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In this paper, we deal with a hydraulic reservoir optimization problem with uncertainty on
inflows in a joint chance constrained programming setting. In particular, we will consider inflows with
a persistency effect, following a causal time series model, and examine the impact of the ”Gaussian”
assumption for such inflows. We present an iterative algorithm for solving similarly structured joint
chance constrained programming problems that requires a Slater point and the computation of gradients.
Several alternatives to the joint chance constraint problem are presented. In particular, we present an
individual chance constraint problem and a robust model. We illustrate the interest of joint chance
constrained programming by comparing results obtained on a realistic hydro-valley with those obtained
from the alternative models. Despite the fact that the alternative models often require less hypothesis
on the law of the inflows, we show that they yield conservative and costly solutions. The simpler models,
such as the individual chance constraint one, are shown to yield insufficient robustness and are therefore
not useful. We therefore conclude that Joint Chance Constrained programming appears as a technique
offering a good trade-off between cost and robustness and can be tractable for complex realistic models.
Analysis of the second phase of the GMRES convergence for a convection-diffusion model problem
(2012)
It is well konwn that GMRES applied to linear algebraic systems arising from a convection-diffusion model problem that has been discretized by the streamline upwind Petrov-Galerkin (SUPG) method, typically displays two distinct phases of convergence: a slow initial phase followed by a convergence acceleration in the second phase. This paper complements the known results on the length of the initial phase by analyzing how the acceleration in the second phase of convergence is related to the mesh Peclet number and the choice of the stabilization parameter in the SUPG discretization. The analysis is based on some new expressions and bounds for the GMRES residuals, which can be of general interest.
The paper examines the applicability of mathematical programming methods to the simultaneous optimization of the structure and the operational parameters of a combined-cycle-based cogeneration plant. The optimization problem is formulated as a nonconvex mixed-integer nonlinear problem (MINLP) and solved by the MINLP solver LaGO. The algorithm generates a convex relaxation of the MINLP and applies a Branch and Cut algorithm to the relaxation. Numerical results for different demands for electric power and process steam are discussed and a sensitivity analysis is performed.
We provide lower estimates for the norm of gradients of Gaussian
distribution functions and apply the results obtained to a special class of
probabilistically constrained optimization problems. In particular, it is shown
how the precision of computing gradients in such problems can be controlled
by the precision of function values for Gaussian distribution functions. Moreover,
a sensitivity result for optimal values with respect to perturbations of the
underlying random vector is derived. It is shown that the so-called maximal
increasing slope of the optimal value with respect to the Kolmogorov distance
between original and perturbed distribution can be estimated explicitly from
the input data of the problem.
Primal heuristics are an important component of state-of-the-art codes for mixed integer programming. In this paper, we focus on primal heuristics that only employ computationally inexpensive procedures such as rounding and logical deductions (propagation). We give an overview of eight different approaches. To assess the impact of these primal heuristics on the ability to find feasible solutions, in particular early during search, we introduce a new performance measure, the primal integral. Computational experiments evaluate this and other measures on MIPLIB~2010 benchmark instances.
Energetic reasoning is one of the most powerful propagation algorithms in cumulative scheduling. In practice, however, it is not commonly used because it has a high running time and its success highly depends on the tightness of the variable bounds. In order to speed up energetic reasoning, we provide an easy-to-check necessary condition for energetic reasoning to detect infeasibilities. We present an implementation of energetic reasoning that employs this condition and that can be parametrically adjusted to handle the trade-off between solving time and propagation overhead. Computational results on instances from the PSPLIB are provided. These results show that using this condition decreases the running time by more than a half, although more search nodes need to be explored.
We present Undercover, a primal heuristic for mixed-integer nonlinear programming (MINLP). The heuristic constructs a mixed-integer linear subproblem (sub-MIP) of a given MINLP by fixing a subset of the variables. We solve a set covering problem to identify a minimal set of variables which need to be fixed in order to linearise each constraint. Subsequently, these variables are fixed to approximate values, e.g. obtained from a linear outer approximation. The resulting sub-MIP is solved by a mixed-integer linear programming solver. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. Although general in nature, the heuristic seems most promising for mixed-integer quadratically constrained programmes (MIQCPs). We present computational results on a general test set of MIQCPs selected from the MINLPLib.
