Coupling schemes and inexact Newton for multi-physics and coupled optimization problems

This work targets mathematical solutions and software for complex numerical simulation and optimization problems. Characteristics are the combination of different models and software modules and the need for massively parallel execution on supercomputers. We consider two different types of multi-component problems in Part I and Part II of the thesis: (i) Surface coupled fluid- structure interactions and (ii) analysis of medical MR imaging data of brain tumor patients. In (i), we establish highly accurate simulations by combining different aspects such as fluid flow and arterial wall deformation in hemodynamics simulations or fluid flow, heat transfer and mechanical stresses in cooling systems. For (ii), we focus on (a) facilitating the transfer of information such as functional brain regions from a statistical healthy atlas brain to the individual patient brain (which is topologically different due to the tumor), and (b) to allow for patient specific tumor progression simulations based on the estimation of biophysical parameters via inverse tumor growth simulation (given a single snapshot in time, only). Applications and specific characteristics of both problems are very distinct, yet both are hallmarked by strong inter-component relations and result in formidable, very large, coupled systems of partial differential equations. Part I targets robust and efficient quasi-Newton methods for black-box surface-coupling of parti- tioned fluid-structure interaction simulations. The partitioned approach allows for great flexibility and exchangeable of sub-components. However, breaking up multi-physics into single components requires advanced coupling strategies to ensure correct inter-component relations and effectively tackle instabilities. Due to the black-box paradigm, solver internals are hidden and information exchange is reduced to input/output relations. We develop advanced quasi-Newton methods that effectively establish the equation coupling of two (or more) solvers based on solving a non-linear fixed-point equation at the interface. Established state of the art methods fall short by either requiring costly tuning of problem dependent parameters, or becoming infeasible for large scale problems. In developing parameter-free, linear-complexity alternatives, we lift the robustness and parallel scalability of quasi-Newton methods for partitioned surface-coupled multi-physics simulations to a new level. The developed methods are implemented in the parallel, general purpose coupling tool preCICE. Part II targets MR image analysis of glioblastoma multiforme pathologies and patient specific simulation of brain tumor progression. We apply a joint medical image registration and biophysical inversion strategy, targeting at facilitating diagnosis, aiding and supporting surgical planning, and improving the efficacy of brain tumor therapy. We propose two problem formulations and decompose the resulting large-scale, highly non-linear and non-convex PDE-constrained optimization problem into two tightly coupled problems: inverse tumor simulation and medical image registration. We deduce a novel, modular Picard iteration-type solution strategy. We are the first to successfully solve the inverse tumor-growth problem based on a single patient snapshot with a gradient-based approach. We present the joint inversion framework SIBIA, which scales to very high image resolutions and parallel execution on tens of thousands of cores. We apply our methodology to synthetic and actual clinical data sets and achieve excellent normal-to-abnormal registration quality and present a proof of concept for a very promising strategy to obtain clinically relevant biophysical information. Advanced inexact-Newton methods are an essential tool for both parts. We connect the two parts by pointing out commonalities and differences of variants used in the two communities in unified notation

Robust Quasi-Newton methods for partitioned fluid-structure simulations

In recent years, quasi-Newton schemes have proven to be a robust and efficient way for the coupling of partitioned multi-physics simulations in particular for fluid-structure interaction. The focus of this work is put on the coupling of partitioned fluid-structure interaction, where minimal interface requirements are assumed for the respective field solvers, thus treated as black box solvers. The coupling is done through communication of boundary values between the solvers. In this thesis a new quasi-Newton variant (IQN-IMVJ) based on a multi-vector update is investigated in combination with serial and parallel coupling systems. Due to implicit incorporation of passed information within the Jacobian update it renders the problem dependent parameter of retained previous time steps unnecessary. Besides, a whole range of coupling schemes are categorized and compared comprehensively with respect to robustness, convergence behaviour and complexity. Those coupling algorithms differ in the structure of the coupling, i.\,e., serial or parallel execution of the field solvers and the used quasi-Newton methods. A more in-depth analysis for a choice of coupling schemes is conducted for a set of strongly coupled FSI benchmark problems, using the in-house coupling library preCICE. The superior convergence behaviour and robust nature of the IQN-IMVJ method compared to well known state of the art methods such as the IQN-ILS method, is demonstrated here. It is confirmed that the multi-vector method works optimal without the need of tuning problem dependent parameters in advance. Furthermore, it appears to be especially suitable in conjunction with the parallel coupling system, in that it yields fairly similar results for parallel and serial coupling. Although we focus on FSI simulation, the considered coupling schemes are supposed to be equally applicable to various kinds of different volume- or surface-coupled problems.In den letzten Jahren haben sich quasi-Newton Verfahren als robuste und effiziente Methode für die Kopplung partitionierter Multiphysik-Simulationen herausgestellt, insbesondere im Bereich der Kopplung von Fluid-Struktur Interaktion. Der Fokus dieser Arbeit liegt auf der Kopplung partitionierten Fluid-Struktur Interaktion mit minimalen Schnittstellenanforderungen an die betreffenden Fluid- oder Struktur-Löser, die daher als black-box Löser angesehen werden können. Die Kopplung an sich wird durch Kommunikation von Randwerten zwischen den Lösern realisiert. Im Rahmen dieser Arbeit wird eine neue quasi-Newton Methode in Kombination mit einem seriellen und parallelen Kopplungssystem untersucht. Diese Variante ist im Gegensatz zu bisherigen Methoden weitestgehend frei von problemabhängigen Parametern wie z. B. der optimalen Menge an wiederverwendeter Information aus vergangenen Zeitschritten, indem alle bisher bekannte Information in einer norm-minimalen und impliziten Art und Weise für die Aufdatierung der Jacobi Matrix verwendet wird. Darüberhinaus betrachten wir in dieser Arbeit ein ganzes Spektrum an Kopplungs-Schemata, die verschiedene quasi-Newton Varianten mit seriellen und parallellen Kopplungsansätzen kombinieren. Eine sorgfältige Klassifizierung sowie ein umfassender Vergleich der Verfahren bezüglich Robustheit, Komplexität und des Konvergenzverhaltens, verschaffen einen guten Überblick. Für eine Auswahl der besten Schemata wird eine eigehendere Analyse anhand einer Reihe anspruchsvoller FSI Anwendungen mit Hilfe der Kopplungs-Bibliothek preCICE durchgeführt. In diesem Zusammenhang gehen wir auf die robuste sowie effiziente Implementierung der Kopplungs-Algorithmen ein. Im Zuge der Experimente zeigt die IQN-IMVJ quasi-Newton Methode ein überlegenes und weitaus robusteres Konvergenzverhalten im Vergleich zu bisherigen Varianten wie beispielsweise die IQN-ILS Methode und arbeitet optimal ohne vorherige Justierung problemabhängiger Parameter. Darüber hinaus ist sie hervorragend für einen parallelen Kopplungsansatz geeeignet und ermöglicht eine effiziente und massiv-parallele Simulation. Die betrachteten Kopplungs-Schemata sind nicht auf die Kopplung von Fluid-Struktur Interaktion Simulation beschränkt, sondern sind vielmehr für alle Arten von oberflächen- oder volumengekoppelten Problemen geeignet

