Bayesian Inference for Scanning Probe Microscopy Forcefield Inversion
The problem of fully Bayesian force-field inversion given observations from scanning probe microscopy will be considered here. The forward model is given by a simple harmonic oscillator perturbed by the nonlinear force-field, which is parametrized by a linear interpolant or expansion in some other basis. This is a novel data assimilation problem  which has not been investigated yet and can potentially transform the field of scanning probe microscopy with full information capture. This field of imagining is involved with fine scale characterization of materials, which can in turn be used for example in design of materials. One can start with a Laplace approximation, which yields good root mean-square error for identical twin numerical experiments with simulated data. For highly nonlinear interactions a fully nonlinear method is required, such as Metropolis- Hastings MCMC. In particular, the newly developed multilevel Monte Carlo methodology is applicable in this context, and is capable of significantly speeding up convergence, both in a monolithic static context , as well as a dynamical context [3, 2]. This is therefore an interesting problem from the modeling perspective, the application perspective, as well as the methodology and algorithms perspective.
 Alexandros Beskos, Ajay Jasra, Kody Law, Raul Tempone, and Yan Zhou. Multilevel sequential Monte Carlo samplers. Stochastic Processes and their Applications, 127(5):1417–1440, 2017.
 H'akon Hoel, Kody Law, and Raul Tempone. Multilevel ensemble Kalman filter. SIAM Journal of Numerical Analysis, 54(3):1813–1839, 2016.
 AjayJasra, KengoKamatani, KodyJHLaw, and YanZhou. Multilevel particle filters. SIAM Journal on Numerical Analysis, 55(6):3068– 3096, 2017.
 Kody Law, Andrew Stuart, and Kostas Zygalakis. Data Assimilation: A Mathematical Introduction, volume preprint. Springer-Verlag Berlin, 2015.