preview

What Is Fixed And Additional Basis

Decent Essays
Open Document

3.1.1 Fixed and additional basis 3.2 Model and prior Consider model Y = D(u(x, t)) + e; e ∼ G and prior κ ∼ π(κ). Let, collectively denote the set of additional basis by {φon,+,ωi,n}, which contains the residual scale information for a particular κ. κ A pde model based selection probability is assumed on the additional basis and subregions as in sequential sampling discussed earlier: P(φon,+,ωi,n = 1|Iωi,n = 1) = αˆon,+,ωi,n and P(Iωi,n = 1) = jj αˆωi,n, where Iωi,n = 1 denotes that basis are added on the ωi computational grid for n th time grid. Prior on κ Prior on κ depends on parametrization and the underlying model. For parabolic equation and positive resolution parameter κ, a popular choice is using Gaussian process on log(κ). In …show more content…

Solving the equation in coarse scale gives an approximate solution and a coarse scale likelihood. If that step is accepted, in the fine scale computation the randomized solution can be used based on the selection probabilities of the additional basis functions and, the multiscale solution and the likelihood can be computed for the acceptance-rejection step. For a given κ, the likelihood around the coarse scale solution is denoted by p1(Y|κ) and the corresponding posterior is denoted by Π1(κ|Y ), and similarly for multiscale basis they are denoted by p2(Y |κ) and Π2(κ|Y ), respectively. 3.3.1 Multilevel MCMC algorithm A brief outline can be given in the following. 1)Construct computational grids. 2) Construct offline space by solving the local problems for κ1, . . . , κN from the prior. 3) for i=2:NMCMC and starting with some initial value of κ at i=1. • For level l = 1, sample κ from proposal q1 = q(κ|κli), where κli is the current value at level l = 1. • Generate u from U [0, 1]), Uniform between 0 and 1. Let α = min{1, Π1 (κ|Y )ql (κli |κ) }. l Π1(κli|Y )ql(κ|κli) • If u<αl then go to l+1 th level. Accept κli+1 =κ if u<αl and go to the l+1 th level. Model based distribution is used to add new basis at this stage and extra basis are added j with probability αﰇk,+ for ωj if ωj is selected, and the forward problem is solved using this basis set and the likelihood and posterior

Get Access
Related
Get Access