SCUBE

Parametric partial differential equations (PDEs) arise in several contexts, e.g., in optimization problems and in mathematical models with inherent uncertainties. In such PDEs, the differential operators usually depend on large, possibly infinite, sets of parameters, so that naive applications of standard numerical methods often lead to the so-called 'curse of dimensionality', a deterioration of the convergence rates and an exponential growth of the computational cost as the dimension of the space increases. In this talk, we give an overview of a specific numerical method for solving such PDEs, the stochastic Galerkin finite element method (SGFEM). First, we introduce a basic version of the method (we call it ’single-level’) for a standard high-dimensional parametric elliptic boundary value problem. Then, we discuss some of our recent results, focusing on a ‘multilevel’ approach, a posteriori error estimation, adaptivity and convergence with optimal algebraic rates. It turns out hat the proposed approaches help to mitigate the curse of dimensionality.

Modeling, analysis and computation are three pillars of computational science. We discuss them within the context of liquid crystal networks (LCNs). These materials couple a nematic liquid crystal with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Thin bodies of LCNs are natural candidates for soft robotics applications. We start from the classical 3D trace energy formula and derive a reduced 2D membrane energy as the formal asymptotic limit of vanishing thickness, including both stretching and bending energies, and characterize the zero energy deformations. We design a sound numerical method and discuss its Gamma convergence. We present computations showing the geometric effects that arise from liquid crystal defects as well as computations of nonisometric origami within and beyond theory. This work is joint with L. Bouck, G. Benavides, and S. Yang.

Modeling, analysis and computation are three pillars of computational science. We discuss them within the context of liquid crystal networks (LCNs). These materials couple a nematic liquid crystal with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Thin bodies of LCNs are natural candidates for soft robotics applications. We start from the classical 3D trace energy formula and derive a reduced 2D membrane energy as the formal asymptotic limit of vanishing thickness, including both stretching and bending energies, and characterize the zero energy deformations. We design a sound numerical method and discuss its Gamma convergence. We present computations showing the geometric effects that arise from liquid crystal defects as well as computations of nonisometric origami within and beyond theory. This work is joint with L. Bouck, G. Benavides, and S. Yang.

The aim of Electrical Impedance Tomography (EIT) is to determine the electrical conductivity distribution inside a domain by applying currents and measuring voltages on its boundary. Mathematically, the EIT reconstruction task can be formulated as a non-linear inverse problem. The Bayesian inverse problems framework has been applied expensively to solutions of the EIT inverse problem, in particular in the cases when the unknown conductivity is believed to be blocky. In this talk, we demonstrate that by exploiting linear algebraic considerations it is possible to organize the calculation for the Bayesian solution of the nonlinear EIT inverse problem via finite element methods with sparsity promoting priors in a computationally efficient manner. The proposed approach uses the Iterative Alternating Sequential (IAS) algorithm for the solution of the linearized problems. Within the IAS algorithm, a substantial reduction in computational complexity is attained by exploiting the low dimensionality of the data. Numerical tests on synthetic and real data illustrate the computational efficiency of the proposed algorithm.

Given a telecommunication network represented by a directed graph, our problem is to route one single stream of packets on the IP network along a min-cost path with a constraint on the maximum delay that any packet may incur. From a mathematical point of view, this problem, known as Delay Constrained Routing (DCR), can be formulated as a Mixed-Integer Second-Order Cone Program (MISOCP), where one needs to simultaneously (and "optimally") compute paths and reserve resources along the paths of the network. The DCR problem presents an interesting mixture of combinatorial and continuous structures and naturally lends itself to decomposition methods. We will discuss formulations, algorithms and computational results on real/realistic network instances.

Peridynamics is a nonlocal version of continuum mechanics theory able to incorporate singularities since it does not take into account spatial partial derivatives. As a consequence, it assumes long-range interactions among material particles and is able to describe the formation and the evolution of fractures. The discretization of such nonlocal model requires the use of raffinate numerical tools for approximating the solutions to the model. Due to the presence of a convolution product in the definition of the nonlocal operator, we propose a spectral collocation method based on the implementation of Fourier and Chebyshev polynomials to discretize the model. The choice can benefit of the FFT algorithm and allow us to deal efficiently with the imposition of non-periodic boundary conditions by a volume penalization technique. We prove the convergence of such methods in the framework of fractional Sobolev space and discuss numerically the stability of the scheme. We also investigate the qualitative aspects of the convolution kernel and of the nonlocality parameters by solving an inverse peridynamic problem by using a Physics-Informed Neural Network activated by suitable Radial Basis functions. Additionally, we propose a virtual element approach to obtain the solution of a nonlocal diffusion problem. The main feature of the proposed technique is that we are able to construct a nonlocal counterpart for the divergence operator in order to obtain a weak formulation of the peridynamic model and exploit the analogies with the known results in the context of Galerkin approximation. We prove the convergence of the proposed method and provide several simulations to validate our results. References: [1] Lopez, L., Pellegrino, S. F. (2021). A spectral method with volume penalization for a nonlinear peridynamic model International Journal for Numerical Methods in Engineering 122(3): 707–725. https://doi.org/10.1002/nme.6555 [2] Lopez, L., Pellegrino, S. F. (2022). A space-time discretization of a nonlinear peridynamic model on a 2D lamina Computers and Mathematics with Applications 116: 161–175. https://doi.org/10.1016/j.camwa.2021.07.0041 [3] Lopez, L., Pellegrino, S. F. (2022). A non-periodic Chebyshev spectral method avoiding penalization techniques for a class of nonlinear peridynamic models International Journal for Numerical Methods in Engineering 123(20): 4859–4876. https://doi.org/10.1002/nme.7058 [4] Difonzo, F. V., Lopez, L., Pellegrino, S. F. (2024). Physics informed neural networks for an inverse problem in peridynamic models Engineering with Computers. https://doi.org/10.1007/s00366-024-01957-5 [5] Difonzo, F. V., Lopez, L., Pellegrino, S. F. (2024). Physics informed neural networks for learning the horizon size in bond-based peridynamic models Computer Methods in Applied Mechanics and Engineering. https://doi.org/10.1016/j.cma.2024.117727

