We solve a parameter estimation problem by computing the conditional expectation

What is it about?

We developed a non-linear extension of the Kalman filter update. Given measurement data we can perform a kind of Bayesian update of the polynomial chaos coefficients. We update not the probability density function (as in the classical Bayesian theory), but Polyomial Chaos Expansion coefficients. The method does not require the prior/posterior to be Gaussian.

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Why is it important?

After you used stochastic Galerkin method (very popular) to solve a PDE with uncertain coefficients, the obtained solution is expressed as a PCE or (KLE+PCE) expansion, i.e. the solution of the stochastic Galerkin system are the PCE (or KLE+PCE) coefficients. Assuming you have some measurements, how to compute the Bayesian update of the prior distribution used in PDE above? One way is to sample the computed PCE, generate a large sample, estimate the probability density, compute the likelihood, apply the classical Bayesian formula etc. Our, the suggested approach, is to derive the Bayesian update formula direct for PDE coefficients. The new Bayesian approach will be New-PCE-coefficients = New-PCE-coefficients + non-linear update.

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