Monte Carlo Algorithms for Linear Problems

MSC Subject Classification: 65C05, 65U05.Monte Carlo methods are a powerful tool in many fields of mathematics, physics and engineering. It is known, that these methods give statistical estimates for the functional of the solution by performing random sampling of a certain chance variable whose mathematical expectation is the desired functional. Monte Carlo methods are methods for solving problems using random variables. In the book [16] edited by Yu. A. Shreider one can find the following definition of the Monte Carlo method

04401 Abstracts Collection -- Algorithms and Complexity for Continuous

From 26.09.04 to 01.10.04, the Dagstuhl Seminar ``Algorithms and Complexity for Continuous Problems\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Input variable selection in time-critical knowledge integration applications: A review, analysis, and recommendation paper

This is the post-print version of the final paper published in Advanced Engineering Informatics. The published article is available from the link below. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. Copyright @ 2013 Elsevier B.V.The purpose of this research is twofold: first, to undertake a thorough appraisal of existing Input Variable Selection (IVS) methods within the context of time-critical and computation resource-limited dimensionality reduction problems; second, to demonstrate improvements to, and the application of, a recently proposed time-critical sensitivity analysis method called EventTracker to an environment science industrial use-case, i.e., sub-surface drilling. Producing time-critical accurate knowledge about the state of a system (effect) under computational and data acquisition (cause) constraints is a major challenge, especially if the knowledge required is critical to the system operation where the safety of operators or integrity of costly equipment is at stake. Understanding and interpreting, a chain of interrelated events, predicted or unpredicted, that may or may not result in a specific state of the system, is the core challenge of this research. The main objective is then to identify which set of input data signals has a significant impact on the set of system state information (i.e. output). Through a cause-effect analysis technique, the proposed technique supports the filtering of unsolicited data that can otherwise clog up the communication and computational capabilities of a standard supervisory control and data acquisition system. The paper analyzes the performance of input variable selection techniques from a series of perspectives. It then expands the categorization and assessment of sensitivity analysis methods in a structured framework that takes into account the relationship between inputs and outputs, the nature of their time series, and the computational effort required. The outcome of this analysis is that established methods have a limited suitability for use by time-critical variable selection applications. By way of a geological drilling monitoring scenario, the suitability of the proposed EventTracker Sensitivity Analysis method for use in high volume and time critical input variable selection problems is demonstrated.E

Optimization governed by stochastic partial differential equations

This thesis provides a rigorous framework for the solution of stochastic elliptic partial differential equation (SPDE) constrained optimization problems. In modeling physical processes with differential equations, much of the input data is uncertain (e.g. measurement errors in the diffusivity coefficients). When uncertainty is present, the governing equations become a family of equations indexed by a stochastic variable. Since solutions of these SPDEs enter the objective function, the objective function usually involves statistical moments. These optimization problems governed by SPDEs are posed as a particular class of optimization problems in Banach spaces. This thesis discusses Monte Carlo, stochastic Galerkin, and stochastic collocation methods for the numerical solution of SPDEs and identifies the stochastic collocation method as particularly useful for the optimization of SPDEs. This thesis extends the stochastic collocation method to the optimization context and explores the decoupling nature of this method for gradient and Hessian computations

Stochastic Techniques for the Solution of Electrostatic Problems With Applications to Electron Optics.

We apply stochastic techniques towards the solution of the two-dimensional Laplace\u27s equation in boundary value problems encountered in the calculation of electrostatic potentials in electron lenses and deflectors. The justification of these techniques arises from an astonishingly simple but far-reaching principle, which has been known for a long time but has been rarely used: the potential at any point in the interior of a charge-free region can be calculated by performing random walks starting at this point and terminating at the boundary of the region--the potential is then the average of the potential boundary values (assumed known) over the random walks. By an optimal combination of the stochastic Monte-Carlo and deterministic Relaxation methods, we show the advantages and competitiveness of our hybrid Monte-Carlo-Relaxation (MCR) technique compared to the conventional numerical techniques used in the previously mentioned problem. In order to enhance the performance of our method, we investigate the convergence, speed and accuracy of MCR versus traditional techniques. We also develop optimized computational techniques that we believe increase MCR\u27s appeal to problems not previously considered amenable to Monte-Carlo type simulations as well as demonstrate its applicability in problems that are intractable by traditional relaxation or analytical techniques. We use MCR to simulate electrostatic lenses and detectors previously presented in the literature. Finally, we demonstrate the application of MCR towards the numerical solution of general elliptic problems in arbitrary domains and we present the generalization of the stochastic method to solve problems with space charge, namely Poisson\u27s equation

Event tracking for real-time unaware sensitivity analysis (EventTracker)

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ 2013 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.This paper introduces a platform for online Sensitivity Analysis (SA) that is applicable in large scale real-time data acquisition (DAQ) systems. Here we use the term real-time in the context of a system that has to respond to externally generated input stimuli within a finite and specified period. Complex industrial systems such as manufacturing, healthcare, transport, and finance require high quality information on which to base timely responses to events occurring in their volatile environments. The motivation for the proposed EventTracker platform is the assumption that modern industrial systems are able to capture data in real-time and have the necessary technological flexibility to adjust to changing system requirements. The flexibility to adapt can only be assured if data is succinctly interpreted and translated into corrective actions in a timely manner. An important factor that facilitates data interpretation and information modelling is an appreciation of the affect system inputs have on each output at the time of occurrence. Many existing sensitivity analysis methods appear to hamper efficient and timely analysis due to a reliance on historical data, or sluggishness in providing a timely solution that would be of use in real-time applications. This inefficiency is further compounded by computational limitations and the complexity of some existing models. In dealing with real-time event driven systems, the underpinning logic of the proposed method is based on the assumption that in the vast majority of cases changes in input variables will trigger events. Every single or combination of events could subsequently result in a change to the system state. The proposed event tracking sensitivity analysis method describes variables and the system state as a collection of events. The higher the numeric occurrence of an input variable at the trigger level during an event monitoring interval, the greater is its impact on the final analysis of the system state. Experiments were designed to compare the proposed event tracking sensitivity analysis method with a comparable method (that of Entropy). An improvement of 10% in computational efficiency without loss in accuracy was observed. The comparison also showed that the time taken to perform the sensitivity analysis was 0.5% of that required when using the comparable Entropy based method.EPSR

Electromagnetic Wave Theory and Applications

Contains reports on twelve research projects.Joint Services Electronics Program (Contract DAALO3-86-K-0002)National Science Foundation (Grant ECS 85-04381)National Aeronautics and Space Administration/Goddard Space Flight Center (Contract NAG5-270)National Aeronautics and Space Administration/Goddard Space Flight Center (Contract NAG5-725)U.S. Navy - Office of Naval Research (Contract N00014-83-K-0258)U.S. Navy - Office of Naval Research (Contract N00014-86-K-0533)U.S. Army - Research Office Durham (Contract DAAG29-85-K-0079)International Business Machines, Inc.National Aeronautics and Space Administration/Goddard Space Flight Center (Contract NAG5-269)Simulation TechnologiesSchlumberger-Doll Researc