Scalable Computational Algorithms for Geo-spatial Covid-19 Spread in High Performance Computing

A nonlinear partial differential equation (PDE) based compartmental model of COVID-19 provides a continuous trace of infection over space and time. Finer resolutions in the spatial discretization, the inclusion of additional model compartments and model stratifications based on clinically relevant categories contribute to an increase in the number of unknowns to the order of millions. We adopt a parallel scalable solver allowing faster solutions for these high fidelity models. The solver combines domain decomposition and algebraic multigrid preconditioners at multiple levels to achieve the desired strong and weak scalability. As a numerical illustration of this general methodology, a five-compartment susceptible-exposed-infected-recovered-deceased (SEIRD) model of COVID-19 is used to demonstrate the scalability and effectiveness of the proposed solver for a large geographical domain (Southern Ontario). It is possible to predict the infections up to three months for a system size of 92 million (using 1780 processes) within 7 hours saving months of computational effort needed for the conventional solvers

A Fully Parallelized and Budgeted Multi-level Monte Carlo Framework for Partial Differential Equations: From Mathematical Theory to Automated Large-Scale Computations

All collected data on any physical, technical or economical process is subject to uncertainty. By incorporating this uncertainty in the model and propagating it through the system, this data error can be controlled. This makes the predictions of the system more trustworthy and reliable. The multi-level Monte Carlo (MLMC) method has proven to be an effective uncertainty quantification tool, requiring little knowledge about the problem while being highly performant. In this doctoral thesis we analyse, implement, develop and apply the MLMC method to partial differential equations (PDEs) subject to high-dimensional random input data. We set up a unified framework based on the software M++ to approximate solutions to elliptic and hyperbolic PDEs with a large selection of finite element methods. We combine this setup with a new variant of the MLMC method. In particular, we propose a budgeted MLMC (BMLMC) method which is capable to optimally invest reserved computing resources in order to minimize the model error while exhausting a given computational budget. This is achieved by developing a new parallelism based on a single distributed data structure, employing ideas of the continuation MLMC method and utilizing dynamic programming techniques. The final method is theoretically motivated, analyzed, and numerically well-tested in an automated benchmarking workflow for highly challenging problems like the approximation of wave equations in randomized media

The ANOVA decomposition and generalized sparse grid methods for the high-dimensional backward Kolmogorov equation

In this thesis, we discuss numerical methods for the solution of the high-dimensional backward Kolmogorov equation, which arises in the pricing of options on multi-dimensional jump-diffusion processes. First, we apply the ANOVA decomposition and approximate the high-dimensional problem by a sum of lower-dimensional ones, which we then discretize by a θ-scheme and generalized sparse grids in time and space, respectively. We solve the resultant systems of linear equations by iterative methods, which requires both preconditioning and fast matrix-vector multiplication algorithms. We make use of a Linear Program and an algebraic formula to compute optimal diagonal scaling parameters. Furthermore, we employ the OptiCom as non-linear preconditioner. We generalize the unidirectional principle to non-local operators and develop a new matrix-vector multiplication algorithm for the OptiCom. As application we focus on the Kou model. Using a new recurrence formula, the computational complexity of the operator application remains linear in the number of degrees of freedom. The combination of the above-mentioned methods allows us to efficiently approximate the solution of the backward Kolmogorov equation for a ten-dimensional Kou model.Die ANOVA-Zerlegung und verallgemeinerte dünne Gitter für die hochdimensionale Kolmogorov-Rückwärtsgleichung In der vorliegenden Arbeit betrachten wir numerische Verfahren zur Lösung der hochdimensionalen Kolmogorov-Rückwärtsgleichung, die beispielsweise bei der Bewertung von Optionen auf mehrdimensionalen Sprung-Diffusionsprozessen auftritt. Zuerst wenden wir eine ANOVA-Zerlegung an und approximieren das hochdimensionale Problem mit einer Summe von niederdimensionalen Problemen, die wir mit einem θ-Verfahren in der Zeit und mit verallgemeinerten dünnen Gittern im Ort diskretisieren. Wir lösen die entstehenden linearen Gleichungssysteme mit iterativen Verfahren, wofür eine Vorkonditionierung als auch schnelle Matrix-Vektor-Multiplikationsalgorithmen nötig sind. Wir entwickeln ein Lineares Programm und eine algebraische Formel, um optimale Diagonalskalierungen zu finden. Des Weiteren setzen wir die OptiCom als nicht-lineares Vorkonditionierungsverfahren ein. Wir verallgemeinern das unidirektionale Prinzip auf nicht-lokale Operatoren und entwickeln einen für die OptiCom optimierten Matrix-Vektor-Multiplikationsalgorithmus. Als Anwendungsbeispiel betrachten wir das Kou-Modell. Mit einer neuen Rekurrenzformel bleibt die Gesamtkomplexität der Operatoranwendung linear in der Anzahl der Freiheitsgrade. Unter Einbeziehung aller genannten Methoden ist es nun möglich, die Lösung der Kolmogorov-Rückwärtsgleichung für ein zehndimensionales Kou-Modell effizient zu approximieren

