Our paper on the novel linearization method for non-linear, pde constrained inverse problems can be considered as an interesting, novel approach, which can open up future research directions. As a first step we split the non-linear statistical problem into a linear-statistical and a non-linear analytical problem. The linear statstical problems can be solved substantially faster than their non-linear counterpart. Furthermore, there is a much broader theoretical underpinning and methodological development for linear inverse problems, which can be applied in the non-linear setting following our strategy. For the non-linear analytic problem one can either analytically compute the explicit solution or use numerical methods approximating it. Combining them results in a faster approach with strong theoretical guarantees, which we have already applied on a range of pde constrained inverse problems. Examples include the time-independent Schrodinger equation (hyperbolic pde), Heat equation with absorption term (parabolic pde), Darcy’s flow problem, …etc. We plan to extend the number of examples, including for instance the Navier-Stokes equation (2d) and non-Abelian X-ray transform on surfaces. Another line of future work is to combine this approach with variational, distributed and other approximation methods to further speed up the (otherwise very time-consuming) computations.
In another line fo research, we have developed a skew symmetric version of the Laplace approximation. We have demonstrated both theoretically and numerically that it provides an order of magnitude better approximation of the posterior than the standard Gaussian one. This motivated our follow up work, where we have developed a skewness inducing factor, which can be used to any symmetric approximation. Examples include Gaussian variational Bayes, expectation propagation, Laplace approximation,…etc. We have demonstrate both theoretically and numerically that the proposed approach can indeed substantially improve the approximation of the posterior while maintaining the (almost same) computational cost as the symmetric approximation. In contrast to higher order approximations (e.g. Edgeworth expansions) our method provides a real, easy to compute and sample from density. We believe that the present approach has the potential to be combined with any standard symmetric approximation used in practice with a simple modification and hence will be implemented in standard software, e.g. INLA, ALA.
