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What Is Offline And Online Scale Computation?

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3.1 Offline and online scale computation For different values of κ the multiscale basis changes as the pde solution changes. To address this issue offline-online computation is used where in offline stage for representative values over a grid of κ’s the offline basis and the corresponding linear space is constructed. The online basis then constructed solving a local problem for each candidate κ∗, using the principal component direction of the spectral problem. The mathematical details of the online-offline technique can be found in Guha and Tan (2016), Efendiev et al( 2011). For a set of values κ1,...,κN solving local eigen value problem to detect the dominant scale (inverse of the eigen value) a offline space is constructed with basis …show more content…

In particular κ ∼ GP(K) and log(κ(x)) = ﰅξ √λ e (x), for x ∈ D where K is covariance kernel on D with iii eigen values λi and eigen vectors ei(x) and ξi ∼ N(0,1). 13 The goal is to derive the posterior distribution of κ’s or equivalently ξi’s given observation and the model. Posterior for κ The posterior for κ can be written as Π(κ|Y ) = p(Y |κ)π(κ) where p(Y |K) is the likelihood of Y given κ which is a normal distribution for the above error distribution assumption around the solution K(u,κ) from the pde. This posterior is not closed form and Metropolis-Hastings MCMC sampling is needed to draw sample from κ. The likelihood computation involves solving for κ and this solution would be calculated using basis function (multiscale basis for example). Using simple coarse scale basis function that has a bilinear form in the coarse grid and zero outside the coarse region, gives a low resolution solution that can be used to filter out the bad proposal value in MCMC set up. 3.3 Multilevel sampling with residue information Selecting the additional basis with this model based distribution the randomized forward solution can be incorporated into a multilevel MCMC methodology (Dodwell et al., 2014). Two level sampling is proposed for convenience but additional levels can be used. Two level sampling are done on the coarse level and fine scale; in coarse scale using coarse scale basis and in the fine level using multiscale basis.

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