Robuste Multilevel-Lösung elliptischer partieller Differentialgleichungen mit springenden Koeffizienten

In dieser Arbeit wird eine robuste Multilevel-Lösung für elliptische partielle Differentialgleichungen mit springenden Koeffizientenfunktionen im Kontext der Partition of Unity Methode realisiert und analysiert. Bei dieser gitterfreien Methode müssen die Koeffizientensprünge nicht auf dem gröbsten Level geometrisch aufgelöst sein, vielmehr kann durch geeignete Anreicherungsfunktionen mittels algebraischer Verfeinerung die Approximationsqualität verbessert werden. Die Implementierung eines stabilen und robusten Multilevel-Lösers sowie die Realisierung verschiedener Anreicherungsfunktionen sind Kernbereich dieser Arbeit. Insbesondere werden verschiedene Anreicherungsfunktionen entwickelt und deren Auswirkung im Hinblick auf die Robustheit des Lösers untersucht. Mögliche Ursachen für nicht robustes Verhalten des Lösers werden in diesem Zusammenhang detailliert diskutiert und Ansätze für Verbesserungen gegeben. Der realisierte Multilevel-Löser zeigt im Vergleich zu früheren Arbeiten effizienteres und für viele Fälle durchaus robustes Verhalten. Einfache Distanzfunktionen führen dabei i. A. zu den besten Ergebnissen jedoch lässt sich durch alleinige Verbesserung der Approximationsqualität der lokalen Anreicherungsräume die Effizienz und Robustheit des Lösers aufgrund schlechterer Glättungseigenschaften nicht beliebig steigern

A High-Performance Inversion Framework for Brain Tumor Growth Models in Personalized Medicine

The precise characterization of aggressive brain tumors remains a challenging problem due to their highly heterogeneous radiographic and molecular presentation. The integration of mathematical models with clini- cal imaging data holds an enormous promise of developing robust predictive and explainable models that quantify cancer growth with the potential to as- sist in diagnosis and treatment. In general, such models are parameterized by many unknown parameters and their estimation can be formally posed as an inverse problem. However, this calibration problem is a formidable task for aggressive brain tumors due to the absence of longitudinal data, resulting in a strongly ill-posed inverse problem. This is further exacerbated by the inherent non-linearity in tumor growth models. Overcoming these difficulties involves the introduction of sophisticated regularization strategies along with compu- tationally efficient algorithms and software. Towards this end, we introduce a fully-automatic inversion framework which provides an entirely new capa- bility to analyze complex brain tumors from a single pretreatment magnetic resonance imaging (MRI) scan. Our framework employs fast algorithms and optimized implementations which exploit distributed-memory parallelism and GPU acceleration to enable reasonable solution times – an important factor for clinical applications. We validate our solver on clinical data and demonstrate its utility in characterizing important biophysics of brain cancer along with its ability to complement other radiographic information in downstream machine learning tasks

A comparison of various quasi-newton schemes for partitioned fluid-structure interaction

During the last 5 years, quasi-Newton schemes have proven to be a robust and efficient way to couple partitioned fluid-structure interaction. We showed in previous work that they also allow to perform a parallel coupling. Bogaers et al. introduced a new variant based on a multi-vector update [14]. This variant renders a tuning of the reuse of old information unnecessary as all old iterations are implicitly covered in a Jacobian update. In this work, we compare this multi-vector variant in an inverse formulation to the classical IQN-ILS algorithm for serial as well as parallel coupling