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Register a new userA Variational Formulation of the BDF2 Method for Metric Gradient Flows
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We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity --- but no smoothness --- of the augmented energy functional, we prove well-posedness of the method and convergence of the discrete approximations to a curve of steepest descent. In a smooth Hilbertian setting, classical theory would predict a convergence order of two in time, we prove convergence order of one-half in the general metric setting and under our weak hypotheses. Further, we illustrate these results with numerical experiments for gradient flows on a compact Riemannian manifold, in a Hilbert space, and in the $L^2$-Wasserstein metric.
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We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion of curves of maximal slope, we introduce the latter to include limits of time-incremental approximations constructed via the Minimizing Movements approach. For both notions of solutions we prove the existence of the global attractor. Since the evolutionary problems we consider may lack uniqueness, we rely on the theory of generalized semiflows introduced by J.M. Ball. The notions of generalized and energy solutions are quite flexible and can be used to address gradient flows in a variety of contexts, ranging from Banach spaces to Wasserstein spaces of probability measures. We present applications of our abstract results by proving the existence of the global attractor for the energy solutions both of abstract doubly nonlinear evolution equations in reflexive Banach spaces, and of a class of evolution equations in Wasserstein spaces, as well as for the generalized solutions of some phase-change evolutions driven by mean curvature.
In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (A) a natural interaction between the duality structure and the coarse-graining, (B) application to systems with non-dissipative effects, and (C) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the small-noise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom.
Numerical simulations of flows are required for numerous applications, and are usually carried out using shallow water equations. We describe the FullSWOF software which is based on up-to-date finite volume methods and well-balanced schemes to solve this kind of equations. It consists of a set of open source C++ codes, freely available to the community, easy to use, and open for further development. Several features make FullSWOF particularly suitable for applications in hydrology: small water heights and wet-dry transitions are robustly handled, rainfall and infiltration are incorporated, and data from grid-based digital topographies can be used directly. A detailed mathematical description is given here, and the capabilities of FullSWOF are illustrated based on analytic solutions and datasets of real cases. The codes, available in 1D and
This paper presents a variational approach to doubly-nonlinear (gradient) flows (P) of nonconvex energies along with nonpotential perturbations (i.e., perturbation terms without any potential structures). An elliptic-in-time regularization of the original equation ${rm (P)}_varepsilon$ is introduced, and then, a variational approach and a fixed-point argument are employed to prove existence of strong solutions to regularized equations. More precisely, we introduce a functional (defined for each entire trajectory and including a small approximation parameter $varepsilon$) whose Euler-Lagrange equation corresponds to the elliptic-in-time regularization of an unperturbed (i.e. without nonpotential perturbations) doubly-nonlinear flow. Secondly, due to the presence of nonpotential perturbation, a fixed-point argument is performed to construct strong solutions $u_varepsilon$ to the elliptic-in-time regularized equations ${rm (P)}_varepsilon$. Here, the minimization problem mentioned above defines an operator $S$ whose fixed point corresponds to a solution $u_varepsilon$ of ${rm (P)}_varepsilon$. Finally, a strong solution to the original equation (P) is obtained by passing to the limit of $u_varepsilon$ as $varepsilon to 0$. Applications of the abstract theory developed in the present paper to concrete PDEs are also exhibited.
A novel computational, non-iterative and noise-robust reconstruction method is introduced for the planar anisotropic inverse conductivity problem. The method is based on bypassing the unstable step of the reconstruction of the values of the isothermal coordinates on the boundary of the domain. Non-uniqueness of the inverse problem is dealt with by recovering the unique isotropic conductivity that can be achieved as a deformation of the measured anisotropic conductivity by emph{isothermal coordinates}. The method shows how isotropic D-bar reconstruction methods have produced reasonable and informative reconstructions even when used on EIT data known to come from anisotropic media, and when the boundary shape is not known precisely. Furthermore, the results pave the way for regularized anisotropic EIT. Key aspects of the approach involve D-bar methods and inverse scattering theory, complex geometrical optics solutions, and quasi-conformal mapping techniques.
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