Scalable computational algorithms for geospatial COVID-19 spread using high performance computing

A nonlinear partial differential equation (PDE) based compartmental model of COVID-19 provides a continuous trace of infection over space and time. Finer resolutions in the spatial discretization, the inclusion of additional model compartments and model stratifications based on clinically relevant categories contribute to an increase in the number of unknowns to the order of millions. We adopt a parallel scalable solver that permits faster solutions for these high fidelity models. The solver combines domain decomposition and algebraic multigrid preconditioners at multiple levels to achieve the desired strong and weak scalabilities. As a numerical illustration of this general methodology, a five-compartment susceptible-exposed-infected-recovered-deceased (SEIRD) model of COVID-19 is used to demonstrate the scalability and effectiveness of the proposed solver for a large geographical domain (Southern Ontario). It is possible to predict the infections for a period of three months for a system size of 186 million (using 3200 processes) within 12 hours saving months of computational effort needed for the conventional solvers

Study of phase noise in optical coherent systems

Le bruit de phase est un problème important dans la conception de systèmes cohérents optiques. Bien que le bruit de phase soit étudié énormément dans les communications sans fil, certains aspects de bruit de phase sont nouveaux dans des systèmes cohérents optiques. Dans cette thèse, nous explorons les statistiques de bruit de phase dans les systèmes optiques cohérentes et proposons une nouvelle technique pour améliorer la robustesse du système envers le bruit de phase. Notre première contribution traite de l’étude des statistiques de bruit de phase en présence de compensation électronique de la dispersion chromatique (CD) dans des systèmes cohérents. Nous montrons que le modèle proposé précédemment pour l’interaction de CD avec bruit de phase doit être modifié à cause d’un modèle trop simple pour la récupération de phase. Nous dérivons une expression plus précise pour le bruit de phase estimé par la récupération de phase avec décision dirigée (DD), et utilisons cette expression pour modifier les statistiques de décision pour les symboles reçus. Nous calculons le taux d’erreur binaire (BER) pour le format de transmission DQPSK semi-analytiquement en utilisant nos statistiques de décision modifiées et montrons que pour la récupération de phase idéale, le BER semi-analytique est bien assorti avec le BER simulé avec la technique Monte-Carlo (MC). Notre deuxième contribution est l’adaptation d’une technique de codage MLCM pour les systèmes cohérents limités par le bruit de phase et le bruit blanc additif Gaussien (AWGN). Nous montrons que la combinaison d’une constellation optimisée pour le bruit de phase avec MLCM offre un système robuste à complexité modérée. Nous vérifions que la performance de MLCM dans des systèmes cohérents avec constellations 16-aires se détériorés par le bruit de phase non-linéaire et de Wiener. Pour le bruit de phase non-linéaire, notre conception de MLCM démontre une performance supérieure par rapport àune conception de MLCM déjà présente dans la littérature. Pour le bruit de phase de Wiener, nous comparons deux format de transmission, constellations carrées et optimisée pour bruit de phase, et deux techniques de codage, MLCM et codage à débit uniforme. Nos résultats expérimentaux pour BER après codage suivent les mêmes tendances que le BER simulé et confirment notre conception.Phase noise is an important issue in designing today’s optical coherent systems. Although phase noise is studied heavily in wireless communications, some aspects of phase noise are novel in optical coherent systems. In this thesis we explore phase noise statistics in optical coherent systems and propose a novel technique to increase system robustness toward phase noise. Our first contribution deals with the study of phase noise statistics in the presence of electronic chromatic dispersion (CD) compensation in coherent systems. We show that previously proposed model for phase noise and CD interaction must be modified due to an overly simple model of carrier phase recovery. We derive a more accurate expression for the estimated phase noise of decision directed (DD) carrier phase recovery, and use this expression to modify the decision statistics of received symbols. We calculate bit error rate (BER) of a differential quadrature phase shift keying (DQPSK) system semi-analytically using our modified decision statistics and show that for ideal DD carrier phase recovery the semi-analytical BER matches the BER simulated via Monte-Carlo (MC) technique. We show that the semi-analytical BER is a lower bound of simulated BER from Viterbi-Viterbi (VV) carrier phase recovery for a wide range of practical system parameters. Our second contribution is concerned with adapting a multi-level coded modulation (MLCM) technique for phase noise and additive white Gaussian noise (AWGN) limited coherent system. We show that the combination of a phase noise optimized constellation with MLCM offers a phase-noise robust system at moderate complexity. We propose a numerical method to design set-partitioning (mapping bits to symbols) and optimizing code rates for minimum block error rate (BLER).We verify MLCM performance in coherent systems of 16-ary constellations impaired by nonlinear and Wiener phase noise. For nonlinear phase noise, superior performance of our MLCM design over a previously designed MLCM system is demonstrated in terms of BLER. For Wiener phase noise, we compare optimized and square 16-QAM constellations assuming either MLCM or uniform rate coding. We compare post forward error correction (FEC) BER in addition to BLER by both simulation and experiment and show that superior BLER performance is translated into post FEC BER. Our experimental post FEC BER results follow the same trends as simulated BER, validating our design