Carlo Marcati

I am a researcher at the Dipartimento di Matematica of the University of Pavia.

See below more information about my research activity. See also the pdf version of my CV and my contact page.

Research

My research activity focuses on

• the regularity of solutions to elliptic problems with point singularities,
• the numerical approximation of a priori electronic structure calculations in quantum chemistry,
• low-rank tensor representation,
• neural networks for PDEs.

Papers and preprints

1. I. Bioli, C. Marcati, G. Sangalli. Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebra, arXiv:2505.11638, 2025 [ abstract | pdf ]

   Natural Gradient Descent (NGD) has emerged as a promising optimization algorithm for training neural network-based solvers for partial differential equations (PDEs), such as Physics-Informed Neural Networks (PINNs). However, its practical use is often limited by the high computational cost of solving linear systems involving the Gramian matrix. While matrix-free NGD methods based on the conjugate gradient (CG) method avoid explicit matrix inversion, the ill-conditioning of the Gramian significantly slows the convergence of CG. In this work, we extend matrix-free NGD to broader classes of problems than previously considered and propose the use of Randomized Nyström preconditioning to accelerate convergence of the inner CG solver. The resulting algorithm demonstrates substantial performance improvements over existing NGD-based methods on a range of PDE problems discretized using neural networks.

2. C. Marcati, Ch. Schwab, J. Zech. Sparsity for Infinite-Parametric Holomorphic Functions on Gaussian Spaces, arXiv:2504.21639, 2025 [ abstract | pdf ]

   We investigate the sparsity of Wiener polynomial chaos expansions of holomorphic maps G on Gaussian Hilbert spaces, as arise in the coefficient-to-solution maps of linear, second order, divergence-form elliptic PDEs with log-Gaussian diffusion coefficient. Representing the Gaussian random field input as an affine-parametric expansion, the nonlinear map becomes a countably-parametric, deterministic holomorphic map of the coordinate sequence y=(yj)j∈ℕ∈ℝ∞. We establish weighted summability results for the Wiener-Hermite coefficient sequences of images of affine-parametric expansions of the log-Gaussian input under G. These results give rise to N-term approximation rate bounds for the full range of input summability exponents p∈(0,2). We show that these approximation rate bounds apply to parameter-to-solution maps for elliptic diffusion PDEs with lognormal coefficients.

3. C. Marcati, Ch. Schwab. Expression rates of neural operators for elliptic PDEs in polytopes, arXiv:2409.17552, 2024 [ abstract | pdf ]

   We study the approximation rates of a class of deep neural network approximations of operators, which arise as data-to-solution maps G† of linear elliptic partial differential equations (PDEs), and act between pairs X,Y of suitable infinite-dimensional spaces. We prove expression rate bounds for approximate neural operators G with the structure D◌A◌E with linear encoders E and decoders D. The constructive proofs are via a recurrent NN structure obtained by unrolling exponentially convergent, self-consistent (``Richardson'') iterations. We bound the operator approximation error with respect to the linear Kolmogorov N-widths of the data and solution sets and in terms of the size of the approximation network. We prove expression rate bounds for approximate, neural solution operators emulating the coefficient-to-solution maps for elliptic PDEs set in d-dimensional polytopes, with d∈{2,3}, and subject to Dirichlet-, Neumann- or mixed boundary conditions. Exploiting weighted norm characterizations of the solution sets of elliptic PDEs in polytopes, we show algebraic rates of expression for problems with data with finite regularity, and exponential operator expression rates for analytic data.

4. M. Faustmann, C. Marcati, J. M. Melenk, Ch. Schwab. Weighted analytic regularity for the integral fractional Laplacian in polyhedra. Analysis and Applications, 23(4), 2025 [ abstract | pdf ]

   On polytopal domains in ℝ³, we prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian with analytic right-hand side. Employing the Caffarelli-Silvestre extension allows to localize the problem and to decompose the regularity estimates into results on vertex, edge, face, vertex-edge, vertex-face, edge-face and vertex-edge-face neighborhoods of the boundary. Using tangential differentiability of the extended solutions, a bootstrapping argument based on Caccioppoli inequalities on dyadic decompositions of the neighborhoods provides weighted, analytic control of higher order solution derivatives.

