Optimal Algorithms for Numerical Integration: Recent Results and Open Problems

We present recent results on optimal algorithms for numerical integration and several open problems. The paper has six parts: 1. Introduction 2. Lower Bounds 3. Universality 4. General Domains 5. iid Information 6. Concluding RemarksComment: Survey written for the MCQMC conference in Linz, 26 pages. arXiv admin note: text overlap with arXiv:2108.0205

Machine Learning Post-Minkowskian Integrals

We study a neural network framework for the numerical evaluation of Feynman loop integrals that are fundamental building blocks for perturbative computations of physical observables in gauge and gravity theories. We show that such a machine learning approach improves the convergence of the Monte Carlo algorithm for high-precision evaluation of multi-dimensional integrals compared to traditional algorithms. In particular, we use a neural network to improve the importance sampling. For a set of representative integrals appearing in the computation of the conservative dynamics for a compact binary system in General Relativity, we perform a quantitative comparison between the Monte Carlo integrators VEGAS and i-flow, an integrator based on neural network sampling.Comment: 26 pages + references, 3 figures, 3 table

Importance Sampling-Based Monte Carlo Methods with Applications to Quantitative Finance

In the present work advanced Monte Carlo methods for discrete-time stochastic processes are developed and investigated. A particular focus is on sequential Monte Carlo methods (particle filters and particle smoothers) which allow the estimation of nonlinear, non-Gaussian state-space models. The key technique which underlies the proposed algorithms is importance sampling. Computationally efficient nonparametric variants of importance sampling which are generally applicable are developed. Asymptotic properties of these methods are analyzed theoretically and it is shown empirically that they improve over existing methods for relevant applications. Particularly, it is shown that they can be applied for financial derivative pricing which constitutes a high-dimensional integration problem and that they can be used to improve sequential Monte Carlo methods. Original models in general state-space form for two important applications are proposed and new sequential Monte Carlo algorithms for their estimation are developed. The first application concerns the on-line estimation of the spot cross-volatility for ultra high-frequency financial data. This is a challenging problem because of the presence of microstructure noise and nonsynchronous trading. For the first time state-space models with non-synchronously evolving states and observations are discussed and a particle filter which can cope with these models is designed. In addition, a new sequential variant of the EM algorithm for parameter estimation is proposed. The second application is a non-linear model for time series with an oscillatory pattern and a phase process in the background. This model can be applied, for instance, to noisy quasiperiodic oscillators occurring in physics and other fields. The estimation of the model is based on an advanced particle smoother and a new nonparametric EM algorithm. The dissertation is accompanied by object-oriented C++ implementations of all proposed algorithms which were developed with a focus on reusability and extendability

Essays on Tree-based Methods for Prediction and Causal Inference

The first chapter of this thesis contains an application of causal forests to a residential electricity smart meter trial dataset. Household specific estimates are obtained for the effect of a Time-of-Use pricing scheme on peak demand. The most and least responsive households differ across education, age, employment status, and past electricity consumption. The results suggest that past consumption information is more useful than pre-trial survey information, which includes building characteristics, household characteristics, and responses to appliance usage questions. The second chapter explores new variations of Bayesian tree-based machine learning algorithms. Bayesian Additive Regression Trees (BART) (Chipman et al. 2010) and Bayesian Causal Forests (BCF) (Hahn et al. 2020) are state-of-the-art machine learning methods for prediction and causal inference. A number of existing implementations of BART make use of Markov Chain Monte Carlo algorithms, which can be computationally expensive when applied to high-dimensional datasets, do not always perform well in terms of mixing of chains, and have limited parallelizability. The second chapter introduces four variations of BART that do not rely on MCMC: 1. An improved implementation of the existing method BART-BMA (Hernandez et al. 2018), which averages over sum-of-tree models found by a model search algorithm, performs well on high-dimensional datasets, and produces more interpretable output than other BART implementations because the output includes a comparatively small number of sum-of-tree models. %, each of which contains (under the default settings) 5 trees. Improvements are made to the model search algorithm, calculation of predictions, and credible intervals.% The algorithm is entirely deterministic. 2. A treatment effect estimation algorithm that combines the model structure of BCF with the implementation of BART-BMA (BCF-BMA). This method successfully accounts for confounding on observables using the BCF parameterization, while retaining the parsimonious model selection approach of BART-BMA. 3. A simple alternative BART implementation algorithm that uses importance sampling of models (BART-IS). This approach contrasts with existing MCMC and model-search based approaches in that BART-IS makes fast data-independent draws of many sum-of-tree models. The advantages of this approach are that it is straightforward to implement, fast, and trivially parallelizable. 4. Bayesian Causal Forests using Importance Sampling (BCF-IS). This is a combination of the BCF model framework with the BART-IS implementation. BART-IS and BCF-IS exhibit comparable performance to BART-MCMC and BCF across a large number of simulated datasets. The second chapter also includes some illustrative applications. The methods are extendable to multiple treatments, multivariate outcomes, and panel data methods. The third chapter of this thesis describes how the methods introduced in the second chapter can be generalized from regression and treatment effect estimation for continuous outcomes, to a range of models with various link functions and outcome variables. As examples of how to apply the general approach, Logit-BART-BMA and Logit-BART-IS are introduced with illustrative applications

