Mario Putti

University of Padova | UNIPD · Department of Mathematics

We present a Lagrangian-Eulerian scheme to solve the shallow water equations in the case of spatially variable bottom geometry. This work was dictated by the fact that geometrically Intrinsic Shallow Water Equations (ISWE) are characterized by non-autonomous fluxes. Handling of non-autonomous fluxes is an open question for schemes based on Riemann...

Network routing approaches are widely used to study the evolution in time of self-adapting systems. However, few advances have been made for problems where adaptation is governed by time-dependent inputs. In this work we study a dynamical systems where the edge conductivities of a network are regulated by time-varying mass loads injected on nodes....

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 numerica...

Understanding the internal structure of permeable and impermeable sediments (e.g. point-bars and tidal-flat deposits) generated by the evolution of meandering tidal channels is essential for accurate modeling of groundwater flow and contaminant transport in coastal areas. The detailed reconstruction of stratal geometry and hydraulic properties from...

We present a Lagrangian-Eulerian scheme to solve the shallow water equations in the case of spatially variable bottom geometry. Using a local curvilinear reference system anchored on the bottom surface, we develop an effective first-order and high-resolution space-time discretization of the no-flow surfaces and solve a Lagrangian initial value prob...

Understanding the internal structure of permeable and impermeable sediments (e.g. point-bars and tidal-flat deposits) generated by the evolution of meandering tidal channels is essential for accurate modeling of groundwater flow and contaminant transport in coastal areas. The detailed reconstruction of stratal geometry and hydraulic properties from...

We introduce the transport energy functional E (a variant of the Bouchitté–Buttazzo–Seppecher shape optimization functional) and prove that its unique minimizer is the optimal transport density μ∗, i.e., the solution of Monge–Kantorovich partial differential equations. We study the gradient flow of E and show that μ∗ is the unique global attractor...

Optimizing passengers routes is crucial to design efficient transportation networks. Recent results show that optimal transport provides an efficient alternative to standard optimization methods. However, it is not yet clear if this formalism has empirical validity on engineering networks. We address this issue by considering different response fun...

Network routing approaches are widely used to study the evolution in time of self-adapting systems. However, few advances have been made for problems where adaptation is governed by time-dependent inputs. In this work, we study a dynamical systems where the edge conductivities -- capacities -- of a network are regulated by time-varying mass loads i...

Optimizing passengers routes is crucial to design efficient transportation networks. Recent results show that optimal transport provides an efficient alternative to standard optimization methods. However, it is not yet clear if this formalism has empirical validity on engineering networks. We address this issue by considering different response fun...

Designing and optimizing different flows in networks is a relevant problem in many contexts. While a number of methods have been proposed in the physics and optimal transport literature for the one-commodity case, we lack similar results for the multicommodity scenario. In this paper we present a model based on optimal transport theory for finding...

We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to...

Recently a Dynamic-Monge-Kantorovich formulation of the PDE-based -optimal transport problem was presented. The model considers a diffusion equation enforcing the balance of the transported masses with a time-varying conductivity that evolves proportionally to the transported flux. In this paper we present an extension of this model that considers...

Model reduction for fluid flow simulation continues to be of great interest across a number of scientific and engineering fields. In a previous work [1], we explored the use of Neural Ordinary Differential Equations (NODE) as a non-intrusive method for propagating the latent-space dynamics in reduced order models. Here, we investigate employing dee...

Model reduction for fluid flow simulation continues to be of great interest across a number of scientific and engineering fields. In a previous work [arXiv:2104.13962], we explored the use of Neural Ordinary Differential Equations (NODE) as a non-intrusive method for propagating the latent-space dynamics in reduced order models. Here, we investigat...

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 numerica...

In this article we study the numerical solution of the $L^1$-Optimal Transport Problem on 2D surfaces embedded in $R^3$, via the DMK formulation introduced in [FaccaCardinPutti:2018]. We extend from the Euclidean into the Riemannian setting the DMK model and conjecture the equivalence with the solution Monge-Kantorovich equations, a PDE-based formu...

We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to...

In this work, we develop Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with a Radial Basis Function (RBF) interpolation method to construct efficient reduced order models for time-dependent problems arising in large scale environmental flow applications. The performance of the POD-RBF NIROM is compar...

