# Hilbert-space

The DAKOTA Project - Home. Polynomial Chaos Methods. BERGISCHE UNIVERSITÄT WUPPERTAL :: FB C :: 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. 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. For example, the Department of Energy Advanced Simulation and Computing (ASC) Program was established in 1995 to accelerate the development of computer simulation capabilities for analyzing and predicting the performance, safety, and reliability of nuclear weapons and certifying their functionality. 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. Introduction to Uncertainty Quantification. Gianluca Iaccarino, Stanford University Introduction an Motivations Why UQ?

: predictive science, reliability, robust design Classification of uncertainties: aleatory, epistemic. 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?

ME470 | Uncertainty Quantification | Stanford University. 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. OpenTURNS | 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. Mathematics | 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. AcademicPublicationsComparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification (Citations: 7) Non-intrusive polynomial chaos expansion (PCE) and stochastic collocation (SC) meth- ods are attractive techniques for uncertainty quantification (UQ) due to their strong math- ematical basis and ability to produce functional representations of stochastic variability.

Evaluation of Non-Intrusive Approaches for Wiener-Askey Generalized Polynomial Chaos. AcademicPublicationsEvaluation of Non-Intrusive Approaches for Wiener-Askey Generalized Polynomial Chaos Evaluation of Non-Intrusive Approaches for Wiener-Askey Generalized Polynomial Chaos (Citations: 6) Polynomial chaos expansions (PCE) are an attractive technique for uncertainty quan- tification (UQ) due to their strong mathematical basis and ability to produce functional representations of stochastic variability.

When tailoring the orthogonal polynomial bases to match the forms of the input uncertainties in a Wiener-Askey scheme, excellent conver- gence properties can be achieved for general probabilistic analysis problems. 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[edit] The integral of a function f between limits a and b can be interpreted as the area under the graph of f. 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.

L2-Space. On a measure space , the set of square integrable L2-functions is an -space. Taken together with the L2-inner product with respect to a measure.