Construction of a Mean Square Error Adaptive Euler--Maruyama Method with Applications in Multilevel Monte Carlo

A formal mean square error expansion (MSE) is derived for Euler--Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise a posteriori adaptive time stepping Euler--Maruyama method for numerical solutions of SDE, and the resulting method is incorporated into a multilevel Monte Carlo (MLMC) method for weak approximations of SDE. This gives an efficient MSE adaptive MLMC method for handling a number of low-regularity approximation problems. In low-regularity numerical example problems, the developed adaptive MLMC method is shown to outperform the uniform time stepping MLMC method by orders of magnitude, producing output whose error with high probability is bounded by TOL>0 at the near-optimal MLMC cost rate O(TOL^{-2}log(TOL)^4).Comment: 43 pages, 12 figure

Grand-canonical molecular dynamics simulations powered by a hybrid 4D nonequilibrium MD/MC method: Implementation in LAMMPS and applications to electrolyte solutions

Molecular simulations in an open environment, involving ion exchange, are necessary to study various systems, from biosystems to confined electrolytes. However, grand-canonical simulations are often computationally demanding in condensed phases. A promising method (L. Belloni, J. Chem. Phys., 2019), one of the hybrid nonequilibrium molecular dynamics/Monte Carlo algorithms, was recently developed, which enables efficient computation of fluctuating number or charge density in dense fluids or ionic solutions. This method facilitates the exchange through an auxiliary dimension, orthogonal to all physical dimensions, by reducing initial steric and electrostatic clashes in three-dimensional systems. Here, we report the implementation of the method in LAMMPS with a Python interface, allowing facile access to grand-canonical molecular dynamics (GCMD) simulations with massively parallelized computation. We validate our implementation with two electrolytes, including a model Lennard-Jones electrolyte similar to a restricted primitive model and aqueous solutions. We find that electrostatic interactions play a crucial role in the overall efficiency due to their long-range nature, particularly for water or ion-pair exchange in aqueous solutions. With properly screened electrostatic interactions and bias-based methods, our approach enhances the efficiency of salt-pair exchange in Lennard-Jones electrolytes by approximately four orders of magnitude, compared to conventional grand-canonical Monte Carlo. Furthermore, the acceptance rate of NaCl-pair exchange in aqueous solutions at moderate concentrations reaches about 3 $\%$ at the maximum efficiency

