Mario Putti
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Mario Putti has a total of 238 co-authors. Here are the top co-authors by number of shared publications:
- Claudio Paniconi
Centre Eau Terre Environnement, Institut National de la Recherche Scientifique, Quebec City, Canada
28 shared publications - Pietro Teatini
Department of Civil, Environmental and Architectural Engineering, University of Padova, 35131 Padova, Italy
21 shared publications - Annamaria Mazzia
Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Padova, via Trieste 63, 35121 Padova, Italy
13 shared publications - Damiano Pasetto
Department of Environmental Sciences, Informatics and Statistics, Ca’ Foscari University of Venice, Venezia Mestre, Italy
12 shared publications G. Gambolati
Department of Mathematical Methods and Models for Scientific Applications University of Padova Padua Italy
10 shared publications- Matteo Camporese
Department of Civil, Environmental and Architectural Engineering, University of Padua, Padua, Italy
8 shared publications
Latest Publications
- 2 January 2026
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 58 (1) , 1-34
The Graph \(\boldsymbol{\infty }\)-Laplacian Eigenvalue ProblemWe analyze various formulations of the [Formula: see text]-Laplacian eigenvalue problem on graphs, comparing their properties and highlighting their respective advantages and limitations. First, we investigate the graph [Formula: see text]-eigenpairs arising as limits of [Formula: see text]-Laplacian eigenpairs, extending key results from the continuous setting to the discrete domain. We prove that every limit of [Formula: see text]-Laplacian eigenpair, for [Formula: see text] going to [Formula: see text], satisfies a limit eigenvalue equation and establishes that the corresponding eigenvalue can be bounded from below by the packing radius of the graph, indexed by the number of nodal domains induced by the eigenfunction. Additionally, we show that the limits, for [Formula: see text] going to [Formula: see text], of the variational [Formula: see text]-Laplacian eigenvalues are bounded from both above and below by the packing radii, achieving equality for the smallest two variational eigenvalues and corresponding packing radii of the graph. In the second part of the paper, we introduce generalized [Formula: see text]-Laplacian eigenpairs as generalized critical points and values of the [Formula: see text]-Rayleigh quotient. We prove that the generalized variational [Formula: see text]-eigenvalues equal the limit of the [Formula: see text]-Laplacian variational eigenvalues and so satisfy the same upper bounds in terms of packing radii. Finally, we establish that any solution to the limit eigenvalue equation is also a generalized eigenpair, while any generalized eigenpair satisfies the limit eigenvalue equation on a suitable subgraph.
- 2 January 2026
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 58 (1) , 1-34
The Graph \(\boldsymbol{\infty }\)-Laplacian Eigenvalue ProblemWe analyze various formulations of the [Formula: see text]-Laplacian eigenvalue problem on graphs, comparing their properties and highlighting their respective advantages and limitations. First, we investigate the graph [Formula: see text]-eigenpairs arising as limits of [Formula: see text]-Laplacian eigenpairs, extending key results from the continuous setting to the discrete domain. We prove that every limit of [Formula: see text]-Laplacian eigenpair, for [Formula: see text] going to [Formula: see text], satisfies a limit eigenvalue equation and establishes that the corresponding eigenvalue can be bounded from below by the packing radius of the graph, indexed by the number of nodal domains induced by the eigenfunction. Additionally, we show that the limits, for [Formula: see text] going to [Formula: see text], of the variational [Formula: see text]-Laplacian eigenvalues are bounded from both above and below by the packing radii, achieving equality for the smallest two variational eigenvalues and corresponding packing radii of the graph. In the second part of the paper, we introduce generalized [Formula: see text]-Laplacian eigenpairs as generalized critical points and values of the [Formula: see text]-Rayleigh quotient. We prove that the generalized variational [Formula: see text]-eigenvalues equal the limit of the [Formula: see text]-Laplacian variational eigenvalues and so satisfy the same upper bounds in terms of packing radii. Finally, we establish that any solution to the limit eigenvalue equation is also a generalized eigenpair, while any generalized eigenpair satisfies the limit eigenvalue equation on a suitable subgraph.
