Download or read book Adaptive Multi Level Monte Carlo and Stochastic Collocation Methods for Hyperbolic Partial Differential Equations with Random Data on Networks written by Elisa Strauch and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Multi fidelity Stochastic Collocation Methods Using Model Reduction Techniques written by Maziar Raissi and published by . This book was released on 2013 with total page 70 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the last few years there have been dramatic advances in our understanding of mathematical and computational models of complex systems in the presence of uncertainty. This has led to a growth in the area of uncertainty quantification as well as the need to develop efficient, scalable, stable and convergent computational methods for solving differential equations with random inputs. Stochastic Galerkin methods based on polynomial chaos expansions have shown superiority to other non-sampling and many sampling techniques. For complicated governing equations numerical implementations of stochastic Galerkin methods can become non-trivial. Monte Carlo and other traditional sampling methods, are straightforward to implement. But they do not offer as fast convergence rates as stochastic Galerkin. Other numerical approaches are the stochastic collocation (SC) methods, which inherit both the ease of implementation of Monte Carlo and the robustness of stochastic Galerkin to a great deal. Stochastic collocation and its powerful extensions, e.g. sparse grid stochastic collocation, can simply fail to handle more levels of complication. The seemingly innocent Burgers equation driven by Brownian motion is such an example. We propose a novel enhancement to stochastic collocation methods using deterministic model reduction techniques that can handle this pathological example and hopefully other more complicated equations like Stochastic Navier Stokes. Our numerical results show the efficiency of the proposed technique. We also perform a mathematically rigorous study of linear parabolic partial differential equations with random forcing terms. Justified by the truncated Karhunen-Loeve expansions, the input data are assumed to be represented by a finite number of random variables. A rigorous convergence analysis of our method applied to parabolic partial differential equations with random forcing terms, supported by numerical results, shows that the proposed technique is not only reliable and robust but also very efficient.
Download or read book Monte Carlo Methods and Stochastic Processes written by Emmanuel Gobet and published by CRC Press. This book was released on 2016-09-15 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: Developed from the author’s course at the Ecole Polytechnique, Monte-Carlo Methods and Stochastic Processes: From Linear to Non-Linear focuses on the simulation of stochastic processes in continuous time and their link with partial differential equations (PDEs). It covers linear and nonlinear problems in biology, finance, geophysics, mechanics, chemistry, and other application areas. The text also thoroughly develops the problem of numerical integration and computation of expectation by the Monte-Carlo method. The book begins with a history of Monte-Carlo methods and an overview of three typical Monte-Carlo problems: numerical integration and computation of expectation, simulation of complex distributions, and stochastic optimization. The remainder of the text is organized in three parts of progressive difficulty. The first part presents basic tools for stochastic simulation and analysis of algorithm convergence. The second part describes Monte-Carlo methods for the simulation of stochastic differential equations. The final part discusses the simulation of non-linear dynamics.
Download or read book An Adaptive Multi Level Monte Carlo Method with Stochastic Bounds for Quantities of Interest in Groundwater Flow with Uncertain Data written by and published by . This book was released on 2015 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt: The focus of this work is the introduction of some computable a posteriori error control to the popular multilevel Monte Carlo sampling for PDE with stochastic data. We are especially interested in applications in the geosciences such as groundwater flow with rather rough stochastic fields for the conductive permeability. With a spatial discretisation based on finite elements, a goal functional is defined which encodes the quantity of interest. The devised goal-oriented error estimator enables to determine guaranteed a posteriori error bounds for this quantity. In particular, it allows for the adaptive refinement of the mesh hierarchy used in the multilevel Monte Carlo simulation. In addition to controlling the deterministic error, we also suggest how to treat the stochastic error in probability. Numerical experiments illustrate the performance of the presented adaptive algorithm for a posteriori error control in multilevel Monte Carlo methods. These include a localised goal with problem-adapted meshes and a slit domain example. The latter demonstrates the refinement of regions with low solution regularity based on an inexpensive explicit error estimator in the multilevel algorithm.
