Foundations of Computational Mathematics最新文献

B-Series and generalizations are a powerful tool for the analysis of numerical integrators. An extension named exotic aromatic B-Series was introduced to study the order conditions for sampling the invariant measure of ergodic SDEs. Introducing a new symmetry normalization coefficient, we analyze the algebraic structures related to exotic B-Series and S-Series. Precisely, we prove the relationship between the Grossman–Larson algebras over exotic and grafted forests and the corresponding duals to the Connes–Kreimer coalgebras and use it to study the natural composition laws on exotic S-Series. Applying this algebraic framework to the derivation of order conditions for a class of stochastic Runge–Kutta methods, we present a multiplicative property that ensures some order conditions to be satisfied automatically.

A precise characterization of the extremal points of sublevel sets of nonsmooth penalties provides both detailed information about minimizers, and optimality conditions in general classes of minimization problems involving them. Moreover, it enables the application of fully corrective generalized conditional gradient methods for their efficient solution. In this manuscript, this program is adapted to the minimization of a smooth convex fidelity term which is augmented with an unbalanced transport regularization term given in the form of a generalized Kantorovich–Rubinstein norm for Radon measures. More precisely, we show that the extremal points associated to the latter are given by all Dirac delta functionals supported in the spatial domain as well as certain dipoles, i.e., pairs of Diracs with the same mass but with different signs. Subsequently, this characterization is used to derive precise first-order optimality conditions as well as an efficient solution algorithm for which linear convergence is proved under natural assumptions. This behavior is also reflected in numerical examples for a model problem.

We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive-semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis, and they have been studied for many decades. On the one hand, we prove that when the rank of the positive-semidefinite matrix in the decomposition is bounded above by an absolute constant, the problem can be solved in polynomial time. On the other hand, we prove that, in general, these problems as well as their certain approximation versions are all NP-hard. Finally, we prove that many of these low-rank decomposition problems are complete in the first-order theory of the reals, i.e., given any system of polynomial equations, we can write down a low-rank decomposition problem in polynomial time so that the original system has a solution iff our corresponding decomposition problem has a feasible solution of certain (lowest) rank.

According to a version of Donsker’s theorem, geodesic random walks on Riemannian manifolds converge to the respective Brownian motion. From a computational perspective, however, evaluating geodesics can be quite costly. We therefore introduce approximate geodesic random walks based on the concept of retractions. We show that these approximate walks converge in distribution to the correct Brownian motion as long as the geodesic equation is approximated up to second order. As a result, we obtain an efficient algorithm for sampling Brownian motion on compact Riemannian manifolds.

In the framework of real Hilbert spaces, we study continuous in time dynamics as well as numerical algorithms for the problem of approaching the set of zeros of a single-valued monotone and continuous operator V. The starting point of our investigations is a second-order dynamical system that combines a vanishing damping term with the time derivative of V along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. Our method exhibits fast convergence rates of order (o left( frac{1}{tbeta (t)} right) ) for (Vert V(z(t))Vert ), where (z(cdot )) denotes the generated trajectory and (beta (cdot )) is a positive nondecreasing function satisfying a growth condition, and also for the restricted gap function, which is a measure of optimality for variational inequalities. We also prove the weak convergence of the trajectory to a zero of V. Temporal discretizations of the dynamical system generate implicit and explicit numerical algorithms, which can be both seen as accelerated versions of the Optimistic Gradient Descent Ascent (OGDA) method for monotone operators, for which we prove that the generated sequence of iterates ((z_k)_{k ge 0}) shares the asymptotic features of the continuous dynamics. In particular we show for the implicit numerical algorithm convergence rates of order (o left( frac{1}{kbeta _k} right) ) for (Vert V(z^k)Vert ) and the restricted gap function, where ((beta _k)_{k ge 0}) is a positive nondecreasing sequence satisfying a growth condition. For the explicit numerical algorithm, we show by additionally assuming that the operator V is Lipschitz continuous convergence rates of order (o left( frac{1}{k} right) ) for (Vert V(z^k)Vert ) and the restricted gap function. All convergence rate statements are last iterate convergence results; in addition to these, we prove for both algorithms the convergence of the iterates to a zero of V. To our knowledge, our study exhibits the best-known convergence rate results for monotone equations. Numerical experiments indicate the overwhelming superiority of our explicit numerical algorithm over other methods designed to solve monotone equations governed by monotone and Lipschitz continuous operators.

