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1 Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen PARAMetric UNCertainties, Budapest STOCHASTIC PROCESSES AND FIELDS Noémi Friedman Institut für Wissenschaftliches Rechnen,
2 Introduction Introduction Random fields and processes Separated representation Karhunen-Loeve expansion Noémi Friedman page 2
3 Random fields and processes Noémi Friedman page 3
4 Introduction to random processes and fields Example: diffusion equation Example: diffusion equation : spatial domain : stochastic domain The conductivity is usually a random field Now we only consider random fields Noémi Friedman page 4
5 Introduction to random processes and fields Description of stochastic field: 1) As a function-valued random variable Fix the probabilistic space to one specific outcome: Realizations/trajectories/ sample paths From the ensemble of realizations we can conclude marginal distributions 2) As a collection of random variables over the spatial domain Fix the spatial coordinate to a point in, and describe the probabilistic behavior at that certain spatial point with a random variables: Noémi Friedman page 5
6 Introduction to random processes and fields The statistics can be given at the fixed points: continious random field: spatial domain is a countable infinite set discrete random field: space is discretized discrete random field Collection of random variables, their distribution is called the marginal distribution Covariance: Noémi Friedman page 6
7 Introduction to random processes and fields The covariance describes the linear dependencies between two spatial points. For full description the joint distribution is also needed! Exception: Gaussian random fields mean and covariance with marginals gives full description Stochastically homogenous field (stationary process) Special case: spatial dependency depends only on the distance White noise: the collection of random variables are independent Often strong spatial dependence (dependence can be described with some functions living in some Sobolev spaces) Noémi Friedman page 7
8 Separated representation of random fields Noémi Friedman page 8
9 Separated representation of random fields Noémi Friedman page 9
10 Separated representation of random fields Noémi Friedman page 10
11 Separated representation of random fields Noémi Friedman page 11
12 Separated representation of random fields Noémi Friedman page 12
13 Separated representation of random fields Noémi Friedman page 13
14 Separated representation of random fields Noémi Friedman page 14
15 The Karhunen- Loeve expansion Noémi Friedman page 15
16 The Karhunen- Loeve expansion Noémi Friedman page 16
17 The Karhunen- Loeve expansion Noémi Friedman page 17
18 The Karhunen- Loeve expansion Noémi Friedman page 18
19 The Karhunen- Loeve expansion Noémi Friedman page 19
20 The Karhunen- Loeve expansion Noémi Friedman page 20
21 The Karhunen- Loeve expansion Noémi Friedman page 21
22 The Karhunen- Loeve expansion Noémi Friedman page 22
23 Thank you for your attention! Noémi Friedman page 23
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