arXiv subjects

Rodrigo Provasi

Publications and source records attributed to Rodrigo Provasi.

5 recordsLinked to original sources

arXiv2026

Anytime PAC-Bayes for Constrained Density-Ratio Networks under Covariate Shift

A unified framework for learning under covariate shift is presented, in which a constrained density-ratio network approximates the Radon-Nikodym derivative $r^\star = dP/dQ$ and feeds an anytime PAC-Bayes generalization certificate. A change-of-measure identity decomposes the gap between target risk and importance-weighted source risk into a ratio-bias term governed by $\|r_\theta - r^\star\|_{L^2(Q)}$ and a generalization-gap term governed by the variability of the weighted loss. Normalization and moment-matching identities are enforced as hard integral constraints through an augmented-Lagrangian scheme, with a second-moment penalty controlling the effective sample size. PAC-Bayes is instantiated on the weighted risk in a fixed-time regime that yields Bernoulli-KL bounds, identifies the network-weighted Gibbs posterior as the unique KL-regularized minimizer, and quantifies stability under $L^2(Q)$ perturbations of the learned ratio, and is then strengthened by geometric peeling to an anytime certificate uniform in $t \geq t_{\min}$. A pre-registered two-campaign protocol combining a patch test against analytic ground truth with a real-data deployment validates the framework: the network produces calibrated ratios, reduces target $0/1$ loss against unweighted ERM and classical direct ratio-estimation baselines, and attains the anytime certificate. A single fixed-time coverage failure is recorded, with per-split coverage aligning one-to-one with the magnitude of the label shift, confirming that the covariate-only assumption is operationally tight rather than a defect of the certificate.

cs.LG↗

arXiv2026

An Investigation of Stabilization Scaling in Finite-Strain Virtual Element Methods for Hyperelasticity

Low-order virtual element methods (VEM) compute a consistent finite-strain contribution through polynomial projections and rely on stabilization to control the unresolved modes in the projector kernel. In current hyperelastic VEM practice, stabilization is often defined by integrating a nonlinear surrogate energy over an auxiliary sub-triangulation and scaled through modified Lam\'e parameters and incompressibility factors; this can introduce sensitivity to the arbitrary internal tessellation, complicate consistent Newton linearization, and, most critically, inject bulk-dependent proxies into shear-type kernel penalties, artificially stiffening isochoric missing modes in the nearly incompressible regime. This work develops a submesh-free, kernel-only stabilization that decouples deviatoric and volumetric channels and is explicitly designed to scale like the current Newton tangent energy on the kernel: the deviatoric term is scaled solely by a shear measure and enhanced by bounded geometry-driven directional weights, while the volumetric term is scaled by an independent bulk measure and can be capped or suppressed as $\nu\to 1/2$. A spectral framework is established in which the canonical VEM stability requirement on the kernel is characterized by generalized Rayleigh quotients and eigenvalue bounds, and it is shown under standard polygon regularity assumptions that the deviatoric stabilization is uniformly equivalent to $\mu_E|\cdot|_{1,E}^2$ on the kernel with constants independent of mesh size and Poisson ratio. Element-level diagnostics confirm that classical surrogate-based stabilizations assign bulk-driven energy to isochoric kernel modes as $\nu\to 1/2$, whereas the proposed decoupled stabilization remains shear-scaled; kernel spectra and Cook's membrane simulations in the nearly incompressible regime further support improved robustness across polygonal mesh families.

math.NA↗

arXiv2025

Mass-Lumped Virtual Element Method with Strong Stability-Preserving Runge-Kutta Time Stepping for Two-Dimensional Parabolic Problems

This paper presents a mass-lumped Virtual Element Method (VEM) with explicit Strong Stability-Preserving Runge--Kutta (SSP-RK) time integration for two-dimensional parabolic problems on general polygonal meshes. A diagonal mass matrix is constructed via row-sum operations combined with flooring to ensure uniform positivity. Stabilization terms vanish identically under row summation, so the lumped weights derive solely from the $L^2$ projector and are computable through a small polynomial system at cost $\mathcal{O}(N_k^3)$ per element. The resulting lumped bilinear form satisfies $L^2$-equivalence with constants independent of the number of element edges, yielding a symmetric positive definite discrete inner product. A mesh-robust spectral estimate is established, showing that the largest eigenvalue of the discrete diffusion operator scales like $h^{-2}$, with constants depending only on the space dimension, polynomial degree, and mesh regularity. This yields the classical diffusion-type CFL condition $\Delta t=\mathcal{O}(h^2)$ for forward Euler stability and extends to higher-order SSP-RK schemes, ensuring the preservation of stability properties inherited from the forward Euler step. Numerical experiments on distorted quadrilateral, serendipity, and Voronoi meshes validate the theoretical predictions: for $k=1$, the lumped VEM attains optimal convergence rates, namely $\mathcal{O}(h)$ in the $H^1$-seminorm and $\mathcal{O}(h^2)$ in the $L^2$-norm, without degradation due to mesh distortion or diagonal mass approximation, while the SSP-RK methods remain stable under the predicted $\Delta t\propto h^2$ scaling. Additional tests on accuracy versus efficiency and on heterogeneous anisotropic diffusion further illustrate the practical competitiveness and robustness of the proposed formulation.

math.NA↗

arXiv2025

Virtual Element Method Applied to Two Dimensional Axisymmetric Elastic Problems

This work presents a Virtual Element Method (VEM) formulation tailored for two-dimensional axisymmetric problems in linear elasticity. By exploiting the rotational symmetry of the geometry and loading conditions, the problem is reduced to a meridional cross-section, where all fields depend only on the radial and axial coordinates. The method incorporates the radial weight $r$ in both the weak formulation and the interpolation estimates to remain consistent with the physical volume measure of cylindrical coordinates. A projection operator onto constant strain fields is constructed via boundary integrals, and a volumetric correction term is introduced to account for the divergence of the stress field arising from axisymmetry. The stabilization term is designed to act only on the kernel of the projection and is implemented using a boundary-based formulation that guarantees stability without affecting polynomial consistency. Furthermore, an a priori interpolation error estimate is established in a weighted Sobolev space, showing optimal convergence rates. The implementation is validated through patch tests that demonstrate the accuracy, consistency, and robustness of the proposed approach.

math.NA↗

arXiv2025

A Hybrid Virtual Element Method and Deep Learning Approach for Solving One-Dimensional Euler-Bernoulli Beams

A hybrid framework integrating the Virtual Element Method (VEM) with deep learning is presented as an initial step toward developing efficient and flexible numerical models for one-dimensional Euler-Bernoulli beams. The primary aim is to explore a data-driven surrogate model capable of predicting displacement fields across varying material and geometric parameters while maintaining computational efficiency. Building upon VEM's ability to handle higher-order polynomials and non-conforming discretizations, the method offers a robust numerical foundation for structural mechanics. A neural network architecture is introduced to separately process nodal and material-specific data, effectively capturing complex interactions with minimal reliance on large datasets. To address challenges in training, the model incorporates Sobolev training and GradNorm techniques, ensuring balanced loss contributions and enhanced generalization. While this framework is in its early stages, it demonstrates the potential for further refinement and development into a scalable alternative to traditional methods. The proposed approach lays the groundwork for advancing numerical and data-driven techniques in beam modeling, offering a foundation for future research in structural mechanics.

cs.LG↗