BGH meeting Feb 2019

Bayesian Inference

P(\mathbf{m}|\mathbf{d}) \propto P(\mathbf{d}|\mathbf{m}) \cdot P(\mathbf{m})

P(\mathbf{m}|\mathbf{d}) \propto P(\mathbf{d}|\mathbf{m}) \cdot P(\mathbf{m})

Posterior ~ likelihood x prior

P(\mathbf{m}|\mathbf{d}) = A \exp (-S(\mathbf{m}))

P(\mathbf{m}|\mathbf{d}) = A \exp (-S(\mathbf{m}))

S(\mathbf{m}) = \sum_i \frac{\lvert g^i(\mathbf{m}) - \mathrm{d}^i \lvert^2}{(2\sigma_{\mathrm{d}}^i)^2} + \sum_{j} \frac{\vert \mathrm{m}^j - \mathrm{m}^j_p \vert^2}{(2\sigma^j_p)^2}

S(\mathbf{m}) = \sum_i \frac{\lvert g^i(\mathbf{m}) - \mathrm{d}^i \lvert^2}{(2\sigma_{\mathrm{d}}^i)^2} + \sum_{j} \frac{\vert \mathrm{m}^j - \mathrm{m}^j_p \vert^2}{(2\sigma^j_p)^2}

-norm objective function

l_2

l_2

Bayes theorem describes the posterior as the probability of a model given the data

The posterior probability can be evaluated with an objective function

We seek the maximum a posteriori (MAP) model, which can be obtained by minimising S(m).

Adjoint inversion

Efficiently invert a large number of parameters

Added complexity to compute forward problem AND gradient vectors

Entrapment by local minima

Number of evaluations increase exponentially with dimensions

Copes better with highly nonlinear problems

Posterior distribution

MCMC

Adjoint

+

-

Thermal solver

\nabla(\mathbf{k} \nabla \mathbf{T}) = - \mathbf{H}

\nabla(\mathbf{k} \nabla \mathbf{T}) = - \mathbf{H}

Objective function

S(\mathbf{m}) = \sum_i \frac{\lvert g^i(\mathbf{m}) - \mathrm{d}^i \lvert^2}{(2\sigma_{\mathrm{d}}^i)^2} + \sum_{j} \frac{\vert \mathrm{m}^j - \mathrm{m}^j_p \vert^2}{(2\sigma^j_p)^2}

S(\mathbf{m}) = \sum_i \frac{\lvert g^i(\mathbf{m}) - \mathrm{d}^i \lvert^2}{(2\sigma_{\mathrm{d}}^i)^2} + \sum_{j} \frac{\vert \mathrm{m}^j - \mathrm{m}^j_p \vert^2}{(2\sigma^j_p)^2}