- Open Access
- 5 November 2025
- Published by American Geophysical Union (AGU) in Water Resources Research
- Vol. 61 (11)
Computing Groundwater Recharge and Saturated Storage Dynamics: A Richards Equation‐Based Recipe
In this study, expressions for saturated zone storage change fluxes are derived directly from the governing equation for flow in variably saturated porous media by treating the saturated zone as a subdomain with a moving boundary and using a combination of Gauss' divergence theorem and the Leibnitz rule (Reynolds transport theorem). This formulation is general and can be implemented in any Richards equation‐based numerical solver, making it broadly applicable across hydrologic modeling platforms. The derived expressions allow for unambiguous tracking of the various contributing factors to saturated storage dynamics and accommodates both water table‐centric definitions of groundwater recharge and broader conceptualizations that can include other sources, sinks, and boundaries. The derived equations are then implemented in a three‐dimensional numerical model for integrated surface–subsurface flow, and their behavior is analyzed in three test cases representing a wide diversity of flow conditions and scenarios. By developing these storage and flux expressions directly from the continuous form of the governing flow equation, unlike previous approaches based on the numerical discretization or a post‐processing analysis from a model simulation, important physical principles such as mass conservation are ensured, and the methodology is independent of parameterizations that are not present in the fundamental equation, such as occurs for instance when the capillary fringe rather than the water table is used as the upper boundary of the saturated zone. This opens new possibilities for accurately quantifying groundwater recharge and aquifer dynamics in diverse hydrological settings and at broader scales.
- 5 November 2025
- Published by American Geophysical Union (AGU) in Water Resources Research
- Vol. 61 (11)
Open AccessComputing Groundwater Recharge and Saturated Storage Dynamics: A Richards Equation‐Based Recipe
In this study, expressions for saturated zone storage change fluxes are derived directly from the governing equation for flow in variably saturated porous media by treating the saturated zone as a subdomain with a moving boundary and using a combination of Gauss' divergence theorem and the Leibnitz rule (Reynolds transport theorem). This formulation is general and can be implemented in any Richards equation‐based numerical solver, making it broadly applicable across hydrologic modeling platforms. The derived expressions allow for unambiguous tracking of the various contributing factors to saturated storage dynamics and accommodates both water table‐centric definitions of groundwater recharge and broader conceptualizations that can include other sources, sinks, and boundaries. The derived equations are then implemented in a three‐dimensional numerical model for integrated surface–subsurface flow, and their behavior is analyzed in three test cases representing a wide diversity of flow conditions and scenarios. By developing these storage and flux expressions directly from the continuous form of the governing flow equation, unlike previous approaches based on the numerical discretization or a post‐processing analysis from a model simulation, important physical principles such as mass conservation are ensured, and the methodology is independent of parameterizations that are not present in the fundamental equation, such as occurs for instance when the capillary fringe rather than the water table is used as the upper boundary of the saturated zone. This opens new possibilities for accurately quantifying groundwater recharge and aquifer dynamics in diverse hydrological settings and at broader scales.
- 9 June 2025
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Matrix Analysis and Applications
- Vol. 46 (2) , 1540-1561
Graph \({p}\)-Laplacian Eigenpairs as Saddle Points of a Family of Spectral Energy FunctionsWe address the problem of computing the graph [Formula: see text]-Laplacian eigenpairs for [Formula: see text]. We propose a reformulation of the graph [Formula: see text]-Laplacian eigenvalue problem in terms of a constrained weighted Laplacian eigenvalue problem and discuss theoretical and computational advantages. We provide a correspondence between [Formula: see text]-Laplacian eigenpairs and linear eigenpairs of a constrained generalized weighted Laplacian eigenvalue problem. As a result, we can assign an index to any [Formula: see text]-Laplacian eigenpair that matches the Morse index of the [Formula: see text]-Rayleigh quotient evaluated at the eigenfunction. In the second part of the paper, we introduce a class of spectral energy functions that depend on edge and node weights. We prove that differentiable saddle points of the [Formula: see text]th energy function correspond to [Formula: see text]-Laplacian eigenpairs having index equal to [Formula: see text]. Moreover, the first energy function is proved to possess a unique saddle point which corresponds to the unique first [Formula: see text]-Laplacian eigenpair. Finally, we develop novel gradient-based numerical methods suited to compute [Formula: see text]-Laplacian eigenpairs for any [Formula: see text] and present some experiments.
