Convergence Rate Of Order Research Articles

Multi-stencils fast marching method for factored eikonal equations with quadratic anisotropy

In this study, we propose a multi-stencils fast marching method to solve the factored eikonal equations for media with quadratic anisotropy, also known as Riemannian anisotropy. Compared with the conventional fast marching method, the new method has an improved accuracy in tracing moving wavefronts via computing the solution at each grid point along several stencils. To deal with the source singularity, we solve for the viscosity solution iteratively by factorize the unknown traveltime function into a function that captures the singularity and a multiplicative factor which is differentiable at the source location. By leveraging the strengths of multi-stencils and factorisation, the newly developed algorithm is highly accurate in solving anisotropic eikonal equations. Through a series of numerical examples, we explicitly show that the algorithm (1) has an obvious first (second) order convergence rate with the first (second) order finite-difference scheme, and (2) is unconditionally stable as long as the initial time table around the source is specified. Finally, we design an eikonal-based anisotropic ray tracing technique. We test the anisotropic ray tracing technique on both radial and azimuthal anisotropy cases and the numerical results demonstrate that the usage of multi-stencils scheme increase the accuracy of the anisotropic ray paths especially for azimuthal anisotropy case. This high-accurate eikonal solver and the associated ray tracing technique can be used in high-resolution seismic anisotropic tomography and other researches in the future.

Convergence analysis for continuous-time Markov chain approximation of stochastic local volatility models: Option pricing and Greeks

This paper establishes the precise second order convergence rates of the continuous-time Markov chain (CTMC) approximation method for pricing options under the general framework of stochastic local volatility (SLV) models. The stochastic local volatility models studied in this paper include Heston model, 4/2 model, α-Hypergeometric model, stochastic alpha beta rho (SABR) model, Heston-SABR model and quadratic SLV model. Using the stochastic flow theorem, the closed-form CTMC approximation formula for the Greeks are obtained and the second order convergence rates are proved. Numerical examples confirm the theoretical findings.

Variational Bayes on manifolds

Variational Bayes (VB) has become a widely-used tool for Bayesian inference in statistics and machine learning. Nonetheless, the development of the existing VB algorithms is so far generally restricted to the case where the variational parameter space is Euclidean, which hinders the potential broad application of VB methods. This paper extends the scope of VB to the case where the variational parameter space is a Riemannian manifold. We develop an efficient manifold-based VB algorithm that exploits both the geometric structure of the constraint parameter space and the information geometry of the manifold of VB approximating probability distributions. Our algorithm is provably convergent and achieves a convergence rate of order $${\mathcal {O}}(1/\sqrt{T})$$ and $$\mathcal O(1/T^{2-2\epsilon })$$ for a non-convex evidence lower bound function and a strongly retraction-convex evidence lower bound function, respectively. We develop in particular two manifold VB algorithms, Manifold Gaussian VB and Manifold Wishart VB, and demonstrate through numerical experiments that the proposed algorithms are stable, less sensitive to initialization and compares favourably to existing VB methods.

Novel approach to solve singularly perturbed boundary value problems with negative shift parameter

Singularly perturbed boundary value problems with negative shift parameter are special types of differential difference equations whose solution exhibits boundary layer behaviour. A simple but novel numerical method is developed to approximate the numerical solution of the problems of these types. The method gives accurate solutions for h≥ε in the inner region of the boundary layer where other classical numerical methods fail to give smooth solution. The present method is proved to be point-wise uniformly convergent with second order rate of convergence.

The strong convergence and stability of explicit approximations for nonlinear stochastic delay differential equations

This paper focuses on explicit approximations for nonlinear stochastic delay differential equations (SDDEs). Under less restrictive conditions, the truncated Euler-Maruyama (TEM) schemes for SDDEs are proposed, which numerical solutions are bounded in the q th moment for q ≥ 2 and converge to the exact solutions strongly in any finite interval. The 1/2 order convergence rate is yielded. Furthermore, the long-time asymptotic behaviors of numerical solutions, such as stability in mean square and $\mathbb {P}-1$ , are examined. Several numerical experiments are carried out to illustrate our results.

Local and parallel stabilized finite element methods based on full domain decomposition for the stationary Stokes equations

Abstract Based upon full domain decomposition, local and parallel stabilized finite element methods for the stationary Stokes equations are proposed and analysed, where the quadratic equal-order finite elements are employed for the velocity and pressure approximations, and a stabilized term based on two local Gauss integrations is used to offset the discrete pressure space to circumvent the discrete inf-sup condition. In the proposed parallel method, all of the computations are performed on the locally refined global grids that are fine around the interested subdomain and coarse elsewhere, making the method easy to implement based on a sequential solver with low communication cost. Stability and optimal error estimates of the present methods are deduced. Numerical results on examples including a problem with known analytic solution, lid-driven cavity flow, backward-facing step flow and flow around a cylinder are given to verify the theoretical predictions and demonstrate the high efficiency of the method. Results show that our parallel method can provide an approximate solution with the convergence rate of the same order as the solution computed by the standard stabilized finite element method, with a substantial reduction in computational time.

