Foundations of Computational Mathematics最新文献

We consider a class of discrete operators introduced by O. Chodosh, acting on infinite sequences and mimicking standard properties of pseudo-differential operators. By using a new approach, we extend this class to finite or periodic sequences, allowing a general representation of discrete pseudo-differential operators obtained by finite differences approximations and easily transferred to time discretizations. In particular we can define the notion of order and regularity, and we recover the fundamental property, well known in pseudo-differential calculus, that the commutator of two discrete operators gains one order of regularity. As examples of practical applications, we revisit standard error estimates for the convergence of splitting methods, obtaining in some Hamiltonian cases no loss of derivative in the error estimates, in particular for discretizations of general waves and/or water-waves equations. Moreover, we give an example of preconditioner constructions inspired by normal form analysis to deal with the similar question for more general cases.

In the present paper, we consider a class of quasilinear wave equations on a smooth, bounded domain. We discretize it in space with isoparametric finite elements and apply a semi-implicit Euler and midpoint rule as well as the exponential Euler and midpoint rule to obtain four fully discrete schemes. We derive rigorous error bounds of optimal order for the semi-discretization in space and the fully discrete methods in norms which are stronger than the classical (H^1times L^2) energy norm under weak CFL-type conditions. To confirm our theoretical findings, we also present numerical experiments.

We propose a modification of the standard linear implicit Euler integrator for the weak approximation of parabolic semilinear stochastic PDEs driven by additive space-time white noise. This new method can easily be combined with a finite difference method for the spatial discretization. The proposed method is shown to have improved qualitative properties compared with the standard method. First, for any time-step size, the spatial regularity of the solution is preserved, at all times. Second, the proposed method preserves the Gaussian invariant distribution of the infinite dimensional Ornstein–Uhlenbeck process obtained when the nonlinearity is removed, for any time-step size. The weak order of convergence of the proposed method is shown to be equal to 1/2 in a general setting, like for the standard Euler scheme. A stronger weak approximation result is obtained when considering the approximation of a Gibbs invariant distribution, when the nonlinearity is a gradient: one obtains an approximation in total variation distance of order 1/2, which does not hold for the standard method. This is the first result of this type in the literature and this is the major and most original result of this article.

Due to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval problem has seen a great deal of attention in recent years. A central problem in STFT phase retrieval concerns the question for which window functions (g in {L^2({mathbb R}^d)}) and which sampling sets (Lambda subseteq {mathbb R}^{2d}) is every (f in {L^2({mathbb R}^d)}) uniquely determined (up to a global phase factor) by phaseless samples of the form

where (V_gf) denotes the STFT of f with respect to g. The investigation of this question constitutes a key step towards making the problem computationally tractable. However, it deviates from ordinary sampling tasks in a fundamental and subtle manner: recent results demonstrate that uniqueness is unachievable if (Lambda ) is a lattice, i.e (Lambda = A{mathbb Z}^{2d}, A in textrm{GL}(2d,{mathbb R})). Driven by this discretization barrier, the present article centers around the initiation of a novel sampling scheme which allows for unique recovery of any square-integrable function via phaseless STFT-sampling. Specifically, we show that square-root lattices, i.e., sets of the form

guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians

This paper studies separating invariants: mappings on D-dimensional domains which are invariant to an appropriate group action and which separate orbits. The motivation for this study comes from the usefulness of separating invariants in proving universality of equivariant neural network architectures. We observe that in several cases the cardinality of separating invariants proposed in the machine learning literature is much larger than the dimension D. As a result, the theoretical universal constructions based on these separating invariants are unrealistically large. Our goal in this paper is to resolve this issue. We show that when a continuous family of semi-algebraic separating invariants is available, separation can be obtained by randomly selecting (2D+1 ) of these invariants. We apply this methodology to obtain an efficient scheme for computing separating invariants for several classical group actions which have been studied in the invariant learning literature. Examples include matrix multiplication actions on point clouds by permutations, rotations, and various other linear groups. Often the requirement of invariant separation is relaxed and only generic separation is required. In this case, we show that only (D+1) invariants are required. More importantly, generic invariants are often significantly easier to compute, as we illustrate by discussing generic and full separation for weighted graphs. Finally we outline an approach for proving that separating invariants can be constructed also when the random parameters have finite precision.

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.