Heat flow and calculus on metric measure spaces with Ricci curvature bounded below - the compact case

We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces over metric measure spaces, the equivalence of the L^2-gradient flow of a suitably defined "Dirichlet energy" and the Wasserstein gradient flow of the relative entropy functional, a metric version of Brenier's Theorem, and a new (stronger) definition of Ricci curvature bound from below for metric measure spaces. This new notion is stable w.r.t. measured Gromov-Hausdorff convergence and it is strictly connected with the linearity of the heat flow.Comment: To the memory of Enrico Magenes, whose exemplar life, research and teaching shaped generations of mathematician

Efficient Resolution of Anisotropic Structures

We highlight some recent new delevelopments concerning the sparse representation of possibly high-dimensional functions exhibiting strong anisotropic features and low regularity in isotropic Sobolev or Besov scales. Specifically, we focus on the solution of transport equations which exhibit propagation of singularities where, additionally, high-dimensionality enters when the convection field, and hence the solutions, depend on parameters varying over some compact set. Important constituents of our approach are directionally adaptive discretization concepts motivated by compactly supported shearlet systems, and well-conditioned stable variational formulations that support trial spaces with anisotropic refinements with arbitrary directionalities. We prove that they provide tight error-residual relations which are used to contrive rigorously founded adaptive refinement schemes which converge in $L_2$. Moreover, in the context of parameter dependent problems we discuss two approaches serving different purposes and working under different regularity assumptions. For frequent query problems, making essential use of the novel well-conditioned variational formulations, a new Reduced Basis Method is outlined which exhibits a certain rate-optimal performance for indefinite, unsymmetric or singularly perturbed problems. For the radiative transfer problem with scattering a sparse tensor method is presented which mitigates or even overcomes the curse of dimensionality under suitable (so far still isotropic) regularity assumptions. Numerical examples for both methods illustrate the theoretical findings

Greedy optimal control for elliptic problems and its application to turnpike problems

This is a post-peer-review, pre-copyedit version of an article published in Numerische Mathematik. The final authenticated version is available online at: https://doi.org/10.1007/s00211-018-1005-zWe adapt and apply greedy methods to approximate in an efficient way the optimal controls for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling the parameter-space to compute controls for each parameter value. The same method can be adapted for parabolic control problems, but this leads to greedy selections of the realizations of the parameters that depend on the initial datum under consideration. The turnpike property (which ensures that parabolic optimal control problems behave nearly in a static manner when the control horizon is long enough) allows using the elliptic greedy choice of the parameters in the parabolic setting too. We present various numerical experiments and an extensive discussion of the efficiency of our methodology for parabolic control and indicate a number of open problems arising when analyzing the convergence of the proposed algorithmsThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694126-DyCon). Part of this research was done while the second author visited DeustoTech and Univesity of Deusto with the support of the DyCon project. The second author was also partially supported by Croatian Science Foundation under ConDyS Project, IP-2016-06-2468. The work of the third author was partially supported by the Grants MTM2014-52347, MTM2017-92996 of MINECO (Spain) and ICON of the French AN

Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs

The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affine parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affine expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold -- only spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic equations confirming the predicted convergence rate

Wavelet boundary element methods – Adaptivity and goal-oriented error estimation

This article is dedicated to the adaptive wavelet boundary element method. It computes an approximation to the unknown solution of the boundary integral equation under consideration with a rate $N^{−s}_{dof}$, whenever the solution can be approximated with this rate in the setting determined by the underlying wavelet basis. The computational cost scale linearly in the number $N_{dof}$ of degrees of freedom. Goal-oriented error estimation for evaluating linear output functionals of the solution is also considered. An algorithm is proposed that approximately evaluates a linear output functional with a rate $N^{−(s+t)}_{dof}$, whenever the primal solution can be approximated with a rate $N^{-s}_{dof}$ and the dual solution can be approximated with a rate $N^{−t}_{dof}$, while the cost still scale linearly in $N_{dof}$. Numerical results for an acoustic scattering problem and for the point evaluation of the potential in case of the Laplace equation are reported to validate and quantify the approach

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references