Giuseppe Savaré

A posteriori error estimates for variable time‐step discretizations of nonlinear evolution equations

We study the backward Euler method with variable time steps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the angle-bounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual. These estimators depend solely on the discrete solution and data and impose no constraints between consecutive time steps. We also prove that they converge to zero with an optimal rate with respect to the regularity of the solution. We apply the abstract results to a number of concrete, strongly nonlinear problems of parabolic type with degenerate or singular character.

Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds

The aim of the present paper is to bridge the gap between the Bakry–Emery and the Lott–Sturm–Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds. We start from a strongly local Dirichlet form E admitting a Carre du champ Γ in a Polish measure space (X,m) and a canonical distance dE that induces the original topology of X. We first characterize the distinguished class of Riemannian Energy measure spaces, where E coincides with the Cheeger energy induced by dE and where every function f with Γ(f)≤1 admits a continuous representative. In such a class, we show that if E satisfies a suitable weak form of the Bakry–Emery curvature dimension condition BE(K,∞) then the metric measure space (X,d,m) satisfies the Riemannian Ricci curvature bound RCD(K,∞) according to [Duke Math. J. 163 (2014) 1405–1490], thus showing the equivalence of the two notions. Two applications are then proved: the tensorization property for Riemannian Energy spaces satisfying the Bakry–Emery BE(K,N) condition (and thus the corresponding one for RCD(K,∞) spaces without assuming nonbranching) and the stability of BE(K,N) with respect to Sturm–Gromov–Hausdorff convergence.

Degenerate Evolution Systems Modeling the Cardiac Electric Field at Micro- and Macroscopic Level

The aim of this paper is to study the reaction-diffusion systems arising from the mathematical models of the cardiac electric activity at the micro- and macroscopic level.

A new class of transport distances between measures

We introduce a new class of distances between nonnegative Radon measures in

Eulerian Calculus for the Displacement Convexity in the Wasserstein Distance

{\mathbb{R}^d}

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

. They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375–393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous

Computational electrocardiology: mathematical and numerical modeling

{W^{-1,p}_\gamma}