Foundations of Computational Mathematics最新文献

We consider the nonlinear Schrödinger equation with a potential, also known as Gross-Pitaevskii equation. By introducing a suitable spectral localization, we prove low regularity error estimates for the time discretization corresponding to an adapted Lie-Trotter splitting scheme. The proof is based on tools from spectral theory and pseudodifferential calculus in order to obtain various estimates on the spectral localization, including discrete Strichartz estimates which support the nonlinear analysis.

We study the global convergence of a Fisher–Rao policy gradient flow for infinite-horizon entropy-regularised Markov decision processes with Polish state and action spaces. The flow is a continuous-time analogue of a policy mirror descent method. We establish the global well-posedness of the gradient flow and demonstrate its exponential convergence to the optimal policy. Moreover, we prove the flow is stable with respect to gradient evaluation, offering insights into the performance of a natural policy gradient flow with log-linear policy parameterisation. To overcome challenges stemming from the lack of the convexity of the objective function and the discontinuity arising from the entropy regulariser, we leverage the performance difference lemma and the duality relationship between the gradient and mirror descent flows. Our analysis provides a theoretical foundation for developing various discrete policy gradient algorithms.

For a given graph whose edges are labeled with general real numbers, we consider the set of functions from the vertex set into the Euclidean plane such that the distance between the images of neighbouring vertices is equal to the corresponding edge label. This set of functions can be expressed as the zero set of quadratic polynomials and our main result characterizes the number of complex irreducible components of this zero set in terms of combinatorial properties of the graph. In case the complex components are three-dimensional, then the graph is minimally rigid and the component number is a well-known invariant from rigidity theory. If the components are four-dimensional, then they correspond to one-dimensional coupler curves of flexible planar mechanisms. As an application, we characterize the degree of irreducible components of such coupler curves combinatorially.

We consider the application of finite element exterior calculus (FEEC) methods to a class of canonical Hamiltonian PDE systems involving differential forms. Solutions to these systems satisfy a local multisymplectic conservation law, which generalizes the more familiar symplectic conservation law for Hamiltonian systems of ODEs, and which is connected with physically-important reciprocity phenomena, such as Lorentz reciprocity in electromagnetics. We characterize hybrid FEEC methods whose numerical traces satisfy a version of the multisymplectic conservation law, and we apply this characterization to several specific classes of FEEC methods, including conforming Arnold–Falk–Winther-type methods and various hybridizable discontinuous Galerkin (HDG) methods. Interestingly, the HDG-type and other nonconforming methods are shown, in general, to be multisymplectic in a stronger sense than the conforming FEEC methods. This substantially generalizes previous work of McLachlan and Stern [Found. Comput. Math., 20 (2020), pp. 35–69] on the more restricted class of canonical Hamiltonian PDEs in the de Donder–Weyl “grad-div” form.

The Wasserstein space of probability measures is known for its intricate Riemannian structure, which underpins the Wasserstein geometry and enables gradient flow algorithms. However, the Wasserstein geometry may not be suitable for certain tasks or data modalities. Motivated by scenarios where the global structure of the data needs to be preserved, this work initiates the study of gradient flows and Riemannian structure in the Gromov-Wasserstein (GW) geometry, which is particularly suited for such purposes. We focus on the inner product GW (IGW) distance between distributions on mathbb {R}^d$mathbb {R}^d$, which preserves the angles within the data and serves as a convenient initial setting due to its analytic tractability. Given a functional textsf{F}:mathcal {P}_2(mathbb {R}^d)rightarrow mathbb {R}$textsf{F}:mathcal {P}_2(mathbb {R}^d)rightarrow mathbb {R}$ to optimize and an initial distribution rho _0in mathcal {P}_2(mathbb {R}^d)$rho _0in mathcal {P}_2(mathbb {R}^d)$, we present an implicit IGW minimizing movement scheme that generates a sequence of distributions {rho _i}_{i=0}^n${rho _i}_{i=0}^n$, which are close in IGW and aligned in the 2-Wasserstein sense. Taking the time step to zero, we prove that the (piecewise constant interpolation of the) discrete solution converges to an IGW generalized minimizing movement (GMM) (rho _t)_t$(rho _t)_t$ that follows the continuity equation with a velocity field v_tin L^2(rho _t;mathbb {R}^d)$v_tin L^2(rho _t;mathbb {R}^d)$, specified by a global transformation of the Wasserstein gradient of textsf{F}$textsf{F}$ (viz., the gradient of its first variation). The transformation is given by a mobility operator that modifies the Wasserstein gradient to encode not only local information, but also global structure, as expected for the IGW gradient flow. Our gradient flow analysis leads us to identify the Riemannian structure that gives rise to the intrinsic IGW geometry, using which we establish a Benamou-Brenier-like formula for IGW. We conclude with a formal derivation, akin to the Otto calculus, of the IGW gradient as the inverse mobility acting on the Wasserstein gradient. Numerical experiments demonstrating the global nature of IGW interpolations are provided to complement the theory.

First order methods endowed with global convergence guarantees operate using global lower bounds on the objective. The tightening of the bounds leads to an increase in theoretical guarantees and in observed practical performance. In this work, we define a global lower bound for smooth objectives that is optimal with respect to the collected oracle information. Our bound can be readily employed by the Gradient Method with Memory to improve its performance. Further using the machinery underlying the optimal bounds, we introduce a modified version of the estimate sequence that we use to construct an Optimized Gradient Method with Memory possessing the best known convergence guarantees for its class of algorithms up to the proportionality constant. We additionally equip the method with an adaptive convergence guarantee adjustment procedure that is an effective replacement for line-search. Simulation results on synthetic but otherwise difficult smooth problems validate the theoretical properties of the bound and of the proposed methods.

In this work we construct many sequences $S=S_{b,d}^{◻}$$S=S^Box _{b,d}$ and $S=S_{b,d}^{⊞}$