Download or read book Adaptive Multilevel Monte Carlo Methods for Random Elliptic Problems written by Evgenia Youett and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Monte Carlo and Quasi Monte Carlo Methods 2004 written by Harald Niederreiter and published by Springer Science & Business Media. This book was released on 2006-02-08 with total page 506 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book represents the refereed proceedings of the Sixth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing and of the Second International Conference on Monte Carlo and Probabilistic Methods for Partial Differential Equations. These conferences were held jointly at Juan-les-Pins (France) in June 2004. The proceedings include carefully selected papers on many aspects of Monte Carlo methods, quasi-Monte Carlo methods, and the numerical solution of partial differential equations. The reader will be informed about current research in these very active areas.
Download or read book Multilevel Monte Carlo Method for Parabolic Stochastic Partial Differential Equations written by Andrea Barth and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Multilevel Monte Carlo Method with Applications to Stocastic Partial Differential Equations written by Andrea Barth and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Monte Carlo Methods written by Malvin H. Kalos and published by John Wiley & Sons. This book was released on 2008-09-26 with total page 195 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introduction to Monte Carlo Methods seeks to identify and study the unifying elements that underlie their effective application. It focuses on two basic themes. The first is the importance of random walks as they occur both in natural stochastic systems and in their relationship to integral and differential equations. The second theme is that of variance reduction in general and importance sampling in particular as a technique for efficient use of the methods. Random walks are introduced with an elementary example in which the modelling of radiation transport arises directly from a schematic probabilistic description of the interaction of radiation with matter. Building on that example, the relationship between random walks and integral equations is outlined. The applicability of these ideas to other problems is shown by a clear and elementary introduction to the solution of the Schrodinger equation by random walks. The detailed discussion of variance reduction includes Monte Carlo evaluation of finite-dimensional integrals. Special attention is given to importance sampling, partly because of its intrinsic interest in quadrature, partly because of its general usefulness in the solution of integral equations. One significant feature is that Monte Carlo Methods treats the "Metropolis algorithm" in the context of sampling methods, clearly distinguishing it from importance sampling. Physicists, chemists, statisticians, mathematicians, and computer scientists will find Monte Carlo Methods a complete and stimulating introduction.
Download or read book Robust Multi level Monte Carlo Finite Volume Methods for Systems of Hyperbolic Conservation Laws with Random Input Data written by Jonas Šukys and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Monte Carlo Methods for Partial Differential Equations With Applications to Electronic Design Automation written by Wenjian Yu and published by Springer. This book was released on 2022-09-06 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Monte Carlo method is one of the top 10 algorithms in the 20th century. This book is focusing on the Monte Carlo method for solving deterministic partial differential equations (PDEs), especially its application to electronic design automation (EDA) problems. Compared with the traditional method, the Monte Carlo method is more efficient when point values or linear functional of the solution are needed, and has the advantages on scalability, parallelism, and stability of accuracy. This book presents a systematic introduction to the Monte Carlo method for solving major kinds of PDEs, and the detailed explanation of relevant techniques for EDA problems especially the cutting-edge algorithms of random walk based capacitance extraction. It includes about 100 figures and 50 tables, and brings the reader a close look to the newest research results and the sophisticated algorithmic skills in Monte Carlo simulation software.