Agriculture is the major user of water resources, accounting for 70% of global freshwater demand. As the demand for clean water increases, so does the need to implement more efficient strategies for water management in irrigated agriculture. While the benefits of precision irrigation in high-value crops, such as cannabis, tomatoes, and potatoes, ar...

Routing optimization is a relevant problem in many contexts. Solving directly this type of optimization problem is often computationally intractable. Recent studies suggest that one can instead turn this problem into one of solving a dynamical system of equations, which can instead be solved efficiently using numerical methods. This results in enab...

We present a model for finding optimal multi-commodity flows on networks based on optimal transport theory. The model relies on solving a dynamical system of equations. We prove that its stationary solution is equivalent to the solution of an optimization problem that generalizes the one-commodity framework. In particular, it generalizes previous r...

We consider a finite element method for Partial Differential Equations (PDEs) on surfaces. Unlike many previous techniques, this approach is based on a geometrically intrinsic formulation. With proper definition of the geometry and transport operators, the resulting finite element method is fully intrinsic to the surface. Here, we lay out in detail...

We illustrate and test an approach grounded on embedding moment equations (MEs) of groundwater flow within a Monte Carlo based modeling strategy to yield a Reduced-Order Model (ROM) that enables the efficient and accurate evaluation of probability distributions of hydraulic heads in randomly heterogeneous transmissivity fields. The projection space...

Routing optimization is a relevant problem in many contexts. Solving directly this type of optimization problem is often computationally unfeasible. Recent studies suggest that one can instead turn this problem into one of solving a dynamical system of equations, which can instead be solved efficiently using numerical methods. This results in enabl...

We extend our previous work on a biologically inspired dynamic Monge–Kantorovich model (Facca et al. in SIAM J Appl Math 78:651–676, 2018) and propose it as an effective tool for the numerical solution of the \(L^{1}\)-PDE based optimal transportation model. We first introduce a new Lyapunov-candidate functional and show that its derivative along t...

In this work, we develop Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with a Radial Basis Function (RBF) interpolation method to construct efficient reduced order models for time-dependent problems arising in large scale environmental flow applications. The performance of the POD-RBF NIROM is compar...

We propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of time-dependent diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. The method of lines (MOL) is used to combine spatial and temporal discretizations. The spatial scheme requires the definition of a high-order approxim...

Internal erosion is the cause of significant damage in dams and river embankments in many countries. In the last 20 years, the use of fiber-optic distributed temperature sensing (DTS) has proved to be an effective tool for the detection and quantification of leakages and internal erosion in dams. This work investigates the effectiveness of DTS for...

There is a growing recognition of the interdependencies among the supply systems that rely upon food, water and energy. Billions of people lack safe and sufficient access to these systems, coupled with a rapidly growing global demand and increasing resource constraints. Modeling frameworks are considered one of the few means available to understand...

We study the connections between Physarum Dynamics and Dynamic Monge Kantorovich (DMK) Optimal Transport algorithms for the solution of Basis Pursuit problems. We show the equivalence between these two models and unveil their dynamic character by showing existence and uniqueness of the solution for all times and constructing a Lyapunov functional w...

In this paper we present an extension of the Dynamic-Monge-Kantorovich model that considers a time derivative of the transport density that grows as a power law of the transport flux counterbalanced by a linear decay term that maintains the density bounded. A sub-linear growth penalizes the flux intensity (i.e. the transport density) and promotes d...

loodplains, and rivers therein, constitute complex systems whose simulation involves modeling of hydrodynamic, morphodynamic, chemical, and biological processes which act over a wide range of time scales (from days to centuries) and affect each other. Self-formed floodplains are produced by the sedimentary processes associated with the migration of...

A formulation of the shallow water equations adapted to general complex terrains is proposed. Its derivation starts from the observation that the typical approach of depth integrating the Navier-Stokes equations along the direction of gravity forces is not exact in the general case of a tilted curved bottom. We claim that an integration path that b...

We consider the efficient solution of sequences of linear systems arising in the numerical solution of a branched transport model whose long time solution for specific parameter settings is equivalent to the solution of the Monge-Kantorovich equations of optimal transport. Galerkin FEM discretization combined with explicit Euler time stepping yield...

The lookup table option, as an alternative to analytical calculation for evaluating the nonlinear heterogeneous soil characteristics, is introduced and compared for both the Picard and Newton iterative schemes in the numerical solution of Richards’ equation. The lookup table method can be a cost-effective alternative to analytical evaluation in the...