Accurately estimating landslides’ failure surface depth is essential for hazard prediction. However, most of the classical methods rely on overly simplistic assumptions [1]. In this work, we will present the landslide thickness estimation problem as an inverse problem Aw = b, obtained from discretization of the thickness equation [2]: ∂(hf vx)/∂x + ∂(hf vy)/∂y = − ∂ζ/∂t , (1) where the forward operator A contains information on the surface velocity (v_x, v_y), the right-hand side b corresponds to the surface elevation change ∂ζ/∂t, and w is the thickness hf . By employing a regularization approach, the inverse problem is reformulated as an optimization problem. In real-world scenarios, often no information on neither the noise type nor the noise level affecting data is available. In this context, the correct choice of the regularization parameter becomes a pressing issue. We propose a method to determine this parameter in a fully automatic way for the thickness inversion problem. Results obtained on both synthetic data generated by landslide simulation software and data measured from real-world landslides will be shown. [1] Jaboyedoff M., Carrea D., Derron M.H., Oppikofer T., Penna I.M., Rudaz B. (2020): A review of methods used to estimate initial landslide failure surface depths and volumes. Engineering Geology, 267, 105478 [2] Booth A. M. ; Lamb M. P. ; Avouac J.P. ; Delacourt C. (2013): Landslide velocity, thickness, and rheology from remote sensing: La Clapière landslide, France. Geophysical Research Letters, Vol. 40, 4299 - 4304.

In this seminar, I will talk about Objective Function Free Optimization (OFFO) in the context of pruning the parameter of a given model. OFFO algorithms are methods where the objective function is never computed; instead, they rely only on derivative information, thus on the gradient in the first-order case. I will give an overview of the main OFFO methods, focusing on adaptive algorithms such as Adagrad, Adam, RMSprop, ADADELTA, which are gradient methods that share the common characteristic of depending only on current and past gradient information to adaptively determine the step size at each iteration. Next, I will briefly discuss the most popular pruning approaches. As the name implies, pruning a model, typically a neural networks, refers to the process of reducing its size and complexity, typically by removing certain parameters that are considered unnecessary for its performance. Pruning emerges as an alternative compression technique for neural networks to matrix and tensor factorization or quantization. Mainly, I will focus on pruning-aware methods that uses specific rules to classify parameters as relevant or irrelevant at each iteration, enhancing convergence to a solution of the problem at hand, which is robust to pruning irrelevant parameters after training.Finally, I will introduce a novel deterministic algorithm which is both adaptive and pruning-aware, based on a modification Adagrad scheme that converges to a solution robust to pruning with complexity of $\log(k) \backslash k$. I will illustrate some preliminary results on different applications.

This seminar presents two recent works focused on sparse signal recovery and inverse problems. The first part introduces the truncated Huber penalty, a non-convex penalty function designed for robust signal recovery. We explore its application in constrained and unconstrained models, proving theoretical properties of the optimal solutions. An efficient algorithm based on the block coordinate descent method is also discussed, along with applications in signal and image processing. The second part covers a generalized Tikhonov regularization framework with spatially varying weights estimated via a neural network. This end-to-end approach integrates adaptive parameter estimation, improving detail preservation.

3D shape analysis tasks often involve characterizing a 3D object by an invariant, computationally efficient, and discriminative numerical representation, called shape descriptors. Among those, spectral-based shape descriptors have become increasingly widespread, since the spectrum is an isometry invariant, and thus is independent of the object’s representation including parametrization and spatial position[1]. However, large spectral decompositions and the choice of the most significant eigen-couples become computationally expensive for large set of data-points. We introduce a concise learning-based shape descriptor, computed through a Generalized Graph Neural Network (G-GNN) [2]. The G-GNN is an unsupervised graph neural network, leveraging spectral-based convolutional operators, derived from a learnable, energy-driven evolution process. Applied to a 3D polygonal mesh, the G-GNN allows to learn features acting as global shape descriptor of the 3D object. Using a 3D mesh related Dirichlet-like energy leads to a spectral and intrinsic shape descriptor, tied to the isometry invariant Laplace-Beltrami operator. Finally, by equipping the G-GNN with a suitable shape retrieval loss, the spectral shape descriptor can be employed in non-linear dimensionality reduction problems since it can define an optimal embedding, squeezing the latent information of a 3D model into a compact low-dimensional shape representation of the 3D object [1] Martin Reuter, Franz-Erich Wolter, Niklas Peinecke, Laplace–Beltrami spectra as ‘Shape-DNA’ of surfaces and solids, Computer-Aided Design, Volume 38, Issue 4, 2006, Pages 342-366, ISSN 0010-4485, https://doi.org/10.1016/j.cad.2005.10.011. [2] D. Lazzaro, S. Morigi, P. Zuzolo, Learning intrinsic shape representations via spectral mesh convolutions, Neurocomputing, Volume 598, 2024, 128152, ISSN 0925-2312, https://doi.org/10.1016/j.neucom.2024.128152.