The DAKOTA Project - Home. Polynomial Chaos Methods. NUMERISCHE MATHEMATIK : : Polynomial Chaos. Uncertainty quantification becomes a more and more important task in many applications.
Given a physical or technical system, the impact of uncertainties in the inputs has to be determined for the outputs. Consequently, the inputs are replaced by random variables in the corresponding mathematical models. The original models may be ordinary differential equations (ODEs), differential algebraic equations (DAEs) or partial differential equations (PDEs). The random process satisfying the stochastic system has to be computed by numerical methods. A Monte-Carlo simulation is able to resolve the stochastic model, for example. The approach of the polynomial chaos can be used in each application, where mathematical models based on ODEs, DAEs or PDEs appear. People in group R. Ongoing projects none Cooperation Literature / Selected Publications: F.
UQ - YouQ. YouQ: A self-guided tour of Uncertainty Quantification This page provides references and links to introductory material on UQ and to computational tools that are freely available on the Web.
Uncertainty Quantification: what is it? The DAKOTA Project - Download Versions. Uncertainty Quantification. Overview The rapid growth of high-performance supercomputing technology and advances in numerical techniques in the last two decades have provided an unprecedented opportunity to explore complex physical phenomena using modeling and simulation.
In the present day, multi-physics simulations involving fluid flow, structural dynamics, chemical kinetics, atomic and nuclear sciences, are ubiquitous. The success of computer simulations also means that they will be increasingly relied upon as important tools for high-consequence predictions and decision making. Sparse Grid Interpolation Toolbox. Sparse Grid Interpolation Toolbox The Sparse Grid Interpolation Toolbox is a Matlab toolbox for recovering (approximating) expensive, possibly high-dimensional multivariate functions.
It was developed by Andreas Klimke at the Institute of Applied Analysis and Numerical Simulation at the High Performance Scientific Computing lab ("Lehrstuhl für Numerische Mathematik für Höchstleistungsrechner"), Universität Stuttgart during his Ph.D. studies. Andreas continues to maintain and improve the toolbox in his spare time since April 2006. He is very grateful to the group and, in particular, Prof. Dr. Introduction to Uncertainty Quantification. UQ - Home. "If a man will begin with certainties, he shall end in doubts; but if he will be content to begin with doubts, he shall end in certainties.
" - F. Bacon - 1605. The availability of powerful computational resources and general purpose numerical algorithms creates increasing opportunities to attempt flow simulations in complex systems. How accurate are the resulting predictions? Are the mathematical and physical models correct? Do we have sufficient information to define relevant operating conditions? Uncertainty Quantification (UQ) aims at developing rigorous methods to characterize the impact of limited knowledge on quantities of interest. Uncertainty Quantification. Imprecise Probability Propagation Toolbox for Matlab. Current version: 1.0 What is IPP Toolbox?
The IPP Toolbox is a collection of methods for uncertainty quantification and propagation using Dempster-Shafer Theory and imprecise probabilities. It runs under MATLAB 6.5 and higher. Why IPP Toolbox? The IPP Toolbox provides all necessary functions to conduct an uncertainty analysis in the imprecise probabilistic framework. IPP Toolbox Resources IPP Toolbox Features Methods for constructing Dempster-Shafer structures (DSS) by different sampling strategies. IPP Toolbox References Fault tree analysis executed using the IPP Toolbox: P. Required Resources Matlab 6.5 or later running under any operating system (Linux, Windows, Unix, Macintosh). The official OpenTURNS Website. Ocw.mit.edu/courses/brain-and-cognitive-sciences/9-520-statistical-learning-theory-and-applications-spring-2003/lecture-notes/mathcamp1slides.pdf.
18.102 Introduction to Functional Analysis, Spring 2009. Applied Functional Analysis: Main Principles and Their Applications (Applied Mathematical Sciences) - library.nu. Polynomial chaos for the approximation of uncertainties: Chances and limits. Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification.
Evaluation of Non-Intrusive Approaches for Wiener-Askey Generalized Polynomial Chaos. Lebesgue integration. The integral of a positive function can be interpreted as the area under a curve.
The Lebesgue integral plays an important role in the branch of mathematics called real analysis and in many other fields in the mathematical sciences, and is named after Henri Lebesgue (1875–1941) who introduced the integral in (Lebesgue 1904). It is also a pivotal portion of the axiomatic theory of probability. Introduction The integral of a function f between limits a and b can be interpreted as the area under the graph of f.
This is easy to understand for familiar functions such as polynomials, but what does it mean for more exotic functions? As part of a general movement toward rigour in mathematics in the nineteenth century, attempts were made to put the integral calculus on a firm foundation. However, Riemann integration does not interact well with taking limits of sequences of functions, making such limiting processes difficult to analyze. Intuitive interpretation Measure (mathematics) Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B.
Furthermore, the measure of the empty set is required to be 0. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral.