Introduction to uncertainty quantification:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham
Springer
[2015]
|
| Series: | Texts in applied mathematics
volume 63 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xii, 342 Seiten Diagramme |
| ISBN: | 9783319233949 |
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Record in the Search Index
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| adam_text | Contents 1 Introduction................................................................................................. 1.1 1.2 1.3 1.4 2 3 Measure and Probability Theory .................................................... 2.1 Measure and Probability Spaces.................................................. 2.2 Random Variables and Stochastic Processes............................. 2.3 Lebesgue Integration...................................................................... 2.4 Decomposition and Total Variation of Signed Measures......... 2.5 The Radon-Nikodým Theorem and Densities......................... 2.6 Product Measures and Independence......................................... 2.7 Gaussian Measures ....................................................................... 2.8 Interpretations of Probability..................................................... 2.9 Bibliography................................................................................... 2.10 Exercises......................................................................................... Banach and Hilbert Spaces................................................................. 3.1 Basic Definitions and Properties................................................. 3.2 Banach and Hilbert Spaces ......................................................... 3.3 3.4 3.5 3.6 3.7 4 What is Uncertainty Quantification?......................................... Mathematical Prerequisites......................................................... Outline of the
Book....................................................................... The Road Not Taken ................................................................... Dual Spaces and Adjoints ........................................................... Orthogonality and Direct Sums ................................................. Tensor Products............................................................................. Bibliography................................................................................... Exercises......................................................................................... Optimization Theory.............................................................................. 4.1 4.2 4.3 Optimization Problems and Terminology................................. Unconstrained Global Optimization........................................... Constrained Optimization........................................................... 1 1 6 7 8 9 9 14 15 19 20 21 23 29 31 32 35 35 39 43 45 50 52 52 55 55 57 60 ix
Contents x Convex Optimization................................................................... Linear Programming..................................................................... Least Squares................................................................................. Bibliography................................................................................... Exercises......................................................................................... 63 68 69 73 74 Measures ofinformation and Uncertainty.................................. 75 75 76 78 81 87 87 4.4 4.5 4.6 4.7 4.8 5 5.1 5.2 5.3 5.4 5.5 5.6 The Existence of Uncertainty....................................................... Interval Estimates........................................................................... Variance, Information and Entropy............................................. Information Gain, Distances and Divergences........................... Bibliography..................................................................................... Exercises........................................................................................... 6 Bayesian Inverse Problems ................................................................ 91 6.1 Inverse Problems and Regularization........................................... 92 6.2 Bayesian Inversion in Banach Spaces........................................... 98 6.3 Well-Posedness and Approximation............................................... 101 6.4 Accessing the Bayesian Posterior Measure................................... 105
6.5 Frequentisi Consistency of Bayesian Methods............................. 107 6.6 Bibliography....................................................................................... 110 6.7 Exercises............................................................................................. 112 7 Filtering and Data Assimilation ........................................................ 113 7.1 State Estimation in Discrete Time................................................. 114 7.2 Linear Kálmán Filter....................................................................... 117 7.3 Extended Kálmán Filter................................................................... 125 7.4 Ensemble Kálmán Filter................................................................... 126 7.5 Bibliography....................................................................................... 128 7.6 Exercises............................................................................................. 129 8 Orthogonal Polynomials and Applications ...................................133 Basic Definitions and Properties.....................................................134 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 Recurrence Relations ....................................................................... 140 Differential Equations....................................................................... 143 Roots of Orthogonal Polynomials................................................... 145 Polynomial Interpolation................................................................. 147 Polynomial
Approximation............................................................. 151 Multivariate Orthogonal Polynomials........................................... 154 Bibliography....................................................................................... 158 Exercises............................................................................................. 158 Tables of Classical Orthogonal Polynomials................................. 161
