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18-03
We consider the use of randomised forward models and log-likelihoods within the Bayesian approach to inverse problems. Such random approximations to the exact forward model or log-likelihood arise naturally when a computationally expensive model is approximated using a cheaper stochastic surrogate, as in Gaussian process emulation (kriging), or in the field of probabilistic numerical methods. We show that the Hellinger distance between the exact and approximate Bayesian posteriors is bounded by moments of the difference between the true and approximate log-likelihoods. Example applications of these stability results are given for randomised misfit models in large data applications and the probabilistic solution of ordinary differential equations.
18-05
The Steiner tree problem in graphs is a classical problem that commonly arises in practical applications as one of many variants. Although the different Steiner tree problem variants are usually strongly related, solution approaches employed so far have been prevalently problem-specific. Against this backdrop, the solver SCIP-Jack was created as a general-purpose framework that can be used to solve the classical Steiner tree problem and 11 of its variants. This versatility is achieved by transforming various problem variants into a general form and solving them by using a state-of-the-art MIP-framework. Furthermore, SCIP-Jack includes various newly developed algorithmic components such as preprocessing routines and heuristics. The result is a high-performance solver that can be employed in massively parallel environments and is capable of solving previously unsolved instances. After the introduction of SCIP-Jack at the 2014 DIMACS Challenge on Steiner problems, the overall performance of the solver has considerably improved. This article provides an overview on the current state.
18-09
Merging criteria for the definition of a local pore and the CSD computation of granular materials
(2018)
18-10
Mathematical models for bioregulatory networks can be based on different formalisms, depending on the quality of available data and the research question to be answered. Discrete boolean models can be constructed based on qualitative data, which are frequently available. On the other hand, continuous models in terms of ordinary differential equations (ODEs) can incorporate time-series data and give more detailed insight into the dynamics of the underlying system. A few years ago, a method based on multivariate polynomial interpolation and Hill functions has been developed for an automatic conversion of boolean models to systems of ordinary differential equations. This method is frequently used by modellers in systems biology today, but there are only a few results available about the conservation of mathematical structures and properties across the formalisms. Here, we consider subsets of the phase space where some components stay fixed, called trap spaces, and demonstrate how boolean trap spaces can be linked to invariant sets in the continuous state space. This knowledge is of practical relevance since finding trap spaces in the boolean setting, which is relatively easy, allows for the construction of reduced ODE models.
18-07
Mitotic and meiotic spindles are microtubule-based structures to faithfully segregate chromosomes. Electron tomography is currently the method of choice to analyze the three-dimensional architecture of both types of spindles. Over the years, we have developed methods and software for automatic segmentation and stitching of microtubules in serial sections for large-scale reconstructions. Three-dimensional reconstruction of microtubules, however, is only the first step towards biological insight. The second step is the analysis of the structural data to derive measurable spindle properties. Here, we present a comprehensive set of techniques to quantify spindle parameters. These techniques provide quantitative analyses of specific microtubule classes and are applicable to a variety of tomographic reconstructions of spindles from different organisms.
18-08
Estimation of time of death based on a single measurement of body
core temperature is a standard procedure in forensic medicine.
Mechanistic models using simulation of heat transport promise
higher accuracy than established phenomenological models in
particular in nonstandard situations, but involve many not exactly
known physical parameters. Identifying both time of death and
physical parameters from multiple temperature measurements is
one possibility to reduce the uncertainty significantly.
In this paper, we consider the inverse problem in a Bayesian setting
and perform both local and sampling-based uncertainty
quantification, where proper orthogonal decomposition is used as
model reduction for fast solution of the forward model. Based on
the local uncertainty quantification, optimal design of experiments
is performed in order to minimize the uncertainty in the time of
death estimate for a given number of measurements. For reasons
of practicability, temperature acquisition points are selected from
a set of candidates in different spatial and temporal locations.
Applied to a real corpse model, a significant accuracy improvement
is obtained already with a small number of measurements.
