Hanne Kekkonen

Martin Burger, Tapio Helin, Hanne Kekkonen

In this paper we consider variational regularization methods for inverse problems with large noise that is in general unbounded in the image space of the forward operator. We introduce a Banach space setting that allows to define a reasonable notion of solutions for more general noise in a larger space provided one has sufficient mapping properties of the forward operators. A key observation, which guides us through the subsequent analysis, is that such a general noise model can be understood with the same setting as approximate source conditions (while a standard model of bounded noise is related directly to classical source conditions). Based on this insight we obtain a quite general existence result for regularized variational problems and derive error estimates in terms of Bregman distances. The latter are specialized for the particularly important cases of one- and p-homogeneous regularization functionals. As a natural further step we study stochastic noise models and in particular white noise, for which we derive error estimates in terms of the expectation of the Bregman distance. The finiteness of certain expectations leads to a novel class of abstract smoothness conditions on the forward operator, which can be easily interpreted in the Hilbert space case. We finally exemplify the approach and in particular the conditions for popular examples of regularization functionals given by squared norm, Besov norm and total variation, respectively.

Jeffrey van der Voort, Martin Verlaan, Hanne Kekkonen

Currently, more and more machine learning (ML) surrogates are being developed for computationally expensive physical models. In this work we investigate the use of a Multi-Fidelity Ensemble Kalman Filter (MF-EnKF) in which the low-fidelity model is such a machine learning surrogate model, instead of a traditional low-resolution or reduced-order model. The idea behind this is to use an ensemble of a few expensive full model runs, together with an ensemble of many cheap but less accurate ML model runs. In this way we hope to reach increased accuracy within the same computational budget. We investigate the performance by testing the approach on two common test problems, namely the Lorenz-2005 model and the Quasi-Geostrophic model. By keeping the original physical model in place, we obtain a higher accuracy than when we completely replace it by the ML model. Furthermore, the MF-EnKF reaches improved accuracy within the same computational budget. The ML surrogate has similar or improved accuracy compared to the low-resolution one, but it can provide a larger speed-up. Our method contributes to increasing the effective ensemble size in the EnKF, which improves the estimation of the initial condition and hence accuracy of the predictions in fields such as meteorology and oceanography.