Artículos

Abstract: We propose a nonsmooth continuum mechanical model for discontinuous shear thickening flow. The model obeys a formulation as energy minimization problem and its solution satisfies a Stokes type system with a nonsmooth constitute relation. Solutions have a free boundary at which the behavior of the fluid changes. We present Sobolev as well as H¨older regularity results and study the limit of the model as the viscosity in the shear thickened volume tends to infinity. A mixed problem formulation is discretized using finite elements and a semismooth Newton method is proposed for the solution of the resulting discrete system. Numerical problems for steady and unsteady shear thickening flows are presented and used to study the solution algorithm, properties of the flow and the free boundary. These numerical problems are motivated by recently reported experimental studies of dispersions with high particle-to-fluid volume fractions, which often show a sudden increase of viscosity at certain strain rates.

Abstract: We propose the use of the Kantorovich--Rubinstein norm from optimal transport in imaging problems. In particular, we discuss a variational regularization model endowed with a Kantorovich--Rubinstein discrepancy term and total variation regularization in the context of image denoising and cartoon-texture decomposition. We point out connections of this approach to several other recently proposed methods such as total generalized variation and norms capturing oscillating patterns. We also show that the respective optimization problem can be turned into a convex-concave saddle point problem with simple constraints and hence can be solved by standard tools. Numerical examples exhibit interesting features and favorable performance for denoising and cartoon-texture decomposition.

Abstract: We state and analyse one-shot methods in function space for the optimal control of nonlinear partial differential equations (PDEs) that can be formulated in terms of a fixed-point operator. A general convergence theorem is proved by generalizing the previously obtained results in finite dimensions. As application examples we consider two nonlinear elliptic model problems: the stationary solid fuel ignition model and the stationary viscous Burgers equation. For these problems we present a more detailed convergence analysis of the method. The resulting algorithms are computationally implemented in combination with an adaptive mesh refinement strategy, which leads to an improvement in the performance of the one-shot approach.

Abstract: We consider the bilevel optimisation approach proposed in [5] for learning the optimal parameters in a Total Variation (TV) denoising model featuring for multiple noise distributions. In applications, the use of databases (dictionaries) allows an accurate estimation of the parameters, but reflects in high computational costs due to the size of the databases and to the nonsmooth nature of the PDE constraints. To overcome this computational barrier we propose an optimisation algorithm that, by sampling dynamically from the set of constraints and using a quasi-Newton method, solves the problem accurately and in an efficient way.

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.

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.

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.

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.