This thesis is devoted to the interdisciplinary work between mathematicians and forensic
experts: the modeling of the human body cooling process after death laying the
foundation for the estimation of the time of death. An inverse problem needs to be
solved. In this thesis the inverse problem computes the time of death given the measured
body temperature and the Forward Model that simulates the body cooling
process. The Forward Model is based on the heat equation established by Fourier.
This differential equation is numerically solved by the discretization over space by the
Finite Element Method and the discretization over time by the Implicit Euler Method.
The applications in this thesis demand a fast computation time. A model reduction is
achieved by the Proper Orthogonal Decomposition in combination with the Galerkin
Method. For reasons of simplification the computations and the measurements are
restricted to a cylindrical phantom that is made out of homogeneous polyethylene.
The estimate of the time of death is accompanied by an uncertainty. The inverse problem
is incorporated by Bayesian inference to interpret the quality of the estimate and
the effciency of the experiment. The uncertainty of the estimate of the time of death
is minimized by approaching the Optimal Design of the Experiment. An objective
function measures the certainty of the data and lays the foundation of the optimization
problem. Solving the optimization problem is successfully done by relaxing the
complex discrete NP-hard problem and applying a gradient-based method.
The results of this thesis clearly show that the design of an experiment has a great in-
uence on the outcome of the quality of the estimate. The comparison of the estimate
and its properties based on different designs and conditions reveals the effciency of
the Design of Experiment in the context of the estimation of the time of death.
The goal of quantitative photoacoustic tomography (qPAT) is to recover maps of the chromophore distributions from multiwavelength images of the initial pressure. Model-based inversions that incorporate the physical processes underlying the photoacoustic (PA) signal generation represent a promising approach. Monte-Carlo models of the light transport are computationally expensive, but provide accurate fluence distributions predictions, especially in the ballistic and quasi-ballistic regimes. Here, we focus on the inverse problem of 3D qPAT of blood oxygenation and investigate the application of the Monte-Carlo method in a model-based inversion scheme. A forward model of the light transport based on the MCX simulator and acoustic propagation modeled by the k-Wave toolbox was used to generate a PA image data set acquired in a tissue phantom over a planar detection geometry. The combination of the optical and acoustic models is shown to account for limited-view artifacts. In addition, the errors in the fluence due to, for example, partial volume artifacts and absorbers immediately adjacent to the region of interest are investigated. To accomplish large-scale inversions in 3D, the number of degrees of freedom is reduced by applying image segmentation to the initial pressure distribution to extract a limited number of regions with homogeneous optical parameters. The absorber concentration in the tissue phantom was estimated using a coordinate descent parameter search based on the comparison between measured and modeled PA spectra. The estimated relative concentrations using this approach lie within 5 % compared to the known concentrations. Finally, we discuss the feasibility of this approach to recover the blood oxygenation from experimental data.
In this article, we introduce parallel mixed integer linear programming (MILP) solvers. MILP solving algorithms have been improved tremendously in the last two decades. Currently, commercial MILP solvers are known as a strong optimization tool. Parallel MILP solver development has started in 1990s. However, since the improvements of solving algorithms have much impact to solve MILP problems than application of parallel computing, there were not many visible successes. With the spread of multi-core CPUs, current state-of-the-art MILP solvers have parallel implementations and researches to apply parallelism in the solving algorithm also getting popular. We summarize current existing parallel MILP solver architectures.
In this paper, the concepts and design for an efficient information service for mathematical software and further mathematical research data are presented. The publication-based approach and the Web-based approach are the main building blocks of the service and will be discussed. Heuristic methods are used for identification, extraction, and ranking of information about software and other mathematical research data. The methods provide not only information about the research data but also link software and mathematical research data to the scientific context.
