redmod.lyx

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\begin_body

\begin_layout Section
\start_of_appendix
Outline
\end_layout

\begin_layout Itemize
Consider model on domain 
\begin_inset Formula $\boldsymbol{x}$
\end_inset

 (e.g.
 positions in space or time) with unknown parameter fields 
\begin_inset Formula $\boldsymbol{a}(\boldsymbol{x})$
\end_inset

 that fulfills
\begin_inset Formula 
\[
\boldsymbol{F}(\boldsymbol{f}(\boldsymbol{x}),\boldsymbol{x};\boldsymbol{a}(\boldsymbol{x}))=0,
\]

\end_inset

yielding fields 
\begin_inset Formula $\boldsymbol{f}(\boldsymbol{x})$
\end_inset

 as a solution implicitly.
\end_layout

\begin_layout Itemize
Replace 
\begin_inset Formula $\boldsymbol{a}(\boldsymbol{x})$
\end_inset

 by (continuous) random fields.
 
\end_layout

\begin_deeper
\begin_layout Itemize
Scalar quantities can be thought of as constant fields, axisymmetric quantities
 as constant over 
\begin_inset Formula $\varphi$
\end_inset

 and plasma profiles as constant over 
\begin_inset Formula $\vartheta$
\end_inset

 and 
\begin_inset Formula $\varphi$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Itemize
Search for 
\begin_inset Formula $\boldsymbol{f}(\boldsymbol{x})$
\end_inset

 as random fields now as an output.
\end_layout

\begin_layout Itemize
Simplification 1: Parametrise 
\begin_inset Formula $\boldsymbol{a}(\boldsymbol{x})$
\end_inset

 by a finite set of scalar random variables 
\begin_inset Formula $\boldsymbol{\alpha}$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Itemize
Choose distribution of random variables according to measurement uncertainties
 via (Bayesian) inference.
\end_layout

\end_deeper
\begin_layout Itemize
Simplification 2: Parametrise 
\begin_inset Formula $\boldsymbol{f}(\boldsymbol{x})$
\end_inset

 by finite set of interpolation parameters 
\begin_inset Formula $\boldsymbol{\beta}$
\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Since 
\begin_inset Formula $\boldsymbol{f}(\boldsymbol{x};\boldsymbol{\alpha})$
\end_inset

 is a random field, 
\begin_inset Formula $\boldsymbol{\beta}(\boldsymbol{\alpha})$
\end_inset

 become random variables.
\end_layout

\begin_layout Itemize
How to translate from 
\begin_inset Formula $\boldsymbol{f}(\boldsymbol{x};\boldsymbol{\alpha})$
\end_inset

 to 
\begin_inset Formula $\boldsymbol{\beta}(\boldsymbol{\alpha})$
\end_inset

? 
\end_layout

\end_deeper
\begin_layout Itemize
Question: In which way are quantities continuous?
\end_layout

\begin_deeper
\begin_layout Itemize
Continuity in space/time: Should be valid up to certain order of differentiation
 in physical quantities.
\end_layout

\begin_layout Itemize
Continuity in parameters: Not necessarily, bifurcations possible.
 Possible to assume locally, otherwise model makes no sense.
\end_layout

\end_deeper
\begin_layout Itemize
Need to discretize each 
\begin_inset Formula $\beta_{k}(\boldsymbol{\alpha})$
\end_inset

 in parameter space to be able to interpolate/extrapolate over 
\begin_inset Formula $\boldsymbol{\alpha}$
\end_inset


\end_layout

\begin_layout Itemize
Methods: PCE and Gaussian processes
\end_layout

\begin_deeper
\begin_layout Itemize
Question: How to treat non-uniformly distributed random variables in PCE?
\end_layout

\begin_deeper
\begin_layout Itemize
Answer: Use Hermite polynomials with support 
\begin_inset Formula $(-\infty,\infty)$
\end_inset

 as a basis for Gaussian random variables (See Xiu).
\end_layout

\end_deeper
\end_deeper
\begin_layout Itemize
Question: How to treat structure-preserving algorithms in a consistent way?
\end_layout

\begin_deeper
\begin_layout Itemize
Simple answer: use structure-preserving FEM degrees of freedom (e.g.
 Raviart-Thomas) as 
\begin_inset Formula $\boldsymbol{\beta}$
\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
Attention: the output is plotted over parameter space.
 Uncertainties of output are evaluated either by Monte Carlo sampling of
 surrogate model, or by explicit sum over squared coefficients in PCE.
 
\end_layout

\begin_deeper
\begin_layout Itemize
Only 
\emph on
after
\emph default
 that, we can plot our random field over the domain 
\begin_inset Formula $\boldsymbol{x}$
\end_inset

 together with error bars.
 The problem of interpolating this field is a different one from the one
 of interpolating in parameter space, and for this, 
\begin_inset Formula $\boldsymbol{\beta}$
\end_inset

 are used.
\end_layout

\begin_layout Itemize
Question: How to translate uncertainties in 
\begin_inset Formula $\boldsymbol{\beta}(\boldsymbol{\alpha})$
\end_inset

 back to uncertainties in 
\begin_inset Formula $\boldsymbol{f}(\boldsymbol{x};\boldsymbol{\alpha})$
\end_inset

?
\end_layout

\end_deeper
\begin_layout Section
Literature
\end_layout

\begin_layout Standard
The Geometry of Random Fields + Random fields and Geometry by Adler
\end_layout

\end_body
\end_document