18-61
The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lovász and Plummer developed a decomposition theory for graphs with perfect matchings and their corresponding polytopes known as the tight cut decomposition which breaks down every graph into a number of indecomposable graphs, so called bricks. For many properties that are of interest on graphs with perfect matchings, including the description of the perfect matching polytope, it suffices to consider these bricks. A key result by Lovász on the tight cut decomposition is that the list of bricks obtained is the same independent of the choice of tight cuts made during the tight cut decomposition procedure. This implies that finding a tight cut decomposition is polynomial time equivalent to finding a single tight cut.
We generalise the notions of a tight cut, a tight cut contraction and a tight cut decomposition to hypergraphs. By providing an example, we show that the outcome of the tight cut decomposition on general hypergraphs is no longer unique. However, we are able to prove that the uniqueness of the tight cut decomposition is preserved on a slight generalisation of uniform hypergraphs. Moreover, we show how the tight cut decomposition leads to a decomposition of the perfect matching polytope of uniformable hypergraphs and that the recognition problem for tight cuts in uniformable hypergraphs is polynomial time solvable.
18-60
Large Neighborhood Search (LNS) heuristics are among the most powerful but also most expensive heuristics for mixed integer programs (MIP). Ideally, a solver learns adaptively which LNS heuristics work best for the MIP problem at hand in order to concentrate its limited computational budget.
To this end, this work introduces Adaptive Large Neighborhood Search (ALNS) for MIP, a primal heuristic that acts a framework for eight popular LNS heuristics such as Local Branching and Relaxation Induced Neighborhood Search (RINS). We distinguish the available LNS heuristics by their individual search domains, which we call neighborhoods. The decision which neighborhood should be executed is guided by selection strategies for the multi armed bandit problem, a related optimization problem during which suitable actions have to be chosen to maximize a reward function. In this paper, we propose an LNS-specific reward function to learn to distinguish between the available neighborhoods based on successful calls and failures. A second, algorithmic enhancement is a generic variable fixing priorization, which ALNS employs to adjust the subproblem complexity as needed. This is particularly useful for some neighborhoods which do not fix variables by themselves. The proposed primal heuristic has been implemented
within the MIP solver SCIP. An extensive computational study is conducted to compare different LNS strategies within our ALNS framework on a large set of publicly available MIP instances from the MIPLIB and Coral benchmark sets. The results of this simulation are used to calibrate the parameters of the bandit selection strategies. A second computational experiment shows the computational benefits of the proposed ALNS framework within the MIP solver SCIP.
18-17
Mixed integer programming is a versatile and valuable optimization tool. However, solving specific problem instances can be computationally demanding even for cutting-edge solvers. Such long running times are often significantly reduced by an appropriate change of the solver's parameters. In this paper we investigate "algorithm selection", the task of choosing among a set of algorithms the ones that are likely to perform best for a particular instance.
In our case, we treat different parameter settings of the MIP solver SCIP as different algorithms to choose from. Two peculiarities of the MIP solving process have our special attention. We address the well-known problem of performance variability by using multiple random seeds. Besides solving time, primal dual integrals are recorded as a second performance measure in order to distinguish solvers that timed out.
We collected feature and performance data for a large set of publicly available MIP instances. The algorithm selection problem is addressed by several popular, feature-based methods, which have been partly extended for our purpose. Finally, an analysis of the feature space and performance results of the selected algorithms are presented.
18-19
We consider the stochastic extensible bin packing problem (SEBP) in which $n$ items of stochastic size are packed into $m$ bins of unit capacity. In contrast to the classical bin packing problem, bins can be extended at extra cost. This problem plays an important role in stochastic environments such as in surgery scheduling: Patients must be assigned to operating rooms beforehand, such that the regular capacity is fully utilized while the amount of overtime is as small as possible.
This paper focuses on essential ratios between different classes of policies: First, we consider the price of non-splittability, in which we compare the optimal non-anticipatory policy against the optimal fractional assignment policy. We show that this ratio has a tight upper bound of $2$. Moreover, we develop an analysis of a fixed assignment variant of the LEPT rule yielding a tight approximation ratio of $1+1/e \approx 1.368$ under a reasonable assumption on the distributions of job durations.
Furthermore, we prove that the price of fixed assignments, which describes the loss when restricting to fixed assignment policies, is within the same factor. This shows that in some sense, LEPT is the best fixed assignment policy we can hope for.