Large neighborhood search (LNS) heuristics are an important component of modern branch-and-cut algorithms for solving mixed-integer linear programs (MIPs). Most of these LNS heuristics use the LP relaxation as the basis for their search, which is a reasonable choice in case of MIPs. However, for more general problem classes, the LP relaxation alone may not contain enough information about the original problem to find feasible solutions with these heuristics, e.g., if the problem is nonlinear or not all constraints are present in the current relaxation. In this paper, we discuss a generic way to extend LNS heuristics that have been developed for MIP to constraint integer programming (CIP), which is a generalization of MIP in the direction of constraint programming (CP). We present computational results of LNS heuristics for three problem classes: mixed-integer quadratically constrained programs, nonlinear pseudo-Boolean optimization instances, and resource-constrained project scheduling problems. Therefore, we have implemented extended versions of the following LNS heuristics in the constraint integer programming framework SCIP: Local Branching, RINS, RENS, Crossover, and DINS. Our results indicate that a generic generalization of LNS heuristics to CIP considerably improves the success rate of these heuristics.
We propose a hybrid approach for solving the resource-constrained project scheduling problem which is an extremely hard to solve combinatorial optimization problem of practical relevance. Jobs have to be scheduled on (renewable) resources subject to precedence constraints such that the resource capacities are never exceeded and the latest completion time of all jobs is minimized. The problem has challenged researchers from different communities, such as integer programming (IP), constraint programming (CP), and satisfiability testing (SAT). Still, there are instances with 60 jobs which have not been solved for many years. The currently best known approach, lazyFD, is a hybrid between CP and SAT techniques. In this paper we propose an even stronger hybridization by integrating all the three areas, IP, CP, and SAT, into a single branch-and-bound scheme. We show that lower bounds from the linear relaxation of the IP formulation and conflict analysis are key ingredients for pruning the search tree. First computational experiments show very promising results. For five instances of the well-known PSPLIB we report an improvement of lower bounds. Our implementation is generic, thus it can be potentially applied to similar problems as well.
Learning during search allows solvers for discrete optimization problems to remember parts of the search that they have already performed and avoid revisiting redundant parts. Learning approaches pioneered by the SAT and CP communities have been successfully incorporated into the SCIP constraint integer programming platform. In this paper we show that performing a heuristic constraint programming search during root node processing of a binary program can rapidly learn useful nogoods, bound changes, primal solutions, and branching statistics that improve the remaining IP search.
Mixed integer programming (MIP) has become one of the most important techniques in Operations Research and Discrete Optimization. SCIP (Solving Constraint Integer Programs) is currently one of the fastest non-commercial MIP solvers. It is based on the branch-and-bound procedure in which the problem is recursively split into smaller subproblems, thereby creating a so-called branching tree. We present ParaSCIP, an extension of SCIP, which realizes a parallelization on a distributed memory computing environment. ParaSCIP uses SCIP solvers as independently running processes to solve subproblems (nodes of the branching tree) locally. This makes the parallelization development independent of the SCIP development. Thus, ParaSCIP directly profits from any algorithmic progress in future versions of SCIP. Using a first implementation of ParaSCIP, we were able to solve two previously unsolved instances from MIPLIB2003, a standard test set library for MIP solvers. For these computations, we used up to 2048 cores of the HLRN~II supercomputer.
We provide a computational study of the performance of a state-of-the-art solver for nonconvex mixed-integer quadratically constrained programs (MIQCPs). Since successful general-purpose solvers for large problem classes necessarily comprise a variety of algorithmic techniques, we focus especially on the impact of the individual solver components. The solver SCIP used for the experiments implements a branch-and-cut algorithm based on linear outer approximation to solve MIQCPs to global optimality. Our analysis is based on a set of 86 publicly available test instances.
MIPLIB 2010
(2012)
This paper reports on the fifth version of the Mixed Integer Programming Library. The MIPLIB 2010 is the first MIPLIB release that has been assembled by a large group from academia and from industry, all of whom work in integer programming. There was mutual consent that the concept of the library had to be expanded in order to fulfill the needs of the community. The new version comprises 361 instances sorted into several groups. This includes the main benchmark test set of 87 instances, which are all solvable by today's codes, and also the challenge test set with 164 instances, many of which are currently unsolved. For the first time, we include scripts to run automated tests in a predefined way. Further, there is a solution checker to test the accuracy of provided solutions using exact arithmetic.