This work addresses the signatures embedded in the planform geometry of meandering rivers consequent to the formation of floodplain heterogeneities as the river bends migrate. Two geomorphic features are specifically considered: scroll bars produced by lateral accretion of point bars at convex banks and oxbow lake fills consequent to neck cutoffs....

Saline–freshwater interaction in porous media is a phenomenon of practical interest particularly for the management of water resources in arid and semi-arid environments, where precious freshwater resources are threatened by seawater intrusion and where storage of freshwater in saline aquifers can be a viable option. Saline–freshwater interactions...

Emphasizing the physical intricacies of integrated hydrology and feedbacks in simulating connected, variably-saturated groundwater-surface water systems, the Integrated Hydrologic Model Intercomparison Project initiated a second phase (IH-MIP2), increasing the complexity of the benchmarks of the first phase. The models that took part in the interco...

In this work we study and expand a model describing the dynamics of a unicellular slime mold, Physarum Polycephalum (PP), which was proposed to simulate the ability of PP to find the shortest path connecting two food sources in a maze. The original model describes the dynamics of the slime mold on a finite dimensional planar graph using a pipe-flow...

Saline-freshwater interaction in porous media is a phenomenon of practical interest particularly for the management of water resources in arid and semi-arid environments, where precious freshwater resources are threatened by seawater intrusion and where storage of freshwater in saline aquifers can be a viable option. Saline-freshwater interactions...

This paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive-definite linear systems arising from the numerical discretization of transient parabolic and self-adjoint partial differential equations. The preconditioners are obtained by nesting appropriate projections of reduced-order models in...

This paper explores the challenges of model parameterization and process representation when simulating multiple hydrologic responses from a highly controlled unsaturated flow and transport experiment with a physically-based model. The experiment, conducted at the Landscape Evolution Observatory (LEO), involved alternate injections of water and deu...

The work proposed deals with the characterization of temporal and spatial variability of water exchange fluxes from/to the Brenta river streambed (Veneto, Italy), critically important to regional water resources management. The aquifer system evolves from a large undifferentiated aquifer close to the adjacent mountain ranges, later developing into...

Coupling hydrological models with plant physiology is crucial to capture the feedback mechanisms occurring within the Soil-Plant-Atmosphere continuum. However, the ability of such models to describe the spatial variability of plant responses to different environmental factors remains to be proven, especially at large scales (field/watershed). Here...

This work deals with the characterization of the spatial and temporal variability of water exchange fluxes from/to the Brenta river streambed (Veneto, Italy), critically important to regional water resources management. The aquifer system is structurally quite complex. It evolves from a large undifferentiated aquifer near adjacent mountain ranges t...

Integrated, process-based numerical models in hydrology are rapidly evolving, spurred by novel theories in mathematical physics, advances in computational methods, insights from laboratory and field experiments, and the need to better understand and predict the potential impacts of population, land use, and climate change on our water resources. At...

Accurate monitoring and modeling of soil-plant systems are a key unresolved issue that currently limits the development of a comprehensive view of the interactions between soil and atmosphere, with a number of practical consequences including the difficulties in predicting climatic change patterns. This paper presents a case study where time-lapse...

Geophysical surveys can provide useful, albeit indirect, information on vadose zone processes. However, the ability to provide a quantitative description of the subsurface hydrological phenomena requires to fully integrate geophysical data into hydrological modeling. Here, we describe a controlled infiltration experiment that was monitored using bo...

A mechanistic model for the soil-plant system is coupled to a conventional slab representation of the atmospheric boundary layer (ABL) to explore the role of groundwater table (WT) variations and free atmospheric (FA) states on convective rainfall predisposition (CRP) at a Loblolly pine plantation site situated in the lower coastal plain of North C...

The modeling of unsaturated groundwater flow is affected by a high degree of uncertainty related to both measurement and model errors. Geophysical methods such as Electrical Resistivity Tomography (ERT) can provide useful indirect information on the hydrological processes occurring in the vadose zone. In this paper, we propose and test an iterated...

We develop and analyze a post processing technique for the family of low-order mimetic discretizations based on vertex unknowns for the numerical treatment of diffusion problems on unstructured polygonal and polyhedral meshes. The post processing works in two steps. First, from the nodal degrees of freedom, we reconstruct an elemental-based vector...