Download or read book Mixing Monte Carlo and Partial Differential Equation Methods For Multi dimensional Optimal Stopping Problems Under Stochastic Volatility written by David Farahany and published by . This book was released on 2019 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we develop a numerical approach for solving multi-dimensional optimal stopping problems (OSPs) under stochastic volatility (SV) that combines least squares Monte Carlo (LSMC) with partial differential equation (PDE) techniques. The algorithm provides dimensional reduction from the PDE and regression perspective along with variance and dimensional reduction from the MC perspective. In Chapter 2, we begin by laying the mathematical foundation of mixed MC-PDE techniques for OSPs. Next, we show the basic mechanics of the algorithm and, under certain mild assumptions, prove it converges almost surely. We apply the algorithm to the one dimensional Heston model and demonstrate that the hybrid algorithm outperforms traditional LSMC techniques in terms of both estimating prices and optimal exercise boundaries (OEBs). In Chapter 3 we describe methods for reducing the complexity and run time of the algorithm along with techniques for computing sensitivities. To reduce the complexity, we apply two methods: clustering via sufficient statistics and multi-level Monte Carlo (mlMC)/multi-grids. While the clustering method allows us to reduce computational run times by a third for high dimensional problems, mlMC provides an order of magnitude reduction in complexity. To compute sensitivities, we employ a grid based method for derivatives with respect to the asset, S, and an MC method that uses initial dispersions for sensitivities with respect to variance, v. To test our approximations and computation of sensitivities, we revisit the one-dimensional Heston model and find our approximations introduce little-to-no error and that our computation of sensitivities is highly accurate in comparison to standard LSMC. To demonstrate the utility of our new computational techniques, we apply the hybrid algorithm to the multi-dimensional Heston model and show that the algorithm is highly accurate in terms of estimating prices, OEBs, and sensitivities, especially in comparison to standard LSMC. In Chapter 4 we highlight the importance of multi-factor SV models and apply our hybrid algorithm to two specific examples: the Double Heston model and a mean-reverting commodity model with jumps. Again, we were able to obtain low variance estimates of the prices, OEBs, and sensitivities.
Download or read book Fast Sequential Monte Carlo Methods for Counting and Optimization written by Reuven Y. Rubinstein and published by John Wiley & Sons. This book was released on 2013-11-13 with total page 177 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive account of the theory and application of Monte Carlo methods Based on years of research in efficient Monte Carlo methods for estimation of rare-event probabilities, counting problems, and combinatorial optimization, Fast Sequential Monte Carlo Methods for Counting and Optimization is a complete illustration of fast sequential Monte Carlo techniques. The book provides an accessible overview of current work in the field of Monte Carlo methods, specifically sequential Monte Carlo techniques, for solving abstract counting and optimization problems. Written by authorities in the field, the book places emphasis on cross-entropy, minimum cross-entropy, splitting, and stochastic enumeration. Focusing on the concepts and application of Monte Carlo techniques, Fast Sequential Monte Carlo Methods for Counting and Optimization includes: Detailed algorithms needed to practice solving real-world problems Numerous examples with Monte Carlo method produced solutions within the 1-2% limit of relative error A new generic sequential importance sampling algorithm alongside extensive numerical results An appendix focused on review material to provide additional background information Fast Sequential Monte Carlo Methods for Counting and Optimization is an excellent resource for engineers, computer scientists, mathematicians, statisticians, and readers interested in efficient simulation techniques. The book is also useful for upper-undergraduate and graduate-level courses on Monte Carlo methods.
Download or read book Introduction to Monte Carlo Methods for Transport and Diffusion Equations written by Bernard Lapeyre and published by OUP Oxford. This book was released on 2003 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is used by for the resolution of partial differential equations, trasnport equations, the Boltzmann equation and the parabolic equations of diffusion.
Download or read book Monte Carlo and Quasi Monte Carlo Methods 2012 written by Josef Dick and published by Springer Science & Business Media. This book was released on 2013-12-05 with total page 680 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book represents the refereed proceedings of the Tenth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing that was held at the University of New South Wales (Australia) in February 2012. These biennial conferences are major events for Monte Carlo and the premiere event for quasi-Monte Carlo research. The proceedings include articles based on invited lectures as well as carefully selected contributed papers on all theoretical aspects and applications of Monte Carlo and quasi-Monte Carlo methods. The reader will be provided with information on latest developments in these very active areas. The book is an excellent reference for theoreticians and practitioners interested in solving high-dimensional computational problems arising, in particular, in finance, statistics and computer graphics.
Download or read book A Multi level Monte Carlo Algorithm for L vy Driven Stochastic Differential Equations written by Steffen Dereich and published by . This book was released on 2009 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Multi level Monte Carlo Finite Difference and Finite Volume Methods for Stochastic Linear Hyperbolic Systems written by Jonas Šukys and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