In this paper we formulate a boundary layer approximation
for an Allen--Cahn-type equation involving a small parameter $\eps$. Here, $\eps$ is related to the thickness of the boundary layer and we are interested in the limit when $\eps$ tends to $0$ in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L^2-gradient flow with the corresponding Allen--Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Gamma- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.
A theorem on error estimates for smooth nonlinear programming
problems in Banach spaces is proved that can be used to derive
optimal error estimates for optimal control problems. This theorem is applied
to a class of optimal control problems for quasilinear elliptic equations.
The state equation is approximated by a finite element scheme, while different
discretization methods are used for the control functions. The distance of
locally optimal controls to their discrete approximations is estimated.
Our model of the bovine estrous cycle is a set of ordinary differential equations which generates hormone profiles of successive estrous cycles with several follicular waves per cycle. It describes the growth and decay of the follicles and the corpus luteum, as well as the change of the key substances over time. In this work we describe recent improvements of this model, including the introduction of new components, and elimination of time delays. We validate our model by showing that the simulations agree with observations from synchronization studies and with measured progesterone data after a single dose administration of synthetic prostaglandin F2alpha.
Modelling, parameter identification, and simulation play an important rôle in Systems Biology. In recent years, various software packages have been established for scientific use in both licencing types, open source as well as commercial. Many of these codes are based on inefficient and mathematically outdated algorithms. By introducing the package BioPARKIN recently developed at ZIB, we want to improve this situation significantly. The development of the software BioPARKIN involves long standing mathematical ideas that, however, have not yet entered the field of Systems Biology, as well as new ideas and tools that are particularly important for the analysis of the dynamics of biological networks. BioPARKIN originates from the package PARKIN, written by P.Deuflhard and U.Nowak, that has been applied successfully for parameter identification in physical chemistry for many years.
This study presents a differential equation model for the feedback mechanisms between
Gonadotropin-releasing Hormone (GnRH), Follicle-Stimulating Hormone (FSH), Luteinizing
Hormone (LH), development of follicles and corpus luteum, and the production of estradiol
(E2), progesterone (P4), inhibin A (IhA), and inhibin B (IhB) during the female menstrual
cycle. In contrast to other models, this model does not involve delay differential equations
and is based on deterministic modelling of the GnRH pulse pattern, which allows for faster
simulation times and efficient parameter identification. These steps were essential to tackle
the task of developing a mathematical model for the administration of GnRH analogues. The
focus of this paper is on model development for GnRH receptor binding and the integration of
a pharmacokinetic/pharmacodynamic model for the GnRH agonist Nafarelin and the GnRH
antagonist Cetrorelix into the menstrual cycle model. The final mathematical model describes
the hormone profiles (LH, FSH, P4, E2) throughout the menstrual cycle in 12 healthy women.
Moreover, it correctly predicts the changes in the cycle following single and multiple dose
administration of Nafarelin or Cetrorelix at different stages in the cycle.
We propose a new approach to competitive analysis by introducing the novel concept of online approximation schemes. Such scheme algorithmically constructs an online algorithm with a competitive ratio arbitrarily close to the best possible competitive ratio for any online algorithm. We study the problem of scheduling jobs online to minimize the weighted sum of completion times on parallel, related, and unrelated machines, and we derive both deterministic and randomized algorithms which are almost best possible among all online algorithms of the respective settings. Our method relies on an abstract characterization of online algorithms combined with various simplifications and transformations. We also contribute algorithmic means to compute the actual value of the best possible competitive ratio up to an arbitrary accuracy. This strongly contrasts all previous manually obtained competitiveness results for algorithms and, most importantly, it reduces the search for the optimal competitive ratio to a question that a computer can answer. We believe that our method can also be applied to many other problems and yields a completely new and interesting view on online algorithms.
Some optimal control problems for linear and nonlinear ordinary differential equations related to the optimal switching between
different magnetic fields are considered. The main aim is to move an electrical initial current by a controllable
voltage in shortest time to a desired terminal current and to hold it afterwards. Necessary optimality conditions are derived by
Pontryagin's principle and a Lagrange technique. In the case of a linear system, the principal structure of time-optimal controls is
discussed. The associated optimality systems are solved by a one-shot strategy
using a multigrid software package. Various numerical examples are discussed.
