One essential problem in quantifying the collective behaviors of molecular systems lies in the accurate construction of free energy surfaces (FESs). The main challenges arise from the prevalence of energy barriers and the high dimensionality. Existing approaches are often based on sophisticated enhanced sampling methods to establish efficient exploration of the full-phase space. On the other hand, the collection of optimal sample points for the numerical approximation of FESs remains largely under-explored, where the discretization error could become dominant for systems with a large number of collective variables (CVs). We propose a consensus sampling-based approach by reformulating the construction as a minimax problem which simultaneously optimizes the function representation and the training set. In particular, the maximization step establishes a stochastic interacting particle system to achieve the adaptive sampling of the max-residue regime by modulating the exploitation of the Laplace approximation of the current loss function and the exploration of the uncharted phase space; the minimization step updates the FES approximation with the new training set. By iteratively solving the minimax problem, the present method essentially achieves an adversarial learning of the FESs with unified tasks for both phase space exploration and posterior error-enhanced sampling. We demonstrate the method by constructing the FESs of molecular systems with a number of CVs up to 30.
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We study Whitney-type estimates for approximation of convex functions in the uniform norm on various convex multivariate domains while paying a particular attention to the dependence of the involved constants on the dimension and the geometry of the domain.
We survey recent contributions to finite element exterior calculus on manifolds and surfaces within a comprehensive formalism for the error analysis of vector-valued partial differential equations on manifolds. Our primary focus is on uniformly bounded commuting projections on manifolds: these projections map from Sobolev de Rham complexes onto finite element de Rham complexes, commute with the differential operators, and satisfy uniform bounds in Lebesgue norms. They enable the Galerkin theory of Hilbert complexes for a large range of intrinsic finite element methods on manifolds. However, these intrinsic finite element methods are generally not computable and thus primarily of theoretical interest. This leads to our second point: estimating the geometric variational crime incurred by transitioning to computable approximate problems. Lastly, our third point addresses how to estimate the approximation error of the intrinsic finite element method in terms of the mesh size. If the solution is not continuous, then such an estimate is achieved via modified Cl\'ement or Scott-Zhang interpolants that facilitate a broken Bramble--Hilbert lemma.
Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree $n$ converge at the root-exponential rate $O(\exp(-2\rho\sqrt{n}))$ with $\rho>0$ when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at $z=-\rho^2$. As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical results.
Multiscale stochastic dynamical systems have been widely adopted to a variety of scientific and engineering problems due to their capability of depicting complex phenomena in many real world applications. This work is devoted to investigating the effective dynamics for slow-fast stochastic dynamical systems. Given observation data on a short-term period satisfying some unknown slow-fast stochastic systems, we propose a novel algorithm including a neural network called Auto-SDE to learn invariant slow manifold. Our approach captures the evolutionary nature of a series of time-dependent autoencoder neural networks with the loss constructed from a discretized stochastic differential equation. Our algorithm is also validated to be accurate, stable and effective through numerical experiments under various evaluation metrics.
In this paper we consider a nonlinear poroelasticity model that describes the quasi-static mechanical behaviour of a fluid-saturated porous medium whose permeability depends on the divergence of the displacement. Such nonlinear models are typically used to study biological structures like tissues, organs, cartilage and bones, which are known for a nonlinear dependence of their permeability/hydraulic conductivity on solid dilation. We formulate (extend to the present situation) one of the most popular splitting schemes, namely the fixed-stress split method for the iterative solution of the coupled problem. The method is proven to converge linearly for sufficiently small time steps under standard assumptions. The error contraction factor then is strictly less than one, independent of the Lam\'{e} parameters, Biot and storage coefficients if the hydraulic conductivity is a strictly positive, bounded and Lipschitz-continuous function.