5. Yanchen He, Carlo Marcati, and Christoph Schwab. Analytic regularity for the incompressible Navier-Stokes equations in polygons with mixed boundary conditions. SIAM J. on Math. Anal., 2024, 56(2), pp. 2488–2520. [ abstract | pdf ]

   We prove weighted analytic regularity of Leray-Hopf variational solutions for the stationary, incompressible Navier-Stokes Equations (NSE) in plane polygonal domains, subject to analytic body forces. We admit mixed boundary conditions which may change type at each vertex, under the assumption that homogeneous Dirichlet (''no-slip'') boundary conditions are prescribed on at least one side at each vertex of the domain. The weighted analytic regularity results are established in Hilbertian Kondrat'ev spaces with homogeneous corner weights. The proofs rely on a priori estimates for the corresponding linearized boundary value problem in sectors in corner-weighted Sobolev spaces and on an induction argument for the weighted norm estimates on the quadratic nonlinear term in the NSE, in a polar frame.

6. M. Faustmann, C. Marcati, J. M. Melenk, Ch. Schwab. Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons. SIAM Journal on Numerical Analysis 61(6), 2023 [ abstract | pdf ]

   We prove exponential convergence in the energy norm of hp finite element discretizations for the integral fractional diffusion operator of order 2s∈(0,2) subject to homogeneous Dirichlet boundary conditions in bounded polygonal domains Ω⊂ℝ². Key ingredient in the analysis are the weighted analytic regularity from our previous work and meshes that feature anisotropic geometric refinement towards ∂Ω.

7. Carlo Marcati and Christoph Schwab. Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations. SIAM Journal on Numerical Analysis, 2023, 61(3), pp. 1513–1545. [ abstract | pdf ]

   We construct deep operator networks (ONets) between infinite-​dimensional spaces that emulate with an exponential rate of convergence the coefficient-​to-solution map of elliptic second-​order PDEs. In particular, we consider problems set in d-​dimensional periodic domains, d=1,2,…, and with analytic right-​hand sides and coefficients. Our analysis covers diffusion-​reaction problems, parametric diffusion equations, and elliptic systems such as linear isotropic elastostatics in heterogeneous materials. We leverage the exponential convergence of spectral collocation methods for boundary value problems whose solutions are analytic. In the present periodic and analytic setting, this follows from classical elliptic regularity. Within the ONet branch and trunk construction of [Chen and Chen, 1993] and of [Lu et al., 2021], we show the existence of deep ONets which emulate the coefficient-​to-solution map to accuracy ε>0 in the H1 norm, uniformly over the coefficient set. We prove that the neural networks in the ONet have size O(|log(ε)|κ) for some κ>0 depending on the physical space dimension.

8. Markus Faustmann, Carlo Marcati, Jens Markus Melenk, and Christoph Schwab. Weighted analytic regularity for the integral fractional Laplacian in polygons. SIAM Journal on Mathematical Analysis 54(6), 2022. [ abstract | pdf ]

   We prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian in polygons with analytic right-​hand side. We localize the problem through the Caffarelli-​Silvestre extension and study the tangential differentiability of the extended solutions, followed by bootstrapping based on Caccioppoli inequalities on dyadic decompositions of vertex, edge, and edge-​vertex neighborhoods.

9. Carlo Marcati, Joost A. A. Opschoor, Philipp C. Petersen, and Christoph Schwab. Exponential ReLU neural network approximation rates for point and edge singularities. Found. Comp. Math., 2023, 23(3), pp. 1043–1127. [ abstract | pdf ]