Global higher integrability of minimizers of variational problems with mixed boundary conditions
(2012)
We consider integral functionals with densities of p-growth, with respect to gradients, on a Lipschitz domain with mixed boundary conditions. The aim of this paper is to prove that, under uniform estimates within certain classes of p-growth and coercivity assumptions on the density, the minimizers are of higher integrability order, meaning that they belong to the space of first order Sobolev functions with an integrability of order $p+\epsilon$ for a uniform $\epsilon >0$. The results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys.
Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints
(2012)
A global higher differentiability result in Besov
spaces is proved for the displacement fields of linear elastic models
with self contact.
Domains with cracks are studied, where nonpenetration
conditions/Signorini conditions are imposed on the crack faces.
It is shown that
in a neighborhood of crack tips (in 2D) or
crack fronts (3D) the displacement fields are
$B^{3/2}_{2,\infty}$ regular.
The proof relies on a difference
quotient argument for the directions tangential to the crack. In order
to obtain the regularity estimates also in the normal direction, an
argument due to
Ebmeyer/Frehse/Kassmann is modified.
The methods are then applied to further examples like
contact problems with nonsmooth rigid foundations, to a model with
Tresca friction and
to minimization problems with
nonsmooth energies and constraints as they occur for instance in the modeling of
shape memory alloys.
Based on Falk's approximation Theorem for variational
inequalities, convergence rates for FE-discretizations of contact
problems are derived relying on the proven regularity properties.
Several numerical examples illustrate the theoretical results.
Existence result for a class of generalized standard materials with thermomechanical coupling
(2012)
This paper deals with the study of a three-dimensional model of thermomechanical coupling for viscous solids exhibiting hysteresis effects. This model is written in accordance with the formalism of generalized standard materials. It is composed by the momentum equilibrium equation combined with the flow rule, which describes some stress-strain dependance, and the heat-transfer equation. An existence result for this thermodynamically consistent problem is obtained by using a fixed-point argument and some qualitative properties of the solutions are established.
In this paper we revisit models for the description of the evolution of crystalline films with anisotropic surface energies.
We prove equivalences of symmetry properties of anisotropic surface energy models commonly used in the literature.
Then we systematically develop a framework for the derivation of surface diffusion models for the self-assembly of quantum dots during Stranski-Krastanov
growth that include surface energies also with large anisotropy as well as the effect of wetting energy,
elastic energy and a randomly perturbed atomic deposition flux.
A linear stability analysis for the resulting sixth-order semilinear evolution equation for the thin film surface shows that that the new model allows
for large anisotropy and gives rise to the formation of anisotropic quantum dots. The nonlinear three-dimensional evolution is investigated via numerical solutions.
These suggest that increasing anisotropy stabilizes the faceted surfaces and may lead to a dramatic slow-down of the coarsening of the dots.
A typical phase field approach for describing phase separation and
coarsening phenomena in alloys is the Cahn-Hilliard model. This
model has been generalized to the so-called Cahn-Larche system
by combining it with elasticity to capture non-neglecting deformation
phenomena,
which occur during phase separation and coarsening processes in the material.
In order to account for damage effects,
we extend the existing framework of Cahn-Hilliard and Cahn-Larche
systems by incorporating an internal damage variable of local character. This
damage variable allows to model the effect that damage of a material point is influenced by its local surrounding.
The damage process is described by a unidirectional rate-dependent evolution inclusion
for the internal variable.
For the introduced Cahn-Larche systems
coupled with rate-dependent
damage processes, we
establish a suitable notion of weak solutions and prove existence of weak solutions.
An electronic model for solar cells including active interfaces and energy resolved defect densities
(2011)
We introduce an electronic model for solar cells taking into account
heterostructures with active
interfaces and energy resolved volume and interface trap densities.
The model consists of continuity equations for electrons and holes with thermionic
emission transfer conditions at the interface and of ODEs for the trap
densities with energy level and spatial position as parameters,
where the right hand sides contain generation-recombination as well as
ionization reactions. This system is coupled with a Poisson
equation for the electrostatic potential.
We show the thermodynamic correctness of the model and prove a priori estimates
for the solutions to the evolution system. Moreover, existence and uniqueness
of weak solutions of the problem are proven. For this purpose we solve a
regularized problem and verify bounds of the corresponding solution
not depending on the regularization level.
