Solving Partial Differential Equations with Monte Carlo / Random Walk on an Analog-Digital Hybrid Computer

Current digital computers are about to hit basic physical boundaries with respect to integration density, clock frequencies, and particularly energy consumption. This requires the application of new computing paradigms, such as quantum and analog computing in the near future. Although neither quantum nor analog computer are general purpose computers they will play an important role as co-processors to offload certain classes of compute intensive tasks from classic digital computers, thereby not only reducing run time but also and foremost power consumption. In this work, we describe a random walk approach to the solution of certain types of partial differential equations which is well suited for combinations of digital and analog computers (hybrid computers). The experiments were performed on an Analog Paradigm Model-1 analog computer attached to a digital computer by means of a hybrid interface. At the end we give some estimates of speedups and power consumption obtainable by using future analog computers on chip.Comment: 9 pages, 7 figures. Proceeding for the MikroSystemTechnik Kongress 2023 (VDE Verlag MST Kongress 2023

Meshless Methods for Option Pricing and Risks Computation

In this thesis we price several financial derivatives by means of radial basis functions. Our main contribution consists in extending the usage of said numerical methods to the pricing of more complex derivatives - such as American and basket options with barriers - and in computing the associated risks. First, we derive the mathematical expressions for the prices and the Greeks of given options; next, we implement the corresponding numerical algorithm in MATLAB and calculate the results. We compare our results to the most common techniques applied in practice such as Finite Differences and Monte Carlo methods. We mostly use real data as input for our examples. We conclude radial basis functions offer a valid alternative to current pricing methods, especially because of the efficiency deriving from the free, direct calculation of risks during the pricing process. Eventually, we provide suggestions for future research by applying radial basis function for an implied volatility surface reconstruction

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Machine Learning-based Generalized Multiscale Finite Element Method and its Application in Reservoir Simulation

In multiscale modeling of subsurface fluid flow in heterogeneous porous media, standard polynomial basis functions are replaced by multiscale basis functions, which are used to predict pressure distribution. To produce such functions in the mixed Generalized Multiscale Finite Element Method (GMsFEM), a number of Partial Differential Equations (PDEs) must be solved, leading to significant computational overhead. The main objective of the work presented in this thesis was to investigate the efficiency of Machine Learning (ML)/Deep Learning (DL) models in reconstructing the multiscale basis functions (Basis 2, 3, 4, and 5) of the mixed GMsFEM. To achieve this, four standard models named SkiplessCNN models were first developed to predict four different multiscale basis functions. These predictions were based on two distinct datasets (initial and extended) generated, with the permeability field being the sole input. Subsequently, focusing on the extended dataset, three distinct skip connection schemes (FirstSkip, MidSkip, and DualSkip) were incorporated into the SkiplessCNN architecture. Following this, the four developed models - SkiplessCNN, FirstSkipCNN, MidSkipCNN, and DualSkipCNN - were separately combined using linear regression and ridge regression within the framework of Deep Ensemble Learning (DEL). Furthermore, the reliability of the DualSkipCNN model was examined using Monte Carlo (MC) dropout. Ultimately, two Fourier Neural Operator (FNO) models, operating on infinite-dimensional spaces, were developed based on a new dataset for directly predicting pressure distribution. Based on the results, sufficient data for the validation and testing subsets could help decrease overfitting. Additionally, all three skip connections were found to be effective in enhancing the performance of SkiplessCNN, with DualSkip being the most effective among them. As evaluated on the testing subset, the combined models using linear regression and ridge regression significantly outperformed the individual models for all basis functions. The results also confirmed the robustness of MC dropout for DualSkipCNN in terms of epistemic uncertainty. Regarding the FNO models, it was discovered that the inclusion of a MultiLayer Perceptron (MLP) in the original Fourier layers significantly improved the prediction performance on the testing subset. Looking at this work as an image (matrix)-to-image (matrix) problem, the developed data-driven models through various techniques could find applications beyond reservoir engineering

Physics-Based Probabilistic Motion Compensation of Elastically Deformable Objects

A predictive tracking approach and a novel method for visual motion compensation are introduced, which accurately reconstruct and compensate the deformation of the elastic object, even in the case of complete measurement information loss. The core of the methods involves a probabilistic physical model of the object, from which all other mathematical models are systematically derived. Due to flexible adaptation of the models, the balance between their complexity and their accuracy is achieved