Reproducing kernel Hilbert spaces and variable metric algorithms in PDE constrained shape optimisation

In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE constrained shape optimisation problems. We show that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations. The shape gradients associated with radial kernels depend on a so called smoothing parameter that allows a smoothness adjustment of the shape during the optimisation process. Besides, this smoothing parameter can be used to modify the movement of the shape. The theoretical findings are verified in a number of numerical experiments

Reproducing kernel Hilbert spaces and variable metric algorithms in PDE constrained shape optimisation

In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE constrained shape optimisation problems. We show that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations. The shape gradients associated with radial kernels depend on a so called smoothing parameter that allows a smoothness adjustment of the shape during the optimisation process. Besides, this smoothing parameter can be used to modify the movement of the shape. The theoretical findings are verified in a number of numerical experiments

Piecewise rigid curve deformation via a Finsler steepest descent

This paper introduces a novel steepest descent flow in Banach spaces. This extends previous works on generalized gradient descent, notably the work of Charpiat et al., to the setting of Finsler metrics. Such a generalized gradient allows one to take into account a prior on deformations (e.g., piecewise rigid) in order to favor some specific evolutions. We define a Finsler gradient descent method to minimize a functional defined on a Banach space and we prove a convergence theorem for such a method. In particular, we show that the use of non-Hilbertian norms on Banach spaces is useful to study non-convex optimization problems where the geometry of the space might play a crucial role to avoid poor local minima. We show some applications to the curve matching problem. In particular, we characterize piecewise rigid deformations on the space of curves and we study several models to perform piecewise rigid evolution of curves

Power disturbance monitoring through techniques for novelty detection on wind power and photovoltaic generation

Novelty detection is a statistical method that verifies new or unknown data, determines whether these data are inliers (within the norm) or outliers (outside the norm), and can be used, for example, in developing classification strategies in machine learning systems for industrial applications. To this end, two types of energy that have evolved over time are solar photovoltaic and wind power generation. Some organizations around the world have developed energy quality standards to avoid known electric disturbances; however, their detection is still a challenge. In this work, several techniques for novelty detection are implemented to detect different electric anomalies (disturbances), which are k-nearest neighbors, Gaussian mixture models, one-class support vector machines, self-organizing maps, stacked autoencoders, and isolation forests. These techniques are applied to signals from real power quality environments of renewable energy systems such as solar photovoltaic and wind power generation. The power disturbances that will be analyzed are considered in the standard IEEE-1159, such as sag, oscillatory transient, flicker, and a condition outside the standard attributed to meteorological conditions. The contribution of the work consists of the development of a methodology based on six techniques for novelty detection of power disturbances, under known and unknown conditions, over real signals in the power quality assessment. The merit of the methodology is a set of techniques that allow to obtain the best performance of each one under different conditions, which constitutes an important contribution to the renewable energy systems.Postprint (published version

New Directions in Geometric and Applied Knot Theory

The aim of this book is to present recent results in both theoretical and applied knot theory—which are at the same time stimulating for leading researchers in the field as well as accessible to non-experts. The book comprises recent research results while covering a wide range of different sub-disciplines, such as the young field of geometric knot theory, combinatorial knot theory, as well as applications in microbiology and theoretical physics

Computing complete Lyapunov functions for discrete-time dynamical systems

A complete Lyapunov function characterizes the behaviour of a general discrete-time dynamical system. In particular, it divides the state space into the chain-recurrent set where the complete Lyapunov function is constant along trajectories and the part where the flow is gradient-like and the complete Lyapunov function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about attractors, repellers, and basins of attraction. We propose two novel classes of methods to compute complete Lyapunov functions for a general discrete-time dynamical system given by an iteration. The first class of methods computes a complete Lyapunov function by approximating the solution of an ill-posed equation for its discrete orbital derivative using meshfree collocation. The second class of methods computes a complete Lyapunov function as solution of a minimization problem in a reproducing kernel Hilbert space. We apply both classes of methods to several examples

Numerical homogenization: multi-resolution and super-localization approaches

Multi-scale problems arise in many scientific and engineering applications, where the effective behavior of a system is determined by the interaction of effects at multiple scales. To accurately simulate such problems without globally resolving all microscopic features, numerical homogenization techniques have been developed. One such technique is the Localized Orthogonal Decomposition (LOD). It provides reliable approximations at coarse discretization levels using problem-adapted basis functions obtained by solving local sub-scale correction problems. This allows the treatment of problems with heterogeneous coefficients without structural assumptions such as periodicity or scale separation. This thesis presents recent achievements in the field of LOD-based numerical homogenization. As a starting point, we introduce a variant of the LOD and provide a rigorous error analysis. This LOD variant is then extended to the multi-resolution setting using the Helmholtz problem as a model problem. The multi-resolution approach allows to improve the accuracy of an existing LOD approximation by adding more discretization levels. All discretization levels are decoupled, resulting in a block-diagonal coarse system matrix. We provide a wavenumber-explicit error analysis that shows convergence under mild assumptions. The fast numerical solution of the block-diagonal coarse system matrix with a standard iterative solver is demonstrated. We further present a novel LOD-based numerical homogenization method named Super-Localized Orthogonal Decomposition (SLOD). The method identifies basis functions that are significantly more local than those of the LOD, resulting in reduced computational cost for the basis computation and improved sparsity of the coarse system matrix. We provide a rigorous error analysis in which the stability of the basis is quantified a posteriori. However, for challenging problems, basis stability issues may arise degrading the approximation quality of the SLOD. To overcome these issues, we combine the SLOD with a partition of unity approach. The resulting method is conceptually simple and easy to implement. Higher order versions of this method, which achieve higher order convergence rates using only the regularity of the source term, are derived. Finally, a local reduced basis (RB) technique is introduced to address the challenges of parameter-dependent multi-scale problems. This method integrates a RB approach into the SLOD framework, enabling an efficient generation of reliable coarse-scale models of the problem. Due to the unique localization properties of the SLOD, the RB snapshot computation can be performed on particularly small patches, reducing the offline and online complexity of the method. All theoretical results of this thesis are supported by numerical experiments

The Gauss-Green theorem in stratified groups

We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the $BV$ fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regularity on sets of finite perimeter. We show several properties of divergence-measure fields in stratified groups, ultimately achieving the related Gauss-Green theorem.Comment: 69 page