been developed that implement qualitative information in different ways, e.g., in purely discrete or hybrid discrete/continuous models. In this paper, we compare the discrete asynchronous logical modeling formalism for gene regulatory networks due to R. Thomas with piecewise affine differential equation models.
We provide a local characterization of the qualitative dynamics of a piecewise affine differential equation model using the discrete dynamics of a corresponding Thomas model. Based on this result, we investigate the consistency of higher-level dynamical properties such as attractor characteristics and reachability. We show that although the two approaches are based on equivalent information, the resulting qualitative dynamics are different. In particular, the dynamics of the piecewise affine differential equation model is not a simple refinement of the dynamics of the Thomas model.
Piecewise linear convex functions arise as integrands in stochastic programs. They are Lipschitz continuous on their domain, but do not belong to tensor product Sobolev spaces. Motivated by applying Quasi-Monte Carlo methods we show that all terms of their ANOVA decomposition, except the one of highest order, are smooth if the underlying densities are smooth and certain geometric condition is satisfied. The latter condition is generically satisfied in the normal case.
We consider convex optimization problems with $k$th order stochastic dominance constraints for $k\ge 2$. We discuss distances of random variables that are relevant for the dominance relation and establish quantitative stability results for optimal values and solution sets in terms of a suitably selected probability metrics.Moreover, we provide conditions ensuring that the optimal value function is Hadamard directionally differentiable. Finally, we discuss some implications of the results for empirical (Monte Carlo,
sample average) approximations of dominance constrained optimization models.
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.
We develop a model for the dynamic evolution of default-free and defaultable interest rates in a LIBOR framework. Utilizing the class of affine processes, this model produces positive LIBOR rates and spreads, while the dynamics are analytically tractable under defaultable forward measures. This leads to explicit formulas for CDS spreads, while semi-analytical formulas are derived for other credit derivatives. Finally, we give an application to counterparty risk.