Fast algorithms for frequency domain wave propagation

textHigh-frequency wave phenomena is observed in many physical settings, most notably in acoustics, electromagnetics, and elasticity. In all of these fields, numerical simulation and modeling of the forward propagation problem is important to the design and analysis of many systems; a few examples which rely on these computations are the development of metamaterial technologies and geophysical prospecting for natural resources. There are two modes of modeling the forward problem: the frequency domain and the time domain. As the title states, this work is concerned with the former regime. The difficulties of solving the high-frequency wave propagation problem accurately lies in the large number of degrees of freedom required. Conventional wisdom in the computational electromagnetics commmunity suggests that about 10 degrees of freedom per wavelength be used in each coordinate direction to resolve each oscillation. If K is the width of the domain in wavelengths, the number of unknowns N grows at least by O(K^2) for surface discretizations and O(K^3) for volume discretizations in 3D. The memory requirements and asymptotic complexity estimates of direct algorithms such as the multifrontal method are too costly for such problems. Thus, iterative solvers must be used. In this dissertation, I will present fast algorithms which, in conjunction with GMRES, allow the solution of the forward problem in O(N) or O(N log N) time.Computational Science, Engineering, and Mathematic

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

Foundations of realistic rendering : a mathematical approach

Die vorliegende Dissertation ist keine gewöhnliche Abhandlung, sondern sie ist als Lehrbuch zum realistischen Rendering für Studenten im zweiten Studienabschnitt, sowie Forscher und am Thema Interessierte konzipiert. Aus mathematischer Sicht versteht man unter realistischem Rendering das Lösen der stationären Lichttransportgleichung, einer komplizierten Fredholm Integralgleichung der 2tenArt, deren exakte Lösung, wenn überhaupt berechenbar, nur in einem unendlich- dimensionalen Funktionenraum existiert. Während in den existierenden Büchern, die sich mit globaler Beleuchtungstheorie beschäftigen, vorwiegend die praktische Implementierung von Lösungsansätzen im Vordergrund steht, sind wir eher daran interessiert, den Leser mit den mathematischen Hilfsmitteln vertraut zu machen, mit welchen das globale Beleuchtungsproblem streng mathematisch formuliert und letzendlich auch gelöst werden kann. Neue, effzientere und elegantere Algorithmen zur Berechnung zumindest approxima- tiver Lösungen der Lichttransportgleichung und ihrer unterschiedlichen Varianten können nur im Kontext mit einem vertieften Verständnis der Lichttransportgleichung entwickelt werden. Da die Probleme des realistischen Renderings tief in verschiedenen mathematis- chen Disziplinen verwurzelt sind, setzt das vollständige Verständnis des globalen Beleuch- tungsproblems Kenntnisse aus verschiedenen Bereichen der Mathematik voraus. Als zen- trale Konzepte kristallisieren sich dabei Prinzipien der Funktionalanalysis, der Theorie der Integralgleichungen, der Maß- und Integrationstheorie sowie der Wahrscheinlichkeitstheo- rie heraus. Wir haben uns zum Ziel gesetzt, dieses Knäuel an mathematischen Konzepten zu entflechten, sie für Studenten verständlich darzustellen und ihnen bei Bedarf und je nach speziellem Interesse erschöpfend Auskunft zu geben.The available doctoral thesis is not a usual paper but it is conceived as a text book for realistic rendering, made for students in upper courses, as well as for researchers and interested people. From mathematical point of view, realistic rendering means solving the stationary light transport equation, a complicated Fredholm Integral equation of 2nd kind. Its exact solution exists|if possible at all|in an infinite dimensional functional space. Whereas practical implementation of approaches for solving problems are in the center of attentionin the existing textbooks that treat global illumination theory, we are more interested in familiarizing our reader with the mathematical tools which permit them to formulate the global illumination problem in accordance with strong mathematical principles and last but not least to solve it. New, more eficient and more elegant algorithms to calculate approximate solutions for the light transport equation and their different variants must be developed in the context of deep and complete understanding of the light transport equation. As the problems of realistic rendering are deeply rooted in different mathematical disciplines, there must precede the complete comprehension of all those areas. There are evolving principles of functional analysis, theory of integral equations, measure and integration theory as well as probability theory. We have set ourselves the target to remerge this bundle of fluff of mathematical concepts and principles, to represent them to the students in an understandable manner, and to give them, if required, exhaustive information