Gabriele Ciaramella

This paper offers a self-contained exposition of the fundamental mathematical and computational tools for interpolation on the Grassmann manifold, including detailed derivations of geodesics and explicit formulations of the exponential and logarithmic maps. The presentation emphasizes intuition and draws continuous parallels with the Euclidean setting. This pedagogical approach facilitates the understanding of linear, piecewise linear, and high-order interpolation algorithms, as well as their extension to more general manifolds. Two numerical examples are finally used to illustrate the potential of these algorithms: one in the context of parametric model order reduction, and another drawn from stationary iterative methods for linear systems.

We thoroughly investigate Discontinuous Galerkin (DG) discretizations as time integrators for second-order oscillatory systems, considering both second-order and first-order formulations of the original problem. Key contributions include new convergence analyses for the second-order formulation and equivalence proofs between DG and classical time-stepping schemes (such as Newmark schemes and general linear methods). In addition, the chapter provides a detailed review and convergence analysis for the first-order formulation, alongside comparisons of the proposed schemes in terms of accuracy, consistency, and computational cost.

This chapter reviews and compares discontinuous Galerkin time-stepping methods for the numerical approximation of second-order ordinary differential equations, particularly those stemming from space finite element discretization of wave propagation problems. Two formulations, tailored for second- and first-order systems of ordinary differential equations, are discussed within a generalized framework, assessing their stability, accuracy, and computational efficiency. Theoretical results are supported by various illustrative examples that validate the findings, enhancing the understanding and applicability of these methods in practical scenarios.

Physical models often contain unknown functions and relations. The goal of our work is to answer the question of how one should excite or control a system under consideration in an appropriate way to be able to reconstruct an unknown nonlinear relation. To answer this question, we propose a greedy reconstruction algorithm within an offline-online strategy. We apply this strategy to a two-dimensional semilinear elliptic model. Our identification is based on the application of several space-dependent excitations (also called controls). These specific controls are designed by the algorithm in order to obtain a deeper insight into the underlying physical problem and a more precise reconstruction of the unknown relation. We perform numerical simulations that demonstrate the effectiveness of our approach which is not limited to the current type of equation. Since our algorithm provides not only a way to determine unknown operators by existing data but also protocols for new experiments, it is a holistic concept to tackle the problem of improving physical models.

Physical models often contain unknown functions and relations. In order to gain more insights into the nature of physical processes, these unknown functions have to be identified or reconstructed. Mathematically, we can formulate this research question within the framework of inverse problems. In this work, we consider optimization techniques to solve the inverse problem using Tikhonov regularization and data from laboratory experiments. We propose an iterative strategy that eliminates the need for laboratory experiments. Our method is applied to identify the coupling and damping coefficients in a system of oscillators, ensuring an efficient and experiment-free approach. We present our results and compare them with those obtained from an alternative, purely experimental approach. By employing our proposed strategy, we demonstrate a significant reduction in the number of laboratory experiments required.

This paper focuses on the adaptive rectangular decomposition (ARD) scheme, a wave-based method employed for acoustic simulations. ARD holds promise for diverse applications. In architectural design, it can forecast acoustical parameters, facilitating the creation of spaces with superior sound quality. Moreover, in the domain of acoustic virtual reality, ARD can offer users a more immersive and lifelike acoustic environment. This enhancement proves advantageous for all of these applications, enabling the simulation of larger environments with heightened precision. Despite its notable efficiency, ARD faces a significant drawback: the absence of atmospheric absorption modeling. The principal aim of this study is to rectify this limitation, thereby augmenting the capabilities of the ARD algorithm.

Wave energy converters (WECs) represent an innovative technology for power generation from renewable sources (marine energy). Although there has been a great deal of research into such devices in recent decades, the power output of a single device has remained low. Therefore, installation in parks is required for economic reasons. The optimal design problem for parks of WECs is challenging since it requires the simultaneous optimization of positions and control parameters. While the literature on this problem usually considers metaheuristic algorithms, we present a novel numerical framework based on a gradient-flow formulation. This framework is capable of solving the optimal design problem for WEC parks. In particular, we use a low-order adaptive Runge-Kutta scheme to integrate the gradient-flow equation and introduce an inexact solution procedure. Here, the tolerances of the linear solver used for projection on the constraint nullspace and of the time-advancing scheme are automatically adapted to avoid over-solving so that the method requires minimal tuning. We then provide the specific details of its application to the considered WEC problem: the goal is to maximize the average power produced by a park, subject to hydrodynamic and dynamic governing equations and to the constraints of available sea area, minimum distance between devices, and limited oscillation amplitude around the undisturbed free surface elevation. A suitable choice of the discrete models allows us to compute analytically the Jacobian of the state problem's residual. Numerical tests with realistic parameters show that the proposed algorithm is efficient, and results of physical interest are obtained.

When considering an unconstrained minimization problem, a standard approach is to solve the optimality system with a Newton method possibly preconditioned by, e.g., nonlinear elimination. In this contribution, we argue that nonlinear elimination could be used to reduce the number of optimization variables by artificially constraining them to satisfy a subset of the optimality conditions. Consequently, a reduced objective function is derived which can now be minimized with any optimization algorithm. By choosing suitable variables to eliminate, the conditioning of the reduced optimization problem is largely improved. We here focus in particular on a right preconditioned gradient descent and show theoretical and numerical results supporting the validity of the presented approach.