arXiv subjects

Gabriele Ciaramella

Publications and source records attributed to Gabriele Ciaramella.

At least 19 recordsLinked to original sources

arXiv2026

A note on the discrete Unmapped Tent Pitching for the heterogeneous wave equation

The Unmapped Tent Pitching (UTP) algorithm is a space-time domain decomposition method for the parallel solution of wave-type problems. We have recently extended UTP to heterogeneous settings and compared, at the continuous level, the computational cost of different space-time decompositions. In this note, we report discrete-level observations showing that the optimal decomposition strategy may differ from the one predicted by the continuous analysis.

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arXiv2026

On the unmapped tent pitching for the heterogeneous wave equation

The Unmapped Tent Pitching (UTP) algorithm is a space--time domain decomposition method for the parallel solution of hyperbolic problems. It was originally introduced for the homogeneous one-dimensional wave equation in [Ciaramella, Gander, Mazzieri, 2024]. UTP is inspired by the Mapped Tent Pitching (MTP) algorithm [Gopalakrishnan, Sch{ö}berl, Wintersteiger, 2017], which constructs the solution by iteratively building polytopal space--time subdomains, referred to as tents. In MTP, each physical tent is mapped onto a space--time rectangle, where local problems are solved before being mapped back to the original domain. In contrast, UTP avoids the nonlinear and potentially singular mapping step by computing the solution directly on a physical space--time rectangle that contains the tent, at the expense of redundant computations in the region outside the tent. In this work, we investigate several strategies to extend UTP to heterogeneous media, where the wave propagation speed is piecewise constant over two subregions of the domain. Among the considered approaches, the most efficient in terms of computational time is the one employing space--time subdomains with identical spatial and temporal dimensions in both regions, determined by the maximum propagation speed.

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arXiv2026

A Scalable Deflated Conjugate Gradient Solver for the Time-Dependent Pseudo-Stress Stokes Problem

We propose a novel iterative solution framework for the unsteady Stokes equations in the pseudo-stress formulation. When solving this class of problems by using implicit time-integration schemes, standard solvers suffer from deteriorating convergence properties for small time steps, independently of the chosen space discretisation method. This is due to the singular modes of the dev-dev operator. For this reason, we introduce a computational framework obtained by combining a deflated Conjugate Gradient method with a W-cycle multigrid scheme that employs a Restricted Additive Schwarz smoother. The key point is to choose the deflation subspace so that the inner system to be solved within a deflated Conjugate Gradient scheme corresponds to a Laplace problem defined on the singular modes of the original dev-dev operator. This results to be independent of the spatial discretisation method and allows one to use efficient multigrid iterative solvers. Numerical experiments show that the proposed strategy significantly accelerates the Conjugate Gradient convergence and provides stable performance with respect to the time step, confirming its robustness for solving linear systems in the pseudo-stress framework.

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arXiv2026

Reconstructing the system coefficients for coupled harmonic oscillators

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 further 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.

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arXiv2026

Numerical Verification of PolyDG Algebraic Solvers for the Pseudo-Stress Stokes Problem

This work focuses on the development of efficient solvers for the pseudo-stress formulation of the unsteady Stokes problem, discretised by means of a discontinuous Galerkin method on polytopal grids (PolyDG). The introduction of the pseudo-stress variable is motivated by the growing interest in non-Newtonian flow models and coupled interface problems, where the stress field plays a fundamental role in the physical description. The space-time discretisation of the problem is obtained by combining the PolyDG approach in space with the implicit Euler method for time integration. The resulting linear system, characterised by a symmetric, positive, definite matrix, exhibits deteriorating convergence with standard solvers as the time step decreases. To address this issue, we investigate two tailored strategies: deflated Conjugate Gradient, which mitigates the effect of the most problematic eigenmodes, and collective Block-Jacobi, which exploits the block structure of the system matrix. Numerical experiments show that both approaches yield iteration counts effectively independent of $Δt$, ensuring robust performance with respect to the time step. Future work will focus on extending this robustness to the spatial discretisation parameter $h$ by integrating multigrid strategies with the time-robust solvers developed in this study.

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arXiv2026

Optimized Schwarz Waveform Relaxation for the Damped Wave Equation

The performance of Schwarz Waveform Relaxation is critically dependent on the choice of transmission conditions. While classical absorbing conditions work well for wave propagation, they prove insufficient for damped wave equations, particularly in viscoelastic damping regimes where convergence becomes prohibitively slow. This paper addresses this limitation by introducing a more general transmission operator with two free parameters for the one-dimensional damped wave equation. Through frequency-domain analysis, we derive an explicit expression for the convergence factor governing the convergence rate. We propose and compare two optimization strategies (L-infinity and L-2 minimization) for determining optimal transmission parameters. Numerical experiments demonstrate that our optimized approach significantly accelerates convergence compared to standard absorbing conditions, especially for viscoelastic damping cases. The method provides a computationally efficient alternative to exhaustive parameter search while maintaining robust performance across different damping regimes.

