We generalize the Brezzi-Rappaz-Raviart approximation theorem, which allows to obtain existence and a priori error estimates for approximations of solutions to some nonlinear partial differential equations. Our contribution lies in the fact that we typically allow for nonlinearities having merely Lipschitz regularity, while previous results required some form of differentiability. This is achieved by making use of the theory of metrically regular mappings, developed in the context of variational analysis. We apply this generalization to derive quasi-optimal error estimates for finite element approximations to solutions of viscous Hamilton-Jacobi equations and second order mean field game systems.
Bifurcation characterizes the qualitative changes in parameterized dynamical systems and is one of the major topics in the field. In this work, we study combinatorial bifurcations within the framework of combinatorial dynamical systems—a young but already well-established theory. We introduce the Conley–Morse persistence barcode, a compact algebraic descriptor of combinatorial bifurcations. This barcode captures structural changes in a dynamical system at the level of Morse decompositions and provides a characterization of the nature of observed transitions in terms of the Conley index. The construction of the Conley–Morse persistence barcode builds upon ideas from topological persistence. Specifically, we consider a persistence module obtained from the Conley index of invariant sets indexed over a poset. Using gentle algebras, we prove that this module decomposes into simple intervals (bars) and compute them by adapting the zigzag persistence algorithm to our purpose.
We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polynomial reproductions—a powerful tool used in the analysis of various mesh-free methods and a mesh-free method in its own right. As a key application, we prove the existence of smooth local polynomial reproductions on compact subsets of algebraic manifolds in ℝ^n with Lipschitz boundary. These results are then applied to derive new findings on the existence, stability, regularity, locality, and approximation properties of shape functions for a coordinate-free moving least squares approximation method on algebraic manifolds, which operates directly on point clouds without requiring tangent plane approximations. There are two appendices: the first derives high order Markov inequalities for polynomials on algebraic manifolds and the second gives instructions for calculating the dimension of the space of degree m polynomials restricted to a real algebraic variety.
In 1946, S. Ulam invented Monte Carlo method, which has since become the standard numerical technique for making statistical predictions for long-term behaviour of dynamical systems. We show that this, or in fact any other numerical approach can fail for the simplest non-linear discrete dynamical systems given by the logistic maps f_a(x)=ax(1-x) of the unit interval. We show that there exist computable real parameters a∈ (0,4) for which almost every orbit of f_a has the same asymptotical statistical distribution in [0,1], but this limiting distribution is not Turing computable.
Coupling arguments are a central tool for bounding the deviation between two stochastic processes, but traditionally have been limited to Wasserstein metrics. In this paper, we apply the shifted composition rule—an information-theoretic principle introduced in our earlier work [3]—in order to adapt coupling arguments to the Kullback–Leibler (KL) divergence. Our framework combines the strengths of two previously disparate approaches: local error analysis and Girsanov’s theorem. Akin to the former, it yields tight bounds by incorporating the so-called weak error, and is user-friendly in that it only requires easily verified local assumptions; and akin to the latter, it yields KL divergence guarantees and applies beyond Wasserstein contractivity. We apply this framework to the problem of sampling from a target distribution π . Here, the two stochastic processes are the Langevin diffusion and an algorithmic discretization thereof. Our framework provides a unified analysis when π is assumed to be strongly log-concave (SLC), weakly log-concave (WLC), or to satisfy a log-Sobolev inequality (LSI). Among other results, this yields KL guarantees for the randomized midpoint discretization of the Langevin diffusion. Notably, our result: (1) yields the optimal O(√(d)/ε ) rate in the SLC and LSI settings; (2) is the first result to hold beyond the 2-Wasserstein metric in the SLC setting; and (3) is the first result to hold in any metric in the WLC and LSI settings.