Contents xi 9 Numerical Integration....................................................................... 165 9.1 Univariate Quadrature.................................................................... 166 9.2 Gaussian Quadrature...................................................................... 169 9.3 Clenshaw-Curtis/Fejér Quadrature.............................................. 173 9.4 Multivariate Quadrature ................................................................ 175 9.5 Monte Carlo Methods...................................................................... 178 9.6 Pseudo-Random Methods .............................................................. 186 9.7 Bibliography...................................................................................... 192 9.8 Exercises............................................................................................ 194 10 Sensitivity Analysis and Model Reduction............................... 197 10.1 Model Reduction for Linear Models.............................................. 198 10.2 Derivatives........................................................................................ 201 10.3 McDiarmid Diameters...................................................................... 206 10.4 ANOVA/HDMR Decompositions.................................................. 210 10.5 Active Subspaces.............................................................................. 213 10.6 Bibliography...................................................................................... 218 10.7
Exercises............................................................................................ 219 11 Spectral Expansions ........................................................................... 223 11.1 Karhunen-Loève Expansions.......................................................... 223 11.2 Wiener-Hermite Polynomial Chaos.............................................. 234 11.3 Generalized Polynomial Chaos Expansions.................................. 237 11.4 Wavelet Expansions.......................................................................... 243 11.5 Bibliography...................................................................................... 247 11.6 Exercises............................................................................................ 248 12 Stochastic Galerkin Methods......................................................... 251 12.1 Weak Formulation of Nonlinearities .............................................. 252 12.2 Random Ordinary Differential Equations.................................... 257 12.3 Lax-Milgram Theory and Random PDEs.................................... 262 12.4 Bibliography...................................................................................... 273 !2.5 Exercises............................................................................................ 273 13 Non-Intrusive Methods..................................................................... 277 13.1 Non-Intrusive Spectral Methods.................................................... 278 13.2 Stochastic Collocation
.................................................................... 282 13.3 Gaussian Process Regression.......................................................... 288 13.4 Bibliography...................................................................................... 292 13.5 Exercises............................................................................................ 292 14 Distributional Uncertainty................................................................295 14.1 Maximum Entropy Distributions.................................................. 296 14.2 Hierarchical Methods...................................................................... 299 14.3 Distributional Robustness .............................................................. 299 14.4 Functional and Distributional Robustness .................................. 311
xii Contents 14.5 Bibliography...................................................................................... 315 14.6 Exercises............................................................................................ 316 References........................................................................................................ 319 Index.................................................................................................................. 339
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| institution | BVB |
| isbn | 9783319233949 |
| language | English |
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| physical | xii, 342 Seiten Diagramme |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | Springer |
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| series | Texts in applied mathematics |
| series2 | Texts in applied mathematics |
| spelling | Sullivan, T. J. 1982- Verfasser (DE-588)1081048786 aut Introduction to uncertainty quantification T.J. Sullivan Cham Springer [2015] © 2015 xii, 342 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics volume 63 Mathematische Methode (DE-588)4155620-3 gnd rswk-swf Unsicheres Schließen (DE-588)4361044-4 gnd rswk-swf Unsicheres Schließen (DE-588)4361044-4 s Mathematische Methode (DE-588)4155620-3 s DE-604 Erscheint auch als Online-Ausgabe 10.1007/978-3-319-23395-6 Erscheint auch als Online-Ausgabe 978-3-319-23395-6 Texts in applied mathematics volume 63 (DE-604)BV002476038 63 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028304574&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sullivan, T. J. 1982- Introduction to uncertainty quantification Texts in applied mathematics Mathematische Methode (DE-588)4155620-3 gnd Unsicheres Schließen (DE-588)4361044-4 gnd |
| subject_GND | (DE-588)4155620-3 (DE-588)4361044-4 |
| title | Introduction to uncertainty quantification |
| title_auth | Introduction to uncertainty quantification |
| title_exact_search | Introduction to uncertainty quantification |
| title_full | Introduction to uncertainty quantification T.J. Sullivan |
| title_fullStr | Introduction to uncertainty quantification T.J. Sullivan |
| title_full_unstemmed | Introduction to uncertainty quantification T.J. Sullivan |
| title_short | Introduction to uncertainty quantification |
| title_sort | introduction to uncertainty quantification |
| topic | Mathematische Methode (DE-588)4155620-3 gnd Unsicheres Schließen (DE-588)4361044-4 gnd |
| topic_facet | Mathematische Methode Unsicheres Schließen |
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