Convergence rate estimation of poly-Sinc-based discontinuous Galerkin methods

The error of approximation of a function using Poly-Sinc approximation was shown to have a convergence rate of exponential order over a global partition. However, it was assumed that the function values at the interpolation points were known. In this paper, we extend our work on poly-Sinc-based discontinuous Galerkin approximation, in which the function values at the interpolation points are replaced by unknown coefficients and solved by the discontinuous Galerkin method. To deal with functions having a singularity at an endpoint, we use Sinc partitions in our discontinuous Galerkin approximation. For such functions, we show that, by using a weighted L2 norm over a Sinc partition in which the weight function is the reciprocal of the asymptotic density of Sinc points in that partition and computing the ℓ2 norm over all Sinc partitions constituting the global partition except the partition containing the singularity, the error of approximation between the exact solution of the ordinary differential equation and its poly-Sinc-based discontinuous Galerkin approximation has a convergence rate of exponential order similar to the one over the global partition. The numerical results are in agreement with our theoretical derivations of the approximation error. We compare our poly-Sinc-based discontinuous Galerkin method with the poly-Sinc collocation method. Our method shows better performance in the L2 norm.

An indirect collocation method for variable-order fractional wave equations on uniform or graded meshes and its optimal error estimates

We develop an indirect collocation method for a variable-order fractional wave equation. Fractional differential equations are well known to exhibit initial weak singularity, which makes it unrealistic to carry out error estimates of their numerical approximations based on the smoothness assumptions of the true solutions. In this paper, we analyze the convergence behaviour of the method without artificially assuming the (often untrue) full regularity of the true solution of the problem, but only based on the behaviour of the coefficients and variable order of the problem. More precisely, we prove the following results: (i) If the variable order has an integer initial value, the method discretized on a uniform partition has an optimal-order convergence rate in the norm. (ii) Otherwise, the method discretized on a uniform mesh has only a sub-optimal order convergence rate. The method discretized on a graded mesh with the mesh grading parameter determined by the initial value of the variable order has an optimal-order convergence rate in the norm. Numerical experiments are performed to substantiate the theoretical results.

First order strong convergence of an explicit scheme for the stochastic SIS epidemic model

A novel explicit time-stepping scheme, called Lamperti smoothing truncation scheme, is devised in this paper to strongly approximate a stochastic SIS epidemic model, whose solution process takes values in a bounded domain and whose coefficients violate the global monotonicity condition. The proposed scheme is based on combining a Lamperti-type transformation with an explicit truncation method. The new scheme results in numerical approximations preserving the domain of the original SDEs and is proved to retain a mean-square convergence rate of order one. Numerical examples are finally reported to confirm our theoretical findings.

Finite stateN-agent and mean field control problems

We examine mean field control problems on a finite state space, in continuous time and over a finite time horizon. We characterize the value function of the mean field control problem as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the simplex. In absence of any convexity assumption, we exploit this characterization to prove convergence, asNgrows, of the value functions of the centralizedN-agent optimal control problem to the limit mean field control problem value function, with a convergence rate of order [see formula in PDF]. Then, assuming convexity, we show that the limit value function is smooth and establish propagation of chaos,i.e.convergence of theN-agent optimal trajectories to the unique limiting optimal trajectory, with an explicit rate.

Fractional order convergence rate estimate of finite-difference method for the heat equation with concentrated capacity

The convergence of difference scheme for initial-boundary value problem for the heat equation with concentrated capacity and time-dependent coefficient of the space derivatives, is considered. Fractional order convergence rate estimate in a special discrete Sobolev norms, compatible with the smoothness of the coefficient and solution, is proved.