- 9 June 2025
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Matrix Analysis and Applications
- Vol. 46 (2) , 1540-1561
Graph \({p}\)-Laplacian Eigenpairs as Saddle Points of a Family of Spectral Energy FunctionsWe address the problem of computing the graph [Formula: see text]-Laplacian eigenpairs for [Formula: see text]. We propose a reformulation of the graph [Formula: see text]-Laplacian eigenvalue problem in terms of a constrained weighted Laplacian eigenvalue problem and discuss theoretical and computational advantages. We provide a correspondence between [Formula: see text]-Laplacian eigenpairs and linear eigenpairs of a constrained generalized weighted Laplacian eigenvalue problem. As a result, we can assign an index to any [Formula: see text]-Laplacian eigenpair that matches the Morse index of the [Formula: see text]-Rayleigh quotient evaluated at the eigenfunction. In the second part of the paper, we introduce a class of spectral energy functions that depend on edge and node weights. We prove that differentiable saddle points of the [Formula: see text]th energy function correspond to [Formula: see text]-Laplacian eigenpairs having index equal to [Formula: see text]. Moreover, the first energy function is proved to possess a unique saddle point which corresponds to the unique first [Formula: see text]-Laplacian eigenpair. Finally, we develop novel gradient-based numerical methods suited to compute [Formula: see text]-Laplacian eigenpairs for any [Formula: see text] and present some experiments.
- Open Access
Computing the L1 optimal transport density: a FEM approach
The $L^1$ optimal transport density $μ^*$ is the unique $L^\infty$ solution of the Monge-Kantorovich equations. It has been recently characterized also as the unique minimizer of the $L^1$ -transport energy functional E. In the present work we develop and we prove convergence of a numerical approxi- mation scheme for $μ^*$ . Our approach relies upon the combination of a FEM- inspired variational approximation of E with a minimization algorithm based on a gradient flow method.Open AccessComputing the L1 optimal transport density: a FEM approach
The $L^1$ optimal transport density $μ^*$ is the unique $L^\infty$ solution of the Monge-Kantorovich equations. It has been recently characterized also as the unique minimizer of the $L^1$ -transport energy functional E. In the present work we develop and we prove convergence of a numerical approxi- mation scheme for $μ^*$ . Our approach relies upon the combination of a FEM- inspired variational approximation of E with a minimization algorithm based on a gradient flow method. - 1 January 2023
- Published by American Institute of Mathematical Sciences (AIMS) in Advances in Computational Science and Engineering
- Vol. 1 (4) , 424-442
Numerical solution of the $ L^1 $-optimal transport problem on surfaces
In this article we study the numerical solution of the $ L^1 $-optimal transport problem on 2D surfaces embedded in $ \mathbb{R}^3 $ by means of the dynamic Monge-Kantorovich(DMK) formulation. Our numerical approach uses the surface finite element method (SFEM) to approximate the elliptic partial differential equation (PDE) and related integrals involved in the surface DMK. The accuracy and efficiency of the proposed numerical scheme is tested against an exact solution on a 2D sphere. When using a mesh with edges aligned with the boundary of the support of the transported measures, the experimental results show errors that scale coherently with our choice of discretization spaces, achieving first order convergence for the calculation of the optimal transport solution. On the other hand, the errors in the computation of the Wasserstein-1 distance (an optimal transport-based distance for nonnegative measures) enjoy faster convergence, scaling almost cubically with $ h $. However, if the mesh used is not adapted to the problem, this super-linear convergence is lost and the convergence toward the optimal transport solution reduces to $ \mathcal{O}(h^{0.6}) $.We also experimentally compared our approach against a state of the art method based on the alternating direction method of multipliers (ADMM). The results obtained in the calculation of the Wasserstein-1 distance show that our proposed approach is more accurate and computationally more efficient than the ADMM-based method, independently on the mesh used.- 1 January 2023
- Published by American Institute of Mathematical Sciences (AIMS) in Advances in Computational Science and Engineering
- Vol. 1 (4) , 424-442
Numerical solution of the $ L^1 $-optimal transport problem on surfaces