Transportation of measure provides a versatile approach for modeling complex probability distributions, with applications in density estimation, Bayesian inference, generative modeling, and beyond. Monotone triangular transport maps—approximations of the Knothe–Rosenblatt (KR) rearrangement—are a canonical choice for these tasks. Yet the representation and parameterization of such maps have a significant impact on their generality and expressiveness, and on properties of the optimization problem that arises in learning a map from data (e.g., via maximum likelihood estimation). We present a general framework for representing monotone triangular maps via invertible transformations of smooth functions. We establish conditions on the transformation such that the associated infinite-dimensional minimization problem has no spurious local minima, i.e., all local minima are global minima; and we show for target distributions satisfying certain tail conditions that the unique global minimizer corresponds to the KR map. Given a sample from the target, we then propose an adaptive algorithm that estimates a sparse semi-parametric approximation of the underlying KR map. We demonstrate how this framework can be applied to joint and conditional density estimation, likelihood-free inference, and structure learning of directed graphical models, with stable generalization performance across a range of sample sizes.

The time-harmonic Maxwell equations at high wavenumber k in domains with an analytic boundary and impedance boundary conditions are considered. A wavenumber-explicit stability and regularity theory is developed that decomposes the solution into a part with finite Sobolev regularity that is controlled uniformly in k and an analytic part. Using this regularity, quasi-optimality of the Galerkin discretization based on Nédélec elements of order p on a mesh with mesh size h is shown under the k-explicit scale resolution condition that (a) kh/p is sufficient small and (b) (p/ln k) is bounded from below.

We develop lower bounds on communication in the memory hierarchy or between processors for nested bilinear algorithms, such as Strassen’s algorithm for matrix multiplication. We build on a previous framework that establishes communication lower bounds by use of the rank expansion, or the minimum rank of any fixed size subset of columns of a matrix, for each of the three matrices encoding a bilinear algorithm. This framework provides lower bounds for a class of dependency directed acyclic graphs (DAGs) corresponding to the execution of a given bilinear algorithm, in contrast to other approaches that yield bounds for specific DAGs. However, our lower bounds only apply to executions that do not compute the same DAG node multiple times. Two bilinear algorithms can be nested by taking Kronecker products between their encoding matrices. Our main result is a lower bound on the rank expansion of a matrix constructed by a Kronecker product derived from lower bounds on the rank expansion of the Kronecker product’s operands. We apply the rank expansion lower bounds to obtain novel communication lower bounds for nested Toom-Cook convolution, Strassen’s algorithm, and fast algorithms for contraction of partially symmetric tensors.

We study a finite-element based space-time discretisation for the 2D stochastic Navier–Stokes equations in a bounded domain supplemented with no-slip boundary conditions. We prove optimal convergence rates in the energy norm with respect to convergence in probability, that is convergence of order (almost) 1/2 in time and 1 in space. This was previously only known in the space-periodic case, where higher order energy estimates for any given (deterministic) time are available. In contrast to this, estimates in the Dirichlet-case are only known for a (possibly large) stopping time. We overcome this problem by introducing an approach based on discrete stopping times. This replaces the localised estimates (with respect to the sample space) from earlier contributions.

We consider deterministic mean field games where the dynamics of a typical agent is non-linear with respect to the state variable and affine with respect to the control variable. Particular instances of the problem considered here are mean field games with control on the acceleration (see Achdou et al. in NoDEA Nonlinear Differ Equ Appl 27(3):33, 2020; Cannarsa and Mendico in Minimax Theory Appl 5(2):221-250, 2020; Cardaliaguet and Mendico in Nonlinear Anal 203: 112185, 2021). We focus our attention on the approximation of such mean field games by analogous problems in discrete time and finite state space which fall in the framework of (Gomes in J Math Pures Appl (9) 93(3):308-328, 2010). For these approximations, we show the existence and, under an additional monotonicity assumption, uniqueness of solutions. In our main result, we establish the convergence of equilibria of the discrete mean field games problems towards equilibria of the continuous one. Finally, we provide some numerical results for two MFG problems. In the first one, the dynamics of a typical player is nonlinear with respect to the state and, in the second one, a typical player controls its acceleration.As per journal style, reference citation should be expanded form in abstract. So kindly check and confirm the reference citation present in the abstract is correct.Please change "Gomes in" below by "Gomes et al. in "