   We prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in H1(Ω) for weighted analytic function classes in certain polytopal domains Ω, in space dimension d=2,3. Functions in these classes are locally analytic on open subdomains D⊂Ω, but may exhibit isolated point singularities in the interior of Ω or corner and edge singularities at the boundary ∂Ω. The exponential expression rate bounds proved here imply uniform exponential expressivity by ReLU NNs of solution families for several elliptic boundary and eigenvalue problems with analytic data. The exponential approximation rates are shown to hold in space dimension d=2 on Lipschitz polygons with straight sides, and in space dimension d=3 on Fichera-type polyhedral domains with plane faces. The constructive proofs indicate in particular that NN depth and size increase poly-logarithmically with respect to the target NN approximation accuracy ε>0 in H1(Ω). The results cover in particular solution sets of linear, second order elliptic PDEs with analytic data and certain nonlinear elliptic eigenvalue problems with analytic nonlinearities and singular, weighted analytic potentials as arise in electron structure models. In the latter case, the functions correspond to electron densities that exhibit isolated point singularities at the positions of the nuclei. Our findings provide in particular mathematical foundation of recently reported, successful uses of deep neural networks in variational electron structure algorithms.

10. Carlo Marcati, Maxim Rakhuba, and Johan E. M. Ulander. Low rank tensor approximation of singularly perturbed partial differential equations in one dimension. Calcolo 59 (2022). [ abstract | pdf ]

    We derive rank bounds on the quantized tensor train (QTT) compressed approximation of singularly perturbed reaction diffusion partial differential equations (PDEs) in one dimension. Specifically, we show that, independently of the scale of the singular perturbation parameter, a numerical solution with accuracy 0<ϵ<1 can be represented in QTT format with a number of parameters that depends only polylogarithmically on ϵ. In other words, QTT compressed solutions converge exponentially to the exact solution, with respect to a root of the number of parameters. We also verify the rank bound estimates numerically, and overcome known stability issues of the QTT based solution of PDEs by adapting a preconditioning strategy to obtain stable schemes at all scales. We find, therefore, that the QTT based strategy is a rapidly converging algorithm for the solution of singularly perturbed PDEs, which does not require prior knowledge on the scale of the singular perturbation and on the shape of the boundary layers.

11. Yvon Maday, Carlo Marcati. Weighted analyticity of Hartree-Fock wavefunctions. arXiv:2010.06923. [ abstract | pdf ]

    We prove analytic-type estimates in weighted Sobolev spaces on the eigenfunctions of a class of elliptic and nonlinear eigenvalue problems with singular potentials, which includes the Hartree-Fock equations. Going beyond classical results on the analyticity of the wavefunctions away from the nuclei, we prove weighted estimates locally at each singular point, with precise control of the derivatives of all orders. Our estimates have far-reaching consequences for the approximation of the eigenfunctions of the problems considered, and they can be used to prove a priori estimates on the numerical solution of such eigenvalue problems.

12. Carlo Marcati, Maxim Rakhuba, and Christoph Schwab. Tensor Rank bounds for Point Singularities in ℝ³. Adv. Comp. Math., 2022, 48(3), 18. [ abstract | pdf ]

    We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in R³. We consider functions in countably normed Sobolev spaces with radial weights and analytic- or Gevrey-type control of weighted semi-norms. Several classes of boundary value and eigenvalue problems from science and engineering are discussed whose solutions belong to the countably normed spaces. It is shown that quantized, tensor-structured approximations of functions in these classes exhibit tensor ranks bounded polylogarithmically with respect to the accuracy ϵ∈(0,1) in the Sobolev space H¹. We prove exponential convergence rates of three specific types of quantized tensor decompositions: quantized tensor train (QTT), transposed QTT and Tucker-QTT. In addition, the bounds for the patchwise decompositions are uniform with respect to the position of the point singularity. An auxiliary result of independent interest is the proof of exponential convergence of hp-finite element approximations for Gevrey-regular functions with point singularities in the unit cube Q=(0,1)³. Numerical examples of function approximations and of Schrödinger-type eigenvalue problems illustrate the theoretical results.

13. Yvon Maday, Carlo Marcati. Analyticity and hp discontinuous Galerkin approximation of nonlinear Schrödinger eigenproblems. Math. Models Methods Appl. Sci., 2023, 33(13), pp. 2657–2701. [ abstract | pdf ]

    We study a class of nonlinear eigenvalue problems of Schrödinger type, where the potential is singular on a set of points. Such proble ms are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and w e propose and analyze the approximation of the solution via an isotropically refined hp discontinuous Galerkin (dG) method. We show that, for weighted analytic potentials and for up-to-quartic nonlinear ities, the eigenfunctions belong to analytic-type non homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the error of the dG finite element method; when using an isotropical ly refined hp space the numerical solution is shown to converge with exponential rate towards the exact eigenfunction. In addition, we investigate the role of pointwise convergence in the doubling of the conve rgence rate for the eigenvalues with respect to the convergence rate of eigenfunctions. We conclude with a series of numerical tests to validate the theoretical results.