Floodplains and sinuous rivers have a close relationship with each other, mutually influencing their evolutions in time and space. The heterogeneity in erosional resistance has a crucial role on meander planform evolution. It depends on external factors, like land use and cover, but also on the composition of the floodplain, which is due to the anc...

The role of root water uptake in regulating soil water saturation in salt marshes is controversial. Modeling studies suggest that soil aeration is improved by transpiration, with implications for the distribution of vegetation species and of the associated topographic features controlling the hydraulic regime of the marshland and eventually its sur...

The Venice coastland, Italy, is a precarious environment jeopardized by both natural and anthropogenic factors. Due to a land elevation below sea level and the presence of sandy paleo-channels, salinization of soil and shallow groundwater is posing a serious threat to the agricultural productivity of the region. In order to identify and quantify th...

Competition for water among multiple tree rooting systems is investigated using a soil-plant model that accounts for soil moisture dynamics and root water uptake (RWU), whole plant transpiration, and leaf-level photosynthesis. The model is based on a numerical solution to the 3D Richards equation modified to account for a 3D RWU, trunk xylem, and s...

The southern portion of the Venice coastland is a very precarious environment and salt contamination of land and groundwater is a severe problem that is seriously impacting the farmland productivity. Geophysical surveys, lab testing and continuous monitoring of hydrological parameters together with crop yield distribution were performed and acquire...

[1] We present a model-order reduction technique that overcomes the computational burden associated with the application of Monte Carlo methods to the solution of the groundwater flow equation with random hydraulic conductivity. The method is based on the Galerkin projection of the high-dimensional model equations onto a subspace, approximated by a...

Preconditioners for the Conjugate Gradient method are studied to solve the Newton system with symmetric positive definite (SPD) Jacobian. Following the theoretical work in Bergamaschi et al. (2011) [4] we start from a given approximation of the inverse of the initial Jacobian, and we construct a sequence of preconditioners by means of a low rank up...

In this work we present the application of time-lapse non-invasive 3D micro- electrical tomography (ERT) to monitor soil-plant interactions in the root zone in the framework of the FP7 Project CLIMB (Climate Induced Changes on the Hydrology of Mediterranean Basins). The goal of the study is to gain a better understanding of the soil-vegetation inte...

Ground-penetrating radar (GPR) and Electrical Resistivity Tomography (ERT) can provide useful indirect information on the dynamic processes occurring in the vadose zone. However, to achieve a quantitative description of soil moisture dynamics, the information content of geophysical observations has to be exploited in a hydrological modeling framewo...

The numerical solution of stochastic groundwater flow problems driven by randomly heterogeneous hydraulic conductivity distributions typically requires performing a set of computationally expensive Monte Carlo (MC) simulations. In this context, reduced order methodologies can be considered as a promising way to obtain accurate and efficient solutio...

We explore the ability of the greedy algorithm to serve as an effective tool for the construction of reduced-order models for the solution of fully saturated groundwater flow in the presence of randomly distributed transmissivities. The use of a reduced model is particularly appealing in the context of numerical Monte Carlo (MC) simulations that ar...

The southern portion of the Venice coastland includes a very precarious environment. Due to an elevation down to 4 m below msl, the Venice Lagoon and Adriatic Sea proximity, and the encroachment of seawater from the mouth of the river network up to 20 km inland, salt contamination of land and groundwater is a severe problem that is seriously impact...

Quantitative hydrogeology often relies on numerical modeling of flow and transport processes in the earth subsurface. Despite the richness of numerical schemes proposed in the literature most applications are performed by using a few very popular codes based on classical finite volume or finite element techniques. An important limitation of these n...

Water resources systems management often requires complex mathematical models as tools for decision making. Most of these models contain parameters that must be correctly identified in order to develop an accurate model; the problem of parameter identification is commonly known as the inverse problem. There exist a multitude of methods designed to...

A three-dimensional (3D) groundwater flow model of the deep multi-aquifer Quaternary deposits of the Po plain sedimentary basin, within a 3,300-km2 area (Veneto, Italy), is developed, tested and applied to aid sustainable large-scale water-resources management. Integrated mathematical modelling proves significantly successful, owing to an unusual w...

In this work, preconditioners for the iterative solution by Krylov methods of the linear systems arising at each Newton iteration are studied. The preconditioner is defined by means of a Broyden-type rank-one update of a given initial preconditioner, at each nonlinear iteration, as described in [5] where convergence properties of the scheme are the...

Simulating fluid-s