In this article we study the numerical solution of the $ L^1 $-optimal transport problem on 2D surfaces embedded in $ \mathbb{R}^3 $ by means of the dynamic Monge-Kantorovich(DMK) formulation. Our numerical approach uses the surface finite element method (SFEM) to approximate the elliptic partial differential equation (PDE) and related integrals involved in the surface DMK. The accuracy and efficiency of the proposed numerical scheme is tested against an exact solution on a 2D sphere. When using a mesh with edges aligned with the boundary of the support of the transported measures, the experimental results show errors that scale coherently with our choice of discretization spaces, achieving first order convergence for the calculation of the optimal transport solution. On the other hand, the errors in the computation of the Wasserstein-1 distance (an optimal transport-based distance for nonnegative measures) enjoy faster convergence, scaling almost cubically with $ h $. However, if the mesh used is not adapted to the problem, this super-linear convergence is lost and the convergence toward the optimal transport solution reduces to $ \mathcal{O}(h^{0.6}) $.We also experimentally compared our approach against a state of the art method based on the alternating direction method of multipliers (ADMM). The results obtained in the calculation of the Wasserstein-1 distance show that our proposed approach is more accurate and computationally more efficient than the ADMM-based method, independently on the mesh used.- 9 December 2022
- Published by Elsevier in Applied and Computational Harmonic Analysis
- Vol. 64, 1-32
Nodal domain count for the generalized graph p-Laplacian
- 9 December 2022
- Published by Elsevier in Applied and Computational Harmonic Analysis
- Vol. 64, 1-32
Nodal domain count for the generalized graph p-Laplacian
- Open Access
Nodal domain count for the generalized graph $p$-Laplacian
Inspired by the linear Schr\"odinger operator, we consider a generalized $p$-Laplacian operator on discrete graphs and present new results that characterize several spectral properties of this operator with particular attention to the nodal domain count of its eigenfunctions. Just like the one-dimensional continuous $p$-Laplacian, we prove that the variational spectrum of the discrete generalized $p$-Laplacian on forests is the entire spectrum. Moreover, we show how to transfer Weyl's inequalities for the Laplacian operator to the nonlinear case and prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized $p$-Laplacian on generic graphs, including variational eigenpairs. In particular, when applied to the linear case $p=2$, in addition to recovering well-known features, the new results provide novel properties of the linear Schr\"odinger operator.Open AccessNodal domain count for the generalized graph $p$-Laplacian
Inspired by the linear Schr\"odinger operator, we consider a generalized $p$-Laplacian operator on discrete graphs and present new results that characterize several spectral properties of this operator with particular attention to the nodal domain count of its eigenfunctions. Just like the one-dimensional continuous $p$-Laplacian, we prove that the variational spectrum of the discrete generalized $p$-Laplacian on forests is the entire spectrum. Moreover, we show how to transfer Weyl's inequalities for the Laplacian operator to the nonlinear case and prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized $p$-Laplacian on generic graphs, including variational eigenpairs. In particular, when applied to the linear case $p=2$, in addition to recovering well-known features, the new results provide novel properties of the linear Schr\"odinger operator. - 6 July 2022
- Published by Springer Nature in Applied Mathematics & Optimization
- Vol. 86 (2) , 1-40
$L^1$ Transport Energy- 6 July 2022
- Published by Springer Nature in Applied Mathematics & Optimization
- Vol. 86 (2) , 1-40
$L^1$ Transport Energy- Open Access
Computing the Cut Locus of a Riemannian Manifold via Optimal Transport
In this paper, we give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge-Kantorovich equations, a PDE formulation of the optimal transport problem with cost equal to the geodesic distance. Combining this result with an optimal transport numerical solver based on the so-called dynamical Monge-Kantorovich approach, we propose a novel framework for the numerical approximation of the cut locus of a point in a manifold. We show the applicability of the proposed method on a few examples settled on 2d-surfaces embedded in $R^{3}$ and discuss advantages and limitations.Open AccessComputing the Cut Locus of a Riemannian Manifold via Optimal Transport
In this paper, we give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge-Kantorovich equations, a PDE formulation of the optimal transport problem with cost equal to the geodesic distance. Combining this result with an optimal transport numerical solver based on the so-called dynamical Monge-Kantorovich approach, we propose a novel framework for the numerical approximation of the cut locus of a point in a manifold. We show the applicability of the proposed method on a few examples settled on 2d-surfaces embedded in $R^{3}$ and discuss advantages and limitations.