Dissipativity is an important property of individual systems
that guarantees a stable interconnected system.
However, due to errors in the modeling process weakly non-dissipative
models may be constructed.
In this paper we introduce a method to perturb a non-dissipative LTI system in order to enforce dissipativity using spectral perturbation results for
para-Hermitian pencils.
Compared to earlier algorithms the new method
is applicable to a wider class of problems,
it utilizes a simpler framework, and
employs a larger class of allowable perturbations
resulting in smaller perturbations. Moreover, system stability can be enforced as well.
Numerical examples are provided to show the effectiveness of the new approach.
Time-stepping procedures for the solution of evolution equations can be performed on parallel architecture by parallelizing the space computation at each time step. This, however, requires heavy communication between processors and becomes inefficient when many time-steps are to be computed and many processors are available. In such
cases parallelization in time is advantageous.
In this paper we present a method for parallelization in time of linear multistep discretizations of linear evolution problems; we consider a model parabolic and a model hyperbolic problem, and their, respectively, A(theta)-stable and A-stable linear multistep discretizations. The method consists of a discrete decoupling procedure, whereby N+1 decoupled Helmholtz problems with complex frequencies are obtained; N being the number of time steps computed in parallel. The usefulness of the method rests on our ability to solve these Helmholtz problems efficiently. We discuss the theory and give numerical examples for multigrid preconditioned iterative solvers of relevant
complex frequency Helmholtz problems. The parallel
approach can easily be combined with a time-stepping procedure, thereby obtaining a block time-stepping method where each block of steps is computed in parallel. In this way we are able to optimize the algorithm with respect to the number of processors available, the difficulty of solving the Helmholtz problems, and the possibility of both time and space adaptivity. Extensions to other linear evolution problems and to Runge-Kutta time discretization
are briefly mentioned.
Affinely-Adjustable Robust Counterparts provide tractable alternatives to (two-stage) robust programs with arbitrary recourse. We apply them to robust network design with polyhedral demand uncertainty, introducing the affine routing principle. We compare the affine routing to the well-studied static and dynamic routing schemes for robust network design. All three schemes are embedded into the general framework of two-stage network design with recourse. It is shown that affine routing can be seen as a generalization of the widely used static routing still being tractable and providing cheaper solutions. We investigate properties on the demand polytope under which affine routings reduce to static routings and also develop conditions on the uncertainty set leading to dynamic routings being affine. We show however that affine routings suffer from the drawback that (even totally) dominated demand vectors are not necessarily supported by affine solutions. Uncertainty sets have to be designed accordingly. Finally, we present computational results on networks from SNDlib. We conclude that for these instances the optimal solutions based on affine routings tend to be as cheap as optimal network designs for dynamic routings. In this respect the affine routing principle can be used to approximate the cost for two-stage solutions with free recourse which are hard to compute.
This paper establishes the equivalence of conforming Courant finite element method and nonconforming Crouzeix-Raviart finite element method in the sense that the respective energy error norms are equivalent up to generic constants and higher-order data oscillations in a Poisson model problem. The Raviart-Thomas mixed finite element method is better than the previous two whereas the conjecture of the converse relation is proved to be false.
This paper completes the analysis of comparison initiated by Braess in Calcolo (2010). Two numerical benchmarks illustrate the comparison theorems and the possible strict superiority of the Raviart-Thomas mixed finite element method. Applications include least-squares finite element methods and equality of approximation classes for concepts of optimality for adaptive finite element methods.
This work concerns the approximation of the shape operator of smooth surfaces in $\mathbb{R}^{3}$ from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximate smooth surfaces and prove two types of approximation estimates: one concerning the approximation of the generalized shape operators in the operator norm and one concerning the pointwise approximation of the (classic) shape operator, including mean and Gaussian curvature, principal curvatures, and principal curvature directions.
The estimates are confirmed by numerical experiments.