DEVELOPMENT AND TESTING OF THE MODIFIED SIMPSON’S RULE FOR DISCRETE ORDINATES TRANSPORT APPLICATIONS

A new angular quadrature type termed Modified Simpson Trapezoidal (MST) is developed based on the conventional Simpson’s 1/3 rule where the angular pattern over polar levels has a trapezoid shape. An adaptive coefficient correction scheme is developed to enable our new quadrature to integrate the angular flux over subintervals separated by the interior jump irregularities. A two-dimensional test problem is employed to verify the angular discretization error in the uncollided SN scalar flux computed with our new quadrature sets, as well as conventional angular quadrature types. Numerical results show that the MST quadrature error in the point-wise scalar flux converges with second order against increasing number of discrete angles, while the error obtained with other conventional quadrature types converges slower than first order depending on the regularity of the exact point-wise uncollided angular flux. In order to reduce the number of discrete points needed, a variant of the MST quadrature, namely MSTP30, is developed by using the Quadruple Range [1] polar quadrature with fixed 30 polar angles and applying the MST quadrature to the azimuthal dependence in each polar level. The angular discretization error in the point-wise SN scalar flux obtained with MSTP30 sets converges with fourth order because the polar discretization error is sufficiently reduced that MSTP30 behaves like a one-dimensional quadrature. Furthermore, because MSTP30 computes the integral over subintervals that keep the true solution’s irregularity at the boundaries, this fourth order convergence rate is unaffected by such inevitable irregularities.

Communication-efficient hierarchical distributed optimization for multi-agent policy evaluation

Policy evaluation problems in multi-agent reinforcement learning (MARL) have attracted growing interest recently. In this setting, agents collaborate to learn the value of a given policy with private local rewards and jointly observed state-action pairs. However, existing fully decentralized algorithms treat each agent equally, without considering the communication structure of the agents over a given network, and the corresponding effects on communication and computation efficiency. In this paper, we propose a hierarchical distributed algorithm that differentiates the roles of each of the agents during the evaluation process. This method allows us to freely choose various mixing schemes (and corresponding mixing matrices that are not necessarily symmetric or doubly stochastic), in order to reduce the communication and computation cost, while still maintaining convergence at rates as fast as or even faster than the previous distributed algorithms. Theoretically, we show the proposed method, which contains existing distributed methods as a special case, achieves the same order of convergence rate as state-of-the-art methods. Extensive numerical experiments on real datasets verify that the performance of our approach indeed improves – sometimes significantly – over other advanced algorithms in terms of convergence and total communication efficiency.

Tailored finite point method for the approximation of diffusion operators with non-symmetric diffusion tensor

We present a tailored finite point method (TFPM) for anisotropic diffusion equations with a non-symmetric diffusion tensor on Cartesian grids. The fluxes on each edge are discretized by using a linear combination of the local basis functions, which come from the exact solution of the diffusion equation with constant coefficients on the local cell. In this way, the scheme is fully consistent and the flux is naturally continuous across the interfaces between the subdomains with a non-symmetric diffusion tensor. Additionally, it is convenient to handle the Neumann boundary condition or a variant of Neumann boundary condition. Numerical results obtained from solving different anisotropic diffusion problems including problems with sharp discontinuity near the interface and boundary, show that this approach is efficient. Second order convergence rate can be obtained with the numerical examples. The new scheme is also tested on a time-dependent problem modeling the Hall effect in the resistive magnetohydrodynamics, the results further show the robustness of this new method.

Quasi-uniform convergence analysis of rectangular Morley element for the singularly perturbed Bi-wave equation

A non-C0 rectangular Morley element method is discussed to solve the singularly perturbed Bi-wave equation and the penalty terms are included to guarantee convergence. The quasi-uniform convergence rate of order O(h) is derived in the energy norm irrelevant to the negative powers of the real perturbation parameter δ appearing in the considered problem, which improves an existing result, here h is the mesh size. Finally, some numerical results are provided to confirm the theoretical analysis.

Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms both biased and unbiased simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory.

Convergence rates of Gaussian ODE filters

A recently introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution x and its first q derivatives a priori as a Gauss–Markov process {varvec{X}}, which is then iteratively conditioned on information about {dot{x}}. This article establishes worst-case local convergence rates of order q+1 for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order q in the case of q=1 and an integrated Brownian motion prior, and analyses how inaccurate information on {dot{x}} coming from approximate evaluations of f affects these rates. Moreover, we show that, in the globally convergent case, the posterior credible intervals are well calibrated in the sense that they globally contract at the same rate as the truncation error. We illustrate these theoretical results by numerical experiments which might indicate their generalizability to q in {2,3,ldots }.

Efficient Simulation of Generalized SABR and Stochastic Local Volatility Models based on Markov Chain Approximations

We propose a novel Monte Carlo simulation method for two-dimensional stochastic differential equation (SDE) systems based on approximation through continuous-time Markov chains (CTMCs). Specifically, we propose an efficient simulation framework for asset prices under general stochastic local volatility (SLV) models arising in finance, which includes the Heston and the stochastic alpha beta rho (SABR) models as special cases. Our simulation algorithm is constructed based on approximating the latent stochastic variance process by a CTMC. Compared with time-discretization schemes, our method exhibits several advantages, including flