uq of spdes driven by multi-dimensional gaussian processes

Multi-dimensional Levy process Gaussian process Jump process Drift Lepage's representati on Levy copula Gaussian copula How to describe the dependence structure ?

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Page 1: UQ of SPDEs driven by multi-dimensional Gaussian processes

Multi-dimensional Levy process

Gaussian process Jump process Drift

Lepage's representati

onLevy copula

Gaussian copula

How to describe the dependence structure ?

Page 2: UQ of SPDEs driven by multi-dimensional Gaussian processes

What is a Gaussian copula ?

In general As an example

margins copula

marginal CDF + copula =

joint PDF

standard normal margins (CDF)

2D Gaussian copula

covariance matrix

joint PDF

Page 3: UQ of SPDEs driven by multi-dimensional Gaussian processes

How many kinds of dependence structures can we describe?

Families of parameterized copulae

One-parameter families

8 families of copulae

Two-parameter families

UQ of SPDEs driven by multi-dimensional Gaussian processes ?

SPDE

Gaussian Margins

Copula w/ parameters

+joint PDF w/ parameters

This is only an integration problem on the sample space w/ known distribution .

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