We propose a framework for deformation-based surface modeling that is interactive, robust and intuitive to use. The deformations are described by a non-linear optimization problem that models static states of elastic shapes under external forces which implement the user input. Interactive response is achieved by a combination of model reduction, a robust energy approximation, and an efficient quasi-Newton solver. Motivated by the observation that a typical modeling session requires only a fraction of the full shape
space of the underlying model, we use second and third derivatives of a deformation energy to construct a low-dimensional shape space that forms the feasible set for the optimization. Based on mesh coarsening, we propose an
energy approximation scheme with adjustable approximation quality. The quasi-Newton solver guarantees superlinear convergence without the need of costly Hessian evaluations during modeling. We demonstrate the effectiveness of the approach on different examples including the test suite introduced in [Botsch and Sorkine 2008].
This work concerns the approximation of the shape operator of smooth surfaces in R^3 from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximate smooth surfaces and prove two types of approximation estimates: one concerning the approximation of the generalized shape operators in the operator norm and one concerning the pointwise approximation of the (classic) shape operator. We show experimental results that confirm our estimates.
Explicit Error Estimates for Courant, Crouzeix-Raviart and Raviart-Thomas Finite Element Methods
(2011)
The elementary analysis of this paper presents explicit expressions of the constants in the a priori error estimates for the lowest-order Courant, Crouzeix-Raviart nonconforming and Raviart-Thomas mixed finite element methods in the Poisson model problem. The three constants and their dependences on some maximal angle in the triangulation are indeed all comparable and allow accurate a priori error control.
We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system $(X,\Phi,K)$ defining the evolution $\dot{U} = -K(U)D\Phi(U)$. Here $\Phi$ is the driving functional, while the Onsager operator $K(U)$ is symmetric and positive semidefinite. If the inverse $G = K^{-1}$ exists, the triple $(X,\Phi,G)$ defines a gradient system. Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential $\Psi^*(U,\Xi) = 1/2 <\Xi,K(U)\Xi>$. Then, the two functionals $\Phi$ and $\Psi^*$ can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts.
A new finite element method computes conductivity in some unstructured particle-reinforced composite material. The 2-phase material under consideration is composed of a poorly conducting matrix material filled by highly conducting circular inclusions which are randomly dispersed. The mathematical model is a Poisson-type problem with discontinuous coefficients. The discontinuities are huge in contrast and quantity.
The proposed method generalizes classical continuous piecewise affine finite elements to special computational meshes which encode the particles in a network structure. Important geometric parameters such as the volume fraction are preserved exactly.
The computational complexity of the resulting method is proportional to the number of inclusions and this is optimal in comparison to data complexity.
The discretization error is proportional to the particle distance and does not depend on the conductivity contrast in the medium.
Duality Results for Stationary Problems of Open Pit Mine Planning in a Continuous Function Framework
(2011)
Open Pit Mine Planning problems are usually considered in a Mixed Integer Programming
context. Characterizing each attainable profile by a continuous function yields a continuous
framework. It allows for a more detailed modeling of slope constraints and other material
properties of slanted layers. Although the resulting nonlinear programming problems are
in general non-convex and non-differentiable, they provide certain advantages as one can
directly compute sensitivities of optimal solutions w.r.t. small data perturbations. In this
work duality results are derived for the stationary problems of the continuous framework
employing an additional condition called convex-likeness.
An optimal control problem to find the fastest collision-free trajectory of a robot surrounded by obstacles is presented.
The collision avoidance is based on linear programming arguments and expressed as state constraints. The optimal control problem is
solved with a sequential programming method. In order to decrease the number of unknowns and constraints a backface culling active set
strategy is added to the resolution technique.
An adaptive a posteriori error estimator based finite element method for the
numerical solution of a coupled Cahn-Hilliard/Navier-Stokes system with a
double-obstacle homogenous free (interfacial) energy density is proposed. A
semi-implicit Euler scheme for the time-integration is applied which results in
a system coupling a quasi-Stokes or Oseen-type problem for the fluid flow to
a variational inequality for the concentration and the chemical potential according
to the Cahn-Hilliard model [13]. A Moreau-Yosida regularization is
employed which relaxes the constraints contained in the variational inequality
and, thus, enables semi-smooth Newton solvers with locally superlinear convergence
in function space. Moreover, upon discretization this yields a mesh
independent method for a fixed relaxation parameter. For the finite dimensional
approximation of the concentration and the chemical potential piecewise
linear and globally continuous finite elements are used, and for the numerical
approximation of the fluid velocity Taylor-Hood finite elements are employed.
