Under mild assumptions, we show that the exact convergence rate in total variation is also exact in weaker Wasserstein distances for the Metropolis–Hastings independence sampler. We develop a new upper and lower bound on the worst-case Wasserstein distance when initialized from points. For an arbitrary point initialization, we show that the convergence rate is the same and matches the convergence rate in total variation. We derive exact convergence expressions for more general Wasserstein distances when initialization is at a specific point. Using optimization, we construct a novel centered independent proposal to develop exact convergence rates in Bayesian quantile regression and many generalized linear model settings. We show that the exact convergence rate can be upper bounded in Bayesian binary response regression (e.g. logistic and probit) when the sample size and dimension grow together.

Motivated by the numerical study of spin-boson dynamics in quantum open systems, we present a convergence analysis of the closure approximation for a class of stochastic differential equations. We show that the naive Monte Carlo simulation of the system by direct temporal discretization is not feasible through variance analysis and numerical experiments. We also show that the Wiener chaos expansion exhibits very slow convergence and high computational cost. Though efficient and accurate, the rationale of the moment closure approach remains mysterious. We rigorously prove that the low moments in the moment closure approximation of the considered model are of exponential convergence to the exact result. It is further extended to more general nonlinear problems and applied to the original spin-boson model with similar structure.

In this paper, a new combined method is presented to simulate saltwater intrusion problem. A splitting positive definite mixed element method is used to solve the water head equation, and a symmetric discontinuous Galerkin (DG) finite element method is used to solve the concentration equation. The introduction of these two numerical methods not only makes the coefficient matrixes symmetric positive definite, but also does well with the discontinuous problem. The convergence of this method is considered and the optimal L2-norm error estimate is also derived.

By introducing a new Gaussian process and a new compensated Poisson random measure, we propose an explicit prediction-correction scheme for solving decoupled forward backward stochastic differential equations with jumps (FBSDEJs). For this scheme, we first theoretically obtain a general error estimate result, which implies that the scheme is stable. Then using this result, we rigorously prove that the accuracy of the explicit scheme can be of second order. Finally, we carry out some numerical experiments to verify our theoretical results.

A novel generalised successive overrelaxation (GSOR) method for solving generalised saddle point problems is proposed, based on splitting the coefficient matrix. The proposed method is shown to converge under suitable restrictions on the iteration parameters, and we present some illustrative numerical results.

Convergence analysis is presented for recently proposed multistep schemes, when applied to a special type of forward-backward stochastic differential equations (FB-SDEs) that arises in finance and stochastic control. The corresponding k-step scheme admits a k-order convergence rate in time, when the exact solution of the forward stochastic differential equation (SDE) is given. Our analysis assumes that the terminal conditions and the FBSDE coefficients are sufficiently regular.

The paper aims to develop an effective preconditioner and conduct the convergence analysis of the corresponding preconditioned GMRES for the solution of discrete problems originating from multi-group radiation diffusion equations. We firstly investigate the performances of the most widely used preconditioners (ILU(k) and AMG) and their combinations (Bco and Bco), and provide drawbacks on their feasibilities. Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzing the features suitable for AMG using more detailed measurements on multiscale nature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we present an adaptive combined preconditioner Bcoα involving an improved ILU(0) along with its convergence constraints. Numerical results demonstrate that Bcoα-GMRES holds the best robustness and efficiency. At last, we analyze the convergence of GMRES with combined preconditioning which not only provides a persuasive support for our proposed algorithms, but also updates the existing estimation theory on condition numbers of combined preconditioned systems.

In this paper, we consider a neural network model for solving the generalized nonlinear complementarity problem (denoted by GNCP) over a polyhedral cone. The neural network is derived from an equivalent unconstrained minimization reformulation of the GNCP, which is based on the penalized Fischer–Burmeister function ${\it\phi}_{{\it\mu}}(a,b)={\it\mu}{\it\phi}_{\mathit{FB}}(a,b)+(1-{\it\mu})a_{+}b_{+}$. We establish the existence and the convergence of the trajectory of the neural network, and study its Lyapunov stability, asymptotic stability and exponential stability. It is found that a larger ${\it\mu}$ leads to a better convergence rate of the trajectory. Simulation results are also reported.

About thirty years ago, Achi Brandt wrote a seminal paper providing a convergence theory for algebraic multigrid methods [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, this theory has been improved and extended in a number of ways, and these results have been used in many works to analyze algebraic multigrid methods and guide their developments. This paper makes a concise exposition of the state of the art. Results for symmetric and nonsymmetric matrices are presented in a unified way, highlighting the influence of the smoothing scheme on the convergence estimates. Attention is also paid to sharp eigenvalue bounds for the case where one uses a single smoothing step, allowing straightforward application to deflation-based preconditioners and two-level domain decomposition methods. Some new results are introduced whenever needed to complete the picture, and the material is self-contained thanks to a collection of new proofs, often shorter than the original ones.

This paper studies convergence analysis of an adaptive finite element algorithm for numerical estimation of some unknown distributed flux in a stationary heat conduction system, namely recovering the unknown Neumann data on interior inaccessible boundary using Dirichlet measurement data on outer accessible boundary. Besides global upper and lower bounds established in [23], a posteriori local upper bounds and quasi-orthogonality results concerning the discretization errors of the state and adjoint variables are derived. Convergence and quasi-optimality of the proposed adaptive algorithm are rigorously proved. Numerical results are presented to illustrate the quasi-optimality of the proposed adaptive method.

