Primal-dual block-proximal splitting for a class of non-convex problems

Abstract: We develop block structure adapted primal-dual algorithms for non-convex non-smoothoptimisation problems whose objectives can be written as compositionsG(x)+F(K(x))of non-smooth block-separable convex functionsGandFwith a non-linear Lipschitz-differentiable op-eratorK. Our methods are renements of the non-linear primal-dual proximal splitting methodfor such problems without the block structure, which itself is based on the primal-dual proximalsplitting method of Chambolle and Pock for convex problems. We propose individual step lengthparameters and acceleration rules for each of the primal and dual blocks of the problem. This allowsthem to convergence faster by adapting to the structure of the problem. For the squared distanceof the iterates to a critical point, we show localO(1/N),O(1/N2)and linear rates under varyingconditions and choices of the step lengths parameters. Finally, we demonstrate the performanceof the methods on practical inverse problems: diffusion tensor imaging and electrical impedancetomography

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

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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On the statistical analysis of ocean wave directional spectra

Abstract: Given the growing availability of directional spectra of ocean waves, we explore two different statistical approaches to mine large spectra databases: Spectral Partitions Statistics (SPS) and Self-Organizing Maps (SOM). The first method is not new in the literature, while the second one is for the first time here applied to directional wave spectra. The main goal is to improve the characterization of the directional wave climate at a site, providing a more complete and consistent description than that obtained from traditional statistical methods based on integral spectral parameters (e.g., $H_s$, $T_m$, $\theta_m$). Indeed, while the use of integral parameters allows a direct application of standard techniques for statistical analysis, important information related to the physics of the processes may be overlooked (e.g., the presence of multiple wave systems, for instance locally and remotely generated). The two proposed methods do not exclude integral parameters analysis, but they further allow accounting for different events (e.g., with different genesis) independently. Although SPS and SOM are equally valid for both numerical model and observational data, we illustrate their potential using a 37-year long (1979–2015) model dataset of directional wave spectra at a study site in the western Mediterranean Sea. We show that standard integral parameters fail to show the complex and even multimodal conditions at this site, that are otherwise revealed by the directional spectra statistical analysis. Although the processing pathways and the resulting indicators of both SPS and SOM are substantially different, we observe that their results are mutually consistent, and provide a better insight into the physical processes at work.

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A Hybrid Physical-Statistical Algorithm for SAR Wave Spectra Quality Assessment

Abstract: A new approach for assessing the quality of Synthetic Aperture Radar (SAR) wave spectra is presented here. The algorithm addresses two specific issues, related to the 180° directional ambiguity in the propagation direction inherent to SAR measurements, and the removal of noise. In spite of several and progressive advances in the mapping and retrieval of SAR wave spectra, these issues are persistent in the existing official databases and hinder the use of these data for practical uses. This new algorithm is based on a recently developed database of long-term global wave spectral characteristics, which allows estimating the occurrence probability of every individual wave system found in the observed spectra, and that of its ambiguous pair. In addition, assuring the spatial consistency of the wave systems along track, helps obtaining more robust results. In this sense, wave spectra within a track are processed in several steps, so that wave systems with more likelihood of being correct are processed first, and consequently used to evaluate the more challenging components. Accordingly, a specific quality label is assigned to every wave system depending on the certainty achieved. An Envisat track over a challenging area with multiple crossing swells is used to illustrate the performance of the algorithm.

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Total generalized variation regularization in data assimilation for Burgers' equation

Abstract: We propose a second-order total generalized variation (TGV) regularization for the reconstruction of the initial condition in variational data assimilation problems. After showing the equivalence between TGV regularization and a Bayesian MAP estimator, we focus on the detailed study of the inviscid Burgers' data assimilation problem. Due to the difficult structure of the governing hyperbolic conservation law, we consider a discretize–then–optimize approach and rigorously derive a first-order optimality condition for the problem. For the numerical solution, we propose a globalized reduced Newton-type method together with a polynomial line-search strategy, and prove convergence of the algorithm to stationary points. The paper finishes with some numerical experiments where, among others, the performance of TGV–regularization compared to TV–regularization is tested.

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A difference-of-convex functions approach for sparse PDE optimal control problems with nonconvex costs

Abstract: We propose a local regularization of elliptic optimal control problems which involves the nonconvex $L^q$ quasi-norm penalization in the cost function. The proposed Huber type regularization allows us to formulate the PDE constrained optimization instance as a DC programming problem (difference of convex functions) that is useful to obtain necessary optimality conditions and tackle its numerical solution by applying the well known DC algorithm used in nonconvex optimization problems. By this procedure we approximate the original problem in terms of a consistent family of parameterized nonsmooth problems for which there are efficient numerical methods available. Finally, we present numerical experiments to illustrate our theory with different configurations associated to the parameters of the problem.

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Acceleration and global convergence of a first-order primal-dual method for nonconvex problems

Abstract: The primal-dual hybrid gradient method, modified (PDHGM, also known as the Chambolle{Pock method), has proved very successful for convex optimization problems involving linear operators arising in image processing and inverse problems. In this paper, we analyze an extension to nonconvex problems that arise if the operator is nonlinear. Based on the idea of testing, we derive new step-length parameter conditions for the convergence in infinite-dimensional Hilbert spaces and provide acceleration rules for suitably (locally and/or partially) monotone problems. Importantly, we prove linear convergence rates as well as global convergence in certain cases. We demonstrate the efficacy of these step-length rules for PDE-constrained optimization problems.

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Block-proximal methods with spatially adapted acceleration

Abstract: We study and develop (stochastic) primal-dual block-coordinate descent methods for convex problems based on the method due to Chambolle and Pock. Our methods have known convergence rates for the iterates and the ergodic gap of $O(1/N^2)$ if each block is strongly convex, $O(1/N)$ if no convexity is present, and more generally a mixed rate $O(1/N2) + O(1/N)$ for strongly convex blocks if only some blocks are strongly convex. Additional novelties of our methods include blockwise-adapted step lengths and acceleration as well as the ability to update both the primal and dual variables randomly in blocks under a very light compatibility condition. In other words, these variants of our methods are doubly-stochastic. We test the proposed methods on various image processing problems, where we employ pixelwise-adapted acceleration.

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