Dieser Bericht beschreibt die Ergebnisse eines Anwendungsprojektes, das parallel zum Aufbau des Berliner Hochgeschwindigkeitsdatennetzes (Berlin Regional Testbed) am ZIB durchgeführt wurde. Es werden allgemeine Werkzeuge und anwendungsspezifische Arbeitsumgebungen zur netzverteilten Visualisierung und Simulation vorgestellt. Die allgemeinen Werkzeuge unterstützen folgende Aufgaben: Kopplung von Simulationen auf (Hochleistungs-)Rechnern an lokale Grafikarbeitsplätze, objektorientierte und verteilte Visualisierung, Remote-Videoaufzeichnung, Bilddatenkompression und digitaler Filmschnitt. Die spezifischen Arbeitsumgebungen wurden für Aufgaben aus den Bereichen Numerische Mathematik, Astrophysik, Strukturforschung, Chemie, Polymerphysik und Strömungsmechanik entwickelt.
The SCIP Optimization Suite is a powerful collection of optimization software that consists of the branch-cut-and-price framework and mixed-integer programming solver SCIP, the linear programming solver SoPlex, the modeling language Zimpl, the parallelization framework UG, and the generic branch-cut-and-price solver GCG. Additionally, it features the extensions SCIP-Jack for solving Steiner tree problems, PolySCIP for solving multi-objective problems, and SCIP-SDP for solving mixed-integer semidefinite programs. The SCIP Optimization Suite has been continuously developed and has now reached version 4.0. The goal of this report is to present the recent changes to the collection. We not only describe the theoretical basis, but focus on implementation aspects and their computational consequences.
Finite reversible Markov chains are characterized by a transition matrix
P that has real eigenvalues and pi-orthogonal eigenvectors, where pi
is the stationary distribution of P. This means, that a transition matrix
with complex eigenvalues corresponds to a non-reversible Markov chain.
This observation leads to the question, whether the imaginary part of that
eigendecomposition corresponds to or indicates the “pattern” of the nonreversibility.
This article shows that the direct relation between imaginary
parts of eigendecompositions and the non-reversibility of a transition matrix
e P is not given. It is proposed to apply the Schur decomposition
of e P instead of the eigendecomposition in order to characterize its nonreversibility.
The Schur decomposition also allows to find the difference
matrix P which turns a non-reversible Markov chain e P into a reversible
one P = e P+lambdaP, such that P and e P have the same stationary distribution
and the same metastabilities. P and e P even have the same eigenvalues, if
e P is diagonalizable in R.
Molecular dynamics (MD) simulations face challenging problems since
the timescales of interest often are much longer than what is possible
to simulate and even if sufficiently long simulation are possible the complex
nature of the resulting simulation data makes interpretation difficult.
Markov State Models (MSMs) help to overcome these problems by making
experimentally relevant timescales accessible via coarse grained representations
that also allows for convenient interpretation. However, standard
set-based MSMs exhibit some caveats limiting their approximation quality
and statistical significance. One of the main caveats results from the fact
that typical MD trajectories repeatedly re-cross the boundary between
the sets used to build the MSM which causes statistical bias in estimating
the transition probabilities between these sets. In this article, we present
a set-free approach to MSM building utilizing smooth overlapping ansatz
functions instead of sets and an adaptive refinement approach. This kind
of meshless discretization helps to overcome the recrossing problem and
yields an adaptive refinement procedure that allows to improve the quality
of the model while exploring state space and inserting new ansatz
functions into the MSM.
In this article we propose an adaptive importance sampling scheme for dynamical quantities of high dimensional complex systems which are metastable. The main idea of this article is to combine a method coming from Molecular Dynamics Simulation, Metadynamics, with a theorem from stochastic analysis, Girsanov's theorem. The proposed algorithm has two advantages compared to a standard estimator of dynamic quantities: firstly, it is possible to produce estimators with a lower variance and, secondly, we can speed up the sampling. One of the main problems for building importance sampling schemes for metastable systems is to find the metastable region in order to manipulate the potential accordingly. Our method circumvents this problem by using an assimilated version of the Metadynamics algorithm and thus creates a non-equilibrium dynamics which is used to sample the equilibrium quantities.