This study proposes a unified optimization-based planning framework that addresses the precise and efficient navigation of a controlled object within a constrained region, while contending with obstacles. We focus on handling two collision avoidance problems, i.e., the object not colliding with obstacles and not colliding with boundaries of the constrained region. The object or obstacle is denoted as a union of convex polytopes and ellipsoids, and the constrained region is denoted as an intersection of such convex sets. Using these representations, collision avoidance can be approached by formulating explicit constraints that separate two convex sets, or ensure that a convex set is contained in another convex set, referred to as separating constraints and containing constraints, respectively. We propose to use the hyperplane separation theorem to formulate differentiable separating constraints, and utilize the S-procedure and geometrical methods to formulate smooth containing constraints. We state that compared to the state of the art, the proposed formulations allow a considerable reduction in nonlinear program size and geometry-based initialization in auxiliary variables used to formulate collision avoidance constraints. Finally, the efficacy of the proposed unified planning framework is evaluated in two contexts, autonomous parking in tractor-trailer vehicles and overtaking on curved lanes. The results in both cases exhibit an improved computational performance compared to existing methods.
The air-gap macro element is reformulated such that rotation, rotor or stator skewing and rotor eccentricity can be incorporated easily. The air-gap element is evaluated using Fast Fourier Transforms which in combination with the Conjugate Gradient algorithm leads to highly efficient and memory inexpensive iterative solution scheme. The improved air-gap element features beneficial approximation properties and is competitive to moving-band and sliding-surface technique.
Measures of association between cortical regions based on activity signals provide useful information for studying brain functional connectivity. Difficulties occur with signals of electric neuronal activity, where an observed signal is a mixture, i.e. an instantaneous weighted average of the true, unobserved signals from all regions, due to volume conduction and low spatial resolution. This is why measures of lagged association are of interest, since at least theoretically, "lagged association" is of physiological origin. In contrast, the actual physiological instantaneous zero-lag association is masked and confounded by the mixing artifact. A minimum requirement for a measure of lagged association is that it must not tend to zero with an increase of strength of true instantaneous physiological association. Such biased measures cannot tell apart if a change in its value is due to a change in lagged or a change in instantaneous association. An explicit testable definition for frequency domain lagged connectivity between two multivariate time series is proposed. It is endowed with two important properties: it is invariant to non-singular linear transformations of each vector time series separately, and it is invariant to instantaneous association. As a first sanity check: in the case of two univariate time series, the new definition leads back to the bivariate lagged coherence of 2007 (eqs 25 and 26 in //doi.org/10.48550/arXiv.0706.1776). As a second stronger sanity check: in the case of a univariate and multivariate vector time series, the new measure presented here leads back to the original multivariate lagged coherence in equation 31 of the same 2007 publication (which trivially includes the bivariate case).
We prove the convergence of meshfree method for solving the elliptic Monge-Ampere equation with Dirichlet boundary on the bounded domain. L2 error is obtained based on the kernel-based trial spaces generated by the compactly supported radial basis functions. We obtain the convergence result when the testing discretization is finer than the trial discretization. The convergence rate depend on the regularity of the solution, the smoothness of the computing domain, and the approximation of scaled kernel-based spaces. The presented convergence theory covers a wide range of kernel-based trial spaces including stationary approximation and non-stationary approximation. An extension to non-Dirichlet boundary condition is in a forthcoming paper.
Recently a nonconforming surface finite element was developed to discretize 2D vector-valued compressible flow problems in a 3D domain. In this contribution we derive an error analysis for this approach on a vector-valued Laplace problem, which is an important operator for fluid-equations on the surface. In our setup, the problem is approximated via edge-integration on local flat triangles using the nonconforming linear Crouzeix-Raviart element. The flat planes coincide with the surface at the edge midpoints. This is also the place, where the Crouzeix-Raviart element requires continuity between two neighbouring elements. The developed Crouzeix-Raviart approximation is a non-parametric approach that works on local coordinate systems, established in each triangle. This setup is numerically efficient and straightforward to implement. For this Crouzeix-Raviart discretization we derive optimal error bounds in the $H^1$-norm and $L^2$-norm and present an estimate for the geometric error. Numerical experiments validate the theoretical results.
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