In this paper, we consider a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine the initial data from a noisy final data. Based on a series expression of the solution, a conditional stability for the initial data is given. Further, we propose a modified quasi-boundary value regularization method to deal with the backward problem and obtain two kinds of convergence rates by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed methods.

In this paper, an extremal eigenvalue problem to the Sturm-Liouville equations with discontinuous coefficients and volume constraint is investigated. Liouville transformation is applied to change the problem into an equivalent minimization problem. Finite element method is proposed and the convergence for the finite element solution is established. A monotonic decreasing algorithm is presented to solve the extremal eigenvalue problem. A global convergence for the algorithm in the continuous case is proved. A few numerical results are given to depict the efficiency of the method.

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval. Using an extension of the analysis of Douglas and Dupont [23] for Dirichlet boundary conditions, we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space. We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROW for solving almost block diagonal linear systems. We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.

Existence of a solution to the quasi-variational inequality problem arising in a modelfor sand surface evolution has been an open problem for a long time. Another long-standingopen problem concerns determining the dual variable, the flux of sand pouring down theevolving sand surface, which is also of practical interest in a variety of applications ofthis model. Previously, these problems were solved for the special case in which theinequality is simply variational. Here, we introduce a regularized mixed formulationinvolving both the primal (sand surface) and dual (sand flux) variables. We derive,analyse and compare two methods for the approximation, and numerical solution, of thismixed problem. We prove subsequence convergence of both approximations, as the meshdiscretization parameters tend to zero; and hence prove existence of a solution to thismixed model and the associated regularized quasi-variational inequality problem. One ofthese numerical approximations, in which the flux is approximated by thedivergence-conforming lowest order Raviart–Thomas element, leads to an efficient algorithmto compute not only the evolving pile surface, but also the flux of pouring sand. Resultsof our numerical experiments confirm the validity of the regularization employed.

We consider the solution of second order elliptic PDEs in Rdwith inhomogeneous Dirichlet data by means of an h–adaptive FEM withfixed polynomial order p ∈ N. As model example serves the Poissonequation with mixed Dirichlet–Neumann boundary conditions, where the inhomogeneousDirichlet data are discretized by use of an H1 / 2–stableprojection, for instance, the L2–projection forp = 1 or the Scott–Zhang projection for general p ≥ 1.For error estimation, we use a residual error estimator which includes the Dirichlet dataoscillations. We prove that each H1 / 2–stable projectionyields convergence of the adaptive algorithm even with quasi–optimal convergence rate.Numerical experiments with the Scott–Zhang projection conclude the work.

This paper is concerned with obtaining an approximate solution and an approximate derivative of the solution for neutral Volterra integro-differential equation with a weakly singular kernel. The solution of this equation, even for analytic data, is not smooth on the entire interval of integration. The Jacobi collocation discretization is proposed for the given equation. A rigorous analysis of error bound is also provided which theoretically justifies that both the error of approximate solution and the error of approximate derivative of the solution decay exponentially in L∞ norm and weighted L2 norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.

The block-by-block method, proposed by Linz for a kind of Volterra integral equations with nonsingular kernels, and extended by Kumar and Agrawal to a class of initial value problems of fractional differential equations (FDEs) with Caputo derivatives, is an efficient and stable scheme. We analytically prove and numerically verify that this method is convergent with order at least 3 for any fractional order index α > 0.

We construct a Galerkin finite element method for the numerical approximation of weaksolutions to a general class of coupled FENE-type finitely extensible nonlinear elasticdumbbell models that arise from the kinetic theory of dilute solutions of polymericliquids with noninteracting polymer chains. The class of models involves the unsteadyincompressible Navier–Stokes equations in a bounded domainΩ ⊂ ℝd, d = 2 or 3, forthe velocity and the pressure of the fluid, with an elastic extra-stress tensor appearingon the right-hand side in the momentum equation. The extra-stress tensor stems from therandom movement of the polymer chains and is defined through the associated probabilitydensity function that satisfies a Fokker–Planck type parabolic equation, a crucial featureof which is the presence of a centre-of-mass diffusion term. We require no structuralassumptions on the drag term in the Fokker–Planck equation; in particular, the drag termneed not be corotational. We perform a rigorous passage to the limit as first the spatialdiscretization parameter, and then the temporal discretization parameter tend to zero, andshow that a (sub)sequence of these finite element approximations converges to a weaksolution of this coupled Navier–Stokes–Fokker–Planck system. The passage to the limit isperformed under minimal regularity assumptions on the data: a square-integrable anddivergence-free initial velocity datum \hbox{$\absundertilde$}${\mathit{u}}_{\mathrm{0}}{\mathrm{~}}_{}$ for the Navier–Stokes equation and a nonnegative initial probabilitydensity function ψ0 for the Fokker–Planck equation, which hasfinite relative entropy with respect to the Maxwellian M.

In this paper, we construct and analyze finite element methods for the three dimensionalMonge-Ampère equation. We derive methods using the Lagrange finite element space such thatthe resulting discrete linearizations are symmetric and stable. With this in hand, we thenprove the well-posedness of the method, as well as derive quasi-optimal error estimates.We also present some numerical experiments that back up the theoretical findings.