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Due to the divergence-instability, low-order conforming finite element methods (FEMs) for nearly incompressible elasticity equations suffer from the so-called locking phenomenon as the Lam\'e parameter $\lambda\to\infty$ and consequently the material becomes more and more incompressible. For the piecewise linear case, the error in the $L^2$-norm of the standard Galerkin conforming FEM is bounded by $C_\lambda h^2$. However, $C_\lambda \to \infty$ as $\lambda \to \infty$, resulting in poor accuracy for practical values of $h$ if $\lambda$ is sufficiently large. In this short paper, we show that for 2D problems the locking phenomenon can be controlled by replacing $\lambda$ with $\lambda^\alpha$ in the stiffness matrix, for a certain choice of $\alpha=\alpha_*(h,\lambda)$ in the range $0<\alpha\le 1$. We prove that for this optimal choice of $\alpha$, the error in the $L^2$-norm is bounded by $Ch$ where $C$ does not depend on $\lambda$. Numerical experiments confirm the expected convergence behaviour and show that, for practical meshes, our locking-free method is more accurate than the standard method if the material is nearly incompressible. Our analysis also shows that the error in the $H^1$-norm is bounded by $Ch^{1/2}$, but our numerical experiments suggest that this bound is not sharp.

We consider a non-linear Bayesian data assimilation model for the periodic two-dimensional Navier-Stokes equations with initial condition modelled by a Gaussian process prior. We show that if the system is updated with sufficiently many discrete noisy measurements of the velocity field, then the posterior distribution eventually concentrates near the ground truth solution of the time evolution equation, and in particular that the initial condition is recovered consistently by the posterior mean vector field. We further show that the convergence rate can in general not be faster than inverse logarithmic in sample size, but describe specific conditions on the initial conditions when faster rates are possible. In the proofs we provide an explicit quantitative estimate for backward uniqueness of solutions of the two-dimensional Navier-Stokes equations.

We discuss a connection between a generative model, called the diffusion model, and nonequilibrium thermodynamics for the Fokker-Planck equation, called stochastic thermodynamics. Based on the techniques of stochastic thermodynamics, we derive the speed-accuracy trade-off for the diffusion models, which is a trade-off relationship between the speed and accuracy of data generation in diffusion models. Our result implies that the entropy production rate in the forward process affects the errors in data generation. From a stochastic thermodynamic perspective, our results provide quantitative insight into how best to generate data in diffusion models. The optimal learning protocol is introduced by the conservative force in stochastic thermodynamics and the geodesic of space by the 2-Wasserstein distance in optimal transport theory. We numerically illustrate the validity of the speed-accuracy trade-off for the diffusion models with different noise schedules such as the cosine schedule, the conditional optimal transport, and the optimal transport.

In this paper, we study a nonlinear cointegration-type model of the form \(Z_t = f_0(X_t) + W_t\) where \(f_0\) is a monotone function and \(X_t\) is a Harris recurrent Markov chain. We use a nonparametric Least Square Estimator to locally estimate \(f_0\), and under mild conditions, we show its strong consistency and obtain its rate of convergence. New results (of the Glivenko-Cantelli type) for localized null recurrent Markov chains are also proved.

This paper shows how to use the shooting method, a classical numerical algorithm for solving boundary value problems, to compute the Riemannian distance on the Stiefel manifold $ \mathrm{St}(n,p) $, the set of $ n \times p $ matrices with orthonormal columns. The proposed method is a shooting method in the sense of the classical shooting methods for solving boundary value problems; see, e.g., Stoer and Bulirsch, 1991. The main feature is that we provide an approximate formula for the Fr\'{e}chet derivative of the geodesic involved in our shooting method. Numerical experiments demonstrate the algorithms' accuracy and performance. Comparisons with existing state-of-the-art algorithms for solving the same problem show that our method is competitive and even beats several algorithms in many cases.

The aim of this paper is to develop a refined error estimate of L1/finite element scheme for a reaction-subdiffusion equation with constant delay $\tau$ and uniform time mesh. Under the non-uniform multi-singularity assumption of exact solution in time, the local truncation errors of the L1 scheme with uniform mesh is investigated. Then we introduce a fully discrete finite element scheme of the considered problem. Next, a novel discrete fractional Gr\"onwall inequality with constant delay term is proposed, which does not include the increasing Mittag-Leffler function comparing with some popular other cases. By applying this Gr\"onwall inequality, we obtain the pointwise-in-time and piecewise-in-time error estimates of the finite element scheme without the Mittag-Leffler function. In particular, the latter shows that, for the considered interval $((i-1)\tau,i\tau]$, although the convergence in time is low for $i=1$, it will be improved as the increasing $i$, which is consistent with the factual assumption that the smoothness of the solution will be improved as the increasing $i$. Finally, we present some numerical tests to verify the developed theory.

We discuss a connection between a generative model, called the diffusion model, and nonequilibrium thermodynamics for the Fokker-Planck equation, called stochastic thermodynamics. Based on the techniques of stochastic thermodynamics, we derive the speed-accuracy trade-off for the diffusion models, which is a trade-off relationship between the speed and accuracy of data generation in diffusion models. Our result implies that the entropy production rate in the forward process affects the errors in data generation. From a stochastic thermodynamic perspective, our results provide quantitative insight into how best to generate data in diffusion models. The optimal learning protocol is introduced by the conservative force in stochastic thermodynamics and the geodesic of space by the 2-Wasserstein distance in optimal transport theory. We numerically illustrate the validity of the speed-accuracy trade-off for the diffusion models with different noise schedules such as the cosine schedule, the conditional optimal transport, and the optimal transport.

We consider the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that inter-variable coupling for characteristic variables is weak relative to intra-variable coupling, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small. For an $\ell$-dimensional system of PDEs, applying the preconditioner consists of solving a sequence of $\ell$ scalar linear(ized)-advection-like problems, each being associated with a different characteristic wave-speed in the underlying linear(ized) PDE. We approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media, and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions.

A numerical experiment based on a particle number-conserving quantum field theory is performed for two initially independent Bose-Einstein condensates that are coherently coupled at two temperatures. The present model illustrates ab initio that the initial relative phase of each of the two condensates doesn't remain random, but is distributed around integer multiple values of $2\pi$ from the interference and thermalization of forward and backward propagating matter waves at the Boltzmann equilibrium, that intrinsically measures zero average phases for each of the two independent condensates. Following this approach, focus is put on the original Gedanken experiment of Anderson on whether a Josephson current between two initially separated Bose-Einstein condensates occurs in a deterministic way or not, depending on the initial phase distribution.