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Adaptive method for indirect identification of the statistical properties of random fields in a Bayesian framework

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Abstract

This work considers the challenging problem of identifying the statistical properties of random fields from indirect observations. To this end, a Bayesian approach is introduced, whose key step is the nonparametric approximation of the likelihood function from limited information. When the likelihood function is based on the evaluation of an expensive computer code, this work also proposes a method to select iteratively new design points to reduce the uncertainties on the results that are due to the approximation of the likelihood. Two applications are finally presented to illustrate the efficiency of the proposed procedure: a first one based on analytic data, and a second one dealing with the identification of the random elasticity field of an heterogeneous microstructure.

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Authors and Affiliations

  1. CEA/DAM/DIF, 91297, Arpajon, France

    Guillaume Perrin

  2. MSME UMR 8208 CNRS, Université Paris-Est, Marne-la-Vallée, France

    Christian Soize

Authors
  1. Guillaume Perrin
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  2. Christian Soize
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Correspondence to Guillaume Perrin.

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Appendix

Appendix

1.1 A.1. Proof of the equality of Eq. 9

Let \(\varvec{A},\varvec{B},\varvec{D}\) be the block decomposition matrices of \(\widehat{\varvec{C}}^{-1}\):

$$\begin{aligned} \widehat{\varvec{C}}^{-1}=\left[ \begin{array}{cc} \varvec{A} &{} \varvec{B} \\ \varvec{B}^T &{} \varvec{D} \end{array} \right] . \end{aligned}$$
(30)

Using the Schur complement, if follows that:

$$\begin{aligned} \left\{ {\begin{matrix} &{} \widehat{\varvec{C}}_{\varvec{ZZ}}^{-1}=\varvec{D}-\varvec{B}^T\varvec{A}^{-1}\varvec{B}, \\ &{} (\widehat{\varvec{C}}_{\varvec{YY}}-\widehat{\varvec{C}}_{\varvec{YZ}}\widehat{\varvec{C}}_{\varvec{ZZ}}^{-1}\widehat{\varvec{C}}_{\varvec{YZ}}^T)^{-1}=\varvec{A}, \\ &{} -\widehat{\varvec{C}}_{\varvec{YZ}}\widehat{\varvec{C}}_{\varvec{ZZ}}^{-1}=\varvec{A}^{-1}\varvec{B}. \end{matrix}} \right. \end{aligned}$$
(31)

It comes

$$\begin{aligned}&\left( (\varvec{y},\varvec{z})-(\varvec{Y}(\omega _m),\varvec{Z}(\omega _m)) \right) ^T (h^2\widehat{\varvec{C}})^{-1} \left( (\varvec{y},\varvec{z})-(\varvec{Y}(\omega _m),\varvec{Z}(\omega _m)) \right) \nonumber \\&\quad =\frac{1}{h^2}\left( {\begin{matrix} &{} (\varvec{y}-\varvec{Y}(\omega _m))^T\varvec{A}(\varvec{y}-\varvec{Y}(\omega _m))+2(\varvec{y}-\varvec{Y}(\omega _m))^T\varvec{A}\varvec{A}^{-1}\varvec{B}\left( \varvec{z}-\varvec{Z}(\omega _m) \right) \\ &{} +(\varvec{z}-\varvec{Z}(\omega _m))^T\varvec{D}(\varvec{z}-\varvec{Z}(\omega _m))\end{matrix}} \right) \nonumber \\&\quad = \frac{1}{h^2}\left( {\begin{matrix}(\varvec{y}-\varvec{Y}(\omega _m)&{} +\varvec{A}^{-1}\varvec{B}\left( \varvec{z}-\varvec{Z}(\omega _m) \right) )^T\varvec{A}(\varvec{y}-\varvec{Y}(\omega _m)+\varvec{A}^{-1}\varvec{B}\left( \varvec{z}-\varvec{Z}(\omega _m) \right) )\\ &{} +(\varvec{z}-\varvec{Z}(\omega _m))^T\left( \varvec{D}-\varvec{B}^T\varvec{A}^{-1}\varvec{B} \right) (\varvec{z}-\varvec{Z}(\omega _m)) \end{matrix}} \right) \nonumber \\&\quad = (\varvec{y}-\varvec{\mu }_n(\varvec{z}))^{T}\varvec{C}_n^{-1}(\varvec{y}-\varvec{\mu }_n(\varvec{z}))+\frac{1}{h^{2}}\left( \varvec{z}-\varvec{Z}(\omega _m) \right) ^T\varvec{C}_{\varvec{ZZ}}^{-1} \left( \varvec{z}-\varvec{Z}(\omega _m) \right) \end{aligned}$$
(32)

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Perrin, G., Soize, C. Adaptive method for indirect identification of the statistical properties of random fields in a Bayesian framework. Comput Stat 35, 111–133 (2020). https://doi.org/10.1007/s00180-019-00936-5

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