The paper ends by a report on numerical examples showing the efficiency of the
new method
We derive a new representation of Lagrangian subspaces in the form
%
\[
{\mathrm Im}\Pi^T [I,X]^T,
\]
%
where $\Pi$ is a symplectic matrix which is the product of a permutation matrix and a real orthogonal diagonal matrix, and $X$ satisfies
%
\[
\abs{X_{ij}} \leq \begin{cases}1 & \text{if $i=j$,}\\ \sqrt{2} & \text{if $i\neq j$.} \end{cases}
\]
%
This representation allows to limit element growth in the context of doubling algorithms
for the computation of Lagrangian subspaces and the solution of Riccati equations.
It is shown that a simple doubling algorithm using this representation can reach full machine accuracy on a wide range of problems, obtaining invariant subspaces of the same quality as those computed by the state-of-the-art algorithms based on orthogonal transformations.
The same idea carries over to representations of arbitrary subspaces and can be used
for other types of structured pencils.
This note constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying diffusion tensor. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially w.r.t. the number of element layers in the patches. Hence, on a uniform mesh of size H, patches of diameter log(1/H) are sufficient to preserve the convergence rates of the classical P1-FEM for the Poisson problem.
The analysis does not rely on regularity of the solution or scale separation in the coefficient.
The result justifies the use of the class of variational multiscale methods, introduced in [Comput. Methods Appl. Mech. Engrg., 196:2313--2324, 2007].
R is a language and environment for statistical computing and graphics. It can be considered an alternative implementation of the S language developed in the 1970s and 1980s for data analysis and graphics (Becker and Chambers, 1984; Becker et al., 1988). The R language is part of the GNU project and offers versions that compile and run on almost every major operating system currently available. We highlight several R packages built specifically for the analysis of neuroimaging data in the context of functional MRI, diffusion tensor imaging, and dynamic contrast-enhanced MRI. We review their methodology and give an overview of their capabilities for neuroimaging. In addition we summarize some of the current activities in the area of neuroimaging software development in R.
The well-known Kalman-Yakubovich-Popov Lemma establishes an equivalence between dissipativity and the solvability of a linear matrix inequality. In this paper we strengthen this result by showing the equivalence of dissipativity to the solvability of a so-called Lur'e equation, which mainly is a linear matrix inequality with a rank minimizing condition. Finally, we apply the result to standard systems to obtain the well-known result about the solvability of the algebraic Riccati equation.
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 introduce functional perturbation results for PDE eigenvalue problems including the functional backward error
and the functional condition number. These results are used to establish a combined a posteriori error estimator embodying the discretization and the approximation error for the simple eigenpair.
Based on known perturbation results in $H^{1}(\Omega)$ and $H^{-1}(\Omega)$ norms and a standard residual a posteriori error estimator, a balancing AFEM algorithm is proposed. The stopping criterion for the eigensolver is based on the equilibrating strategy, i.e., iterations proceed as long as the discrete part of the error estimator dominates the continuous part. All our statements are illustrated with several numerical examples.
We consider anisotropic Allen--Cahn equations with interfacial energy
induced by an anisotropic surface energy density $\gamma$.
Assuming that $\gamma$
is positive, positively homogeneous of degree one,
strictly convex in tangential directions to the unit sphere,
and sufficiently smooth, we show stability of
various time discretizations. In particular,
we consider a fully implicit and a linearized time discretization
of the interfacial energy combined with implicit
and semi-implicit time discretizations
of the double-well potential. In the semi-implicit variant,
concave terms are taken explicitly.
The arising discrete spatial problems are solved by
globally convergent truncated nonsmooth Newton multigrid methods.
Numerical experiments show the accuracy of the different
discretizations.
We also illustrate that pinch-off under anisotropic
mean curvature flow is no longer frame invariant,
but depends on the orientation of the initial configuration.
This paper deals with a three-dimensional mixture model describing materials undergoing phase transition with thermal expansion. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. A global solution for this thermodynamically consistent problem is obtained by using a fixed-point argument combined with global energy estimates.
We consider three-dimensional models for rate-independent processes describing materials undergoing phase transformations with heat transfer. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. Under appropriate regularity assumptions on the initial data, we prove the existence a global solution for this thermodynamically consistent system, by using a fixed-point argument combined with global energy estimates.