A bilevel learning approach for optimal observation placement in variational data assimilation

Abstract: Inverse Problems ACCEPTED MANUSCRIPT A bilevel learning approach for optimal observation placement in variational data assimilation Paula Castro1 and Juan Carlos De los Reyes2 Accepted Manuscript online 8 October 2019 • © 2019 IOP Publishing Ltd What is an Accepted Manuscript? Download Accepted Manuscript PDF 5 Total downloads Turn on MathJax Get permission to re-use this article Share this article Share this content via email Share on Facebook Share on Twitter Share on Google+ Share on Mendeley Article information Abstract In this paper we propose a bilevel optimization approach for the placement of observations in variational data assimilation problems. Within the framework of supervised learning, we consider a bilevel problem where the lower level task is the variational reconstruction of the initial condition of a semilinear system, and the upper level problem solves the optimal placement with help of a sparsity inducing norm. Due to the pointwise nature of the observations, an optimality system with regular Borel measures on the right-hand side is obtained as necessary optimality condition for the lower level problem. The latter is then considered as constraint for the upper level instance, yielding an optimization problem constrained by a multi-state system with measures. We demonstrate existence of Lagrange multipliers and derive a necessary optimality system characterizing the optimal solution of the bilevel problem. The numerical solution is carried out also on two levels. The lower level problem is solved using a standard BFGS method, while the upper level one is solved by means of a projected BFGS algorithm based on the estimation of $\epsilon$-active sets. A penalty function is also considered for enhancing sparsity of the location weights. Finally some numerical experiments are presented to illustrate the main features of our approach.

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Abstract:

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Analyzing the dynamics of deterministic systems from a hypergraph theoretical point of view.

Abstract: To model the dynamics of discrete deterministic systems, we extend the Petri nets framework by a priority relation between conflicting transitions, which is encoded by orienting the edges of a transition conflict graph. The aim of this paper is to gain some insight into the structure of this conflict graph and to characterize a class of suitable orientations by an analysis in the context of hypergraph theory.

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Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization.

Abstract: We propose a nonsmooth PDE-constrained optimization approach for the determination of the correct noise model in total variation (TV) image denoising. An optimization problem for the determination of the weights corresponding to different types of noise distributions is stated and existence of an optimal solution is proved. A tailored regularization approach for the approximation of the optimal parameter values is proposed thereafter and its consistency studied. Additionally, the differentiability of the solution operator is proved and an optimality system characterizing the optimal solutions of each regularized problem is derived. The optimal parameter values are numerically computed by using a quasi-Newton method, together with semismooth Newton type algorithms for the solution of the TV-subproblems.

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Extending the applicability of Newton’s method for k-Fréchet differentiable operators in Banach spaces

Abstract: We extend the applicability of Newton’s method for k-Fréchet differentiable operators in a Banach space setting by using a more flexible way of computing upper bounds on the inverses of the operators involved. In particular, we improve and extend the recent works by Ezquerro et al. (2012, 2013) [13,15]. Moreover, we illustrate our study with some numerical examples involving Hammerstein integral equations.

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Extending the applicability of Newton’s method by improving a local result due to Dennis and Schnabel

Abstract: We present a tighter local convergence result for Newton’s method under generalized conditions of Kantorovich type than the one given by Dennis and Schnabel (Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia, 1996) and by Ezquerro et al. (J Comput Appl Math 236:2246–2258, 2012) by using more precise majorizing functions and sequences. These improvements are obtained under the same computational cost as in the earlier studies. Numerical examples are also provided to show that the new convergence radii are larger and the new error bounds are tighter than the older ones.

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