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arXiv2025

Solving Semi-Linear Elliptic Optimal Control Problems with $L^1$-Cost via Regularization and RAS-Preconditioned Newton Methods

We present a new parallel computational framework for the efficient solution of a class of $L^2$/$L^1$-regularized optimal control problems governed by semi-linear elliptic partial differential equations (PDEs). The main difficulty in solving this type of problem is the nonlinearity and non-smoothness of the $L^1$-term in the cost functional, which we address by employing a combination of several tools. First, we approximate the non-differentiable projection operator appearing in the optimality system by an appropriately chosen regularized operator and establish convergence of the resulting system's solutions. Second, we apply a continuation strategy to control the regularization parameter to improve the behavior of (damped) Newton methods. Third, we combine Newton's method with a domain-decomposition-based nonlinear preconditioning, which improves its robustness properties and allows for parallelization. The efficiency of the proposed numerical framework is demonstrated by extensive numerical experiments.

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arXiv2025

A review of discontinuous Galerkin time-stepping methods for wave propagation problems

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.

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arXiv2025

Discontinuous Galerkin time integration for second-order differential problems: formulations, analysis, and analogies

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.

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arXiv2025

Adjoint-based optimal control of jump-diffusion processes

Stochastic differential equations (SDEs) using jump-diffusion processes describe many natural phenomena at the microscopic level. Since they are commonly used to model economic and financial evolutions, the calibration and optimal control of such processes are of interest to many communities and have been the subject of extensive research. In this work, we develop an optimization method working at the microscopic level. This allows us also to reduce computational time since we can parallelize the calculations and do not encounter the so-called curse of dimensionality that occurs when lifting the problem to its macroscopic counterpart using partial differential equations (PDEs). Using a discretize-then-optimize approach, we derive an adjoint process and an optimality system in the Lagrange framework. Then, we apply Monte Carlo methods to solve all the arising equations. We validate our optimization strategy by extensive numerical experiments. We also successfully test a optimization procedure that avoids storing the information of the forward equation.

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arXiv2025

Reconstruction of unknown monotone nonlinear operators in semilinear elliptic models using optimal inputs

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.

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arXiv2024

A gradient flow approach for combined layout-control design of wave energy parks

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.

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arXiv2024

A multigrid solver for PDE-constrained optimization with uncertain inputs

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.

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arXiv2023

Robust optimization of control parameters for WEC arrays using stochastic methods

This work presents a new computational optimization framework for the robust control of parks of Wave Energy Converters (WEC) in irregular waves. The power of WEC parks is maximized with respect to the individual control damping and stiffness coefficients of each device. The results are robust with respect to the incident wave direction, which is treated as a random variable. Hydrodynamic properties are computed using the linear potential model, and the dynamics of the system is computed in the frequency domain. A slamming constraint is enforced to ensure that the results are physically realistic. We show that the stochastic optimization problem is well posed. Two optimization approaches for dealing with stochasticity are then considered: stochastic approximation and sample average approximation. The outcomes of the above mentioned methods in terms of accuracy and computational time are presented. The results of the optimization for complex and realistic array configurations of possible engineering interest are then discussed. Results of extensive numerical experiments demonstrate the efficiency of the proposed computational framework.

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arXiv2021

Spectral substructured two-level domain decomposition methods

Two-level domain decomposition (DD) methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume. In this paper, a new class of substructured two-level methods is introduced,for which both domain decomposition smoothers and coarse correction steps are defined on the interfaces (or skeletons). This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, it allows one to use some of the well-known efficient coarse spaces proposed in the literature. While analyzing in detail the new substructured methods, we present a new convergence analysis for two-level iterative methods, which covers the proposed substructured framework. Further, we study the asymptotic optimality of coarse spaces both theoretically and numerically using deep neural networks. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

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arXiv2021

Nonlinear optimized Schwarz preconditioner for elliptic optimal control problems

We introduce a domain decomposition-based nonlinear preconditioned iteration for solving nonlinear, nonsmooth elliptic optimal control problems, with a nonlinear reaction term, $L^1$ regularization and box constraints on the control function. The method is obtained by applying semismooth Newton to the fixed-point equation of the parallel optimized Schwarz iteration. As a proof of concept, numerical experiments are performed on two subdomains, as well as on a multi-subdomain test case. The results show that it is possible to obtain substantial improvements in robustness and efficiency with the new method, relative to semismooth Newton applied directly to the full optimization problem, provided appropriate Robin parameters and a good continuation strategy are chosen.

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arXiv2021

On the effect of boundary conditions on the scalability of Schwarz methods

In contrast with classical Schwarz theory, recent results have shown that for special domain geometries, one-level Schwarz methods can be scalable. This property has been proved for the Laplace equation and external Dirichlet boundary conditions. Much less is known if mixed boundary conditions are considered. This short manuscript focuses on the convergence and scalability analysis of one-level parallel Schwarz method and optimized Schwarz method for several different external configurations of boundary conditions, i.e., mixed Dirichlet, Neumann and Robin conditions.

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arXiv2021

An overlapping waveform-relaxation preconditioner for economic optimal control problems with state constraints

In this work, a class of parabolic economic optimal control problems is considered. These problems are characterized by pointwise state constraints regularized by a parameter, which transforms the pure state constraints in mixed control-state ones. However, the convergence of classical (semismooth) Newton methods deteriorates for decreasing values of the regularization parameter. To tackle this problem, a nonlinear preconditioner is introduced. This is based on an overlapping optimized waveform-relaxation method characterized by Robin transmission conditions. Numerical experiments show that appropriate choices of the overlap and of the Robin parameter lead to a preconditioned Newton method with a robust convergence against the state constraints regularization parameter.

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