14. Alexey Chernov, Carlo Marcati, and Lorenzo Mascotto. p- and hp- virtual elements for the Stokes problem. Adv. Comput. Math., 47 (2021). [ abstract | pdf ]

    We analyse the p- and hp-versions of the virtual element method (VEM) for the the Stokes problem on a polygonal domain. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from [Beirão da Veiga, L., Chernov, A., Mascotto, L., Russo, A.: Exponential convergence of the hp virtual element method with corner singularity. Numer. Math. 138(3), 581–613 (2018)] an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy of the method. We prove exponential convergence of the hp-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the p- and hp-versions of the method.

15. Dominik Schötzau, Carlo Marcati, and Christoph Schwab. Exponential convergence of mixed hp-DGFEM for the incompressible Navier-Stokes equations in ℝ². IMA J. Numer. Anal., 2021, 41(3), pp. 1966–1999. [ abstract | pdf ]

    In a polygon Ω⊂ℝ², we consider mixed hp-discontinuous Galerkin approximations of the stationary, incompressible Navier-Stokes equations, subject to no-slip boundary conditions. We use geometrically corner-refined meshes and hp spaces with linearly increasing polynomial degrees. Based on recent results on analytic regularity of velocity field and pressure of Leray solutions in Ω, we prove exponential rates of convergence of the mixed hp-discontinuous Galerkin finite element method (hp-DGFEM), with respect to the number of degrees of freedom, for small data which is piecewise analytic.

16. Carlo Marcati, Christoph Schwab. Analytic Regularity for the incompressible Navier-Stokes Equations in polygons. SIAM J. Math. Anal. 52 (2020), no. 3, 2945–2968. [ abstract | pdf ]

    In a plane polygon P with straight sides, we prove analytic regularity of the Leray-Hopf solution of the stationary, viscous and incompressible Navier-Stokes equations. We assume small data, analytic volume force and no-slip boundary conditions. Analytic regularity is quantified in so-called countably normed, corner-weighted spaces with homogeneous norms. Implications of this analytic regularity include exponential convergence rates of so-called hp-Finite Element and Spectral Element discretizations.

17. Yvon Maday, Carlo Marcati. Regularity and hp discontinuous Galerkin finite element approximation of linear elliptic eigenvalue problems with singular potentials. Math. Models Methods Appl. Sci. 29 (2019), no. 8, 1585–1617. [ abstract | pdf ]

    We study the regularity in weighted Sobolev spaces of Schrödinger-type eigenvalue problems, and we analyse their approximation via a discontinuous Galerkin (dG) hp finite element method. In particular, we show that, for a class of singular potentials, the eigenfunctions of the operator belong to analytic-type non homogeneous weighted Sobolev spaces. Using this result, we prove that the an isotropically graded hp dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the the behaviour of the method for varying discretisation parameters.

18. Paola F. Antonietti, Carlo Marcati, Ilario Mazzieri, and Alfio Quarteroni. High order discontinuous Galerkin methods on simplicial elements for the elastodynamics equation. Numerical Algorithms, 71(1):181–206, 2015. [ abstract | http ]

    In this work we apply the discontinuous Galerkin (dG) spectral element method on meshes made of simplicial elements for the approximation of the elastodynamics equation. Our approach combines the high accuracy of spectral methods, the geometrical flexibility of simplicial elements and the computational efficiency of dG methods. We analyze the dissipation, dispersion and stability properties of the resulting scheme, with a focus on the choice of different sets of basis functions. Finally, we apply the method on benchmark as well as realistic test cases.

PhD Thesis

Carlo Marcati. Discontinuous hp finite element methods for elliptic eigenvalue problems with singular potentials — with applications to quantum chemistry. Sorbonne Université, 2018. [ abstract ]