G. Mathematics of Computing
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It is shown how piecewise differentiable functions \(F: R^n → R^m\) that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points x̌ and x̂. We show that the discrepancy between function and model at any point x is of the bilinear order O(||x − x̌|| ||x − x̂||). This is a little surprising since x ∈ R^n may vary over the whole Euclidean space, and we utilize only two function samples F̌ = F(x̌) and F̂ = F(x̂), as well as the intermediates computed during their evaluation. As an application of the piecewise linearization procedure we devise a generalized Newton’s method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equaling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.
Modern MIP solvers employ dozens of auxiliary algorithmic components to support the branch-and-bound search in finding and improving primal solutions and in strengthening the dual bound.
Typically, all components are tuned to minimize the average running time to prove optimality. In this article, we take a different look at the run of a MIP solver. We argue that the solution process consists of three different phases, namely achieving feasibility, improving the incumbent solution, and proving optimality. We first show that the entire solving process can be improved by adapting the search strategy with respect to the phase-specific aims using different control tunings. Afterwards, we provide criteria to predict the transition between the individual phases and evaluate the performance impact of altering the algorithmic behavior of the MIP solver SCIP at the predicted phase transition points.
Portfolio parallelization is an approach that runs several solver instances in parallel and terminates when one of them succeeds in solving the problem. Despite it's simplicity portfolio parallelization has been shown to perform well for modern mixed-integer programming (MIP) and boolean satisfiability problem (SAT) solvers. Domain propagation has also been shown to be a simple technique in modern MIP and SAT solvers that effectively finds additional domain reductions after a variables domain has been reduced. This paper investigates the impact of distributed domain propagation in modern MIP solvers that employ portfolio parallelization. Computational experiments were conducted for two implementations of this parallelization approach. While both share global variable bounds and solutions they communicate differently. In one implementation the communication is performed only at designated points in the solving process and in the other it is performed completely asynchronously. Computational experiments show a positive performance impact of communicating global variable bounds and provide valuable insights in communication strategies for parallel solvers.
This paper investigates the criterion of long-term average costs for a Markov decision process (MDP) which is not permanently observable. Each observation of the process produces a fixed amount of \textit{information costs} which enter the considered performance criterion and preclude from arbitrarily frequent state testing. Choosing the \textit{rare} observation times is part of the control procedure. In contrast to the theory of partially observable Markov decision processes, we consider an arbitrary continuous-time Markov process on a finite state space without further restrictions on the dynamics or the type of interaction.
Based on the original Markov control theory, we redefine the control model and the average cost criterion for the setting of information costs. We analyze the constant of average costs for the case of ergodic dynamics and present an optimality equation which characterizes the optimal choice of control actions and observation times. For this purpose, we construct an equivalent freely observable MDP and translate the well-known results from the original theory to the new setting.
Optimization-based bound tightening (OBBT) is one of the most effective procedures to reduce variable domains of nonconvex mixed-integer nonlinear programs (MINLPs). At the same time it is one of the most expensive bound tightening procedures, since it solves auxiliary linear programs (LPs)—up to twice the number of variables many. The main goal of this paper is to discuss algorithmic techniques for an efficient implementation of OBBT.
Most state-of-the-art MINLP solvers apply some restricted version of OBBT and it seems to be common belief that OBBT is beneficial if only one is able to keep its computational cost under control. To this end, we introduce three techniques to increase the efficiency of OBBT: filtering strategies to reduce the number of solved LPs, ordering heuristics to exploit simplex warm starts, and the generation of Lagrangian variable bounds (LVBs). The propagation of LVBs during tree search is a fast approximation to OBBT without the need to solve auxiliary LPs.
We conduct extensive computational experiments on MINLPLib2. Our results indicate that OBBT is most beneficial on hard instances, for which we observe a speedup of 17% to 19% on average. Most importantly, more instances can be solved when using OBBT.
Reconstruction of photonic crystal geometries using a reduced basis method for nonlinear outputs
(2016)
Maxwell solvers based on the hp-adaptive finite element method allow for accurate geometrical modeling and high numerical accuracy. These features are indispensable for the optimization of optical properties or reconstruction of parameters through inverse processes. High computational complexity prohibits the evaluation of the solution for many parameters. We present a reduced basis method (RBM) for the time-harmonic electromagnetic scattering problem allowing to compute solutions for a parameter configuration orders of magnitude faster. The RBM allows to evaluate linear and nonlinear outputs of interest like Fourier transform or the enhancement of the electromagnetic field in milliseconds. We apply the RBM to compute light-scattering off two dimensional photonic crystal structures made of silicon and reconstruct geometrical parameters.
Model order reduction for the time-harmonic Maxwell equation applied to complex nanostructures
(2016)
Fields such as optical metrology and computational lithography require fast and efficient methods for solving
the time-harmonic Maxwell’s equation. Highly accurate geometrical modeling and numerical accuracy atcomputational costs are a prerequisite for any simulation study of complex nano-structured photonic devices.
We present a reduced basis method (RBM) for the time-harmonic electromagnetic scattering problem based
on the hp-adaptive finite element solver JCMsuite capable of handling geometric and non-geometric parameter
dependencies allowing for online evaluations in milliseconds. We apply the RBM to compute light-scatteringoptical wavelengths off periodic arrays of fin field-effect transistors (FinFETs) where geometrical properties such
as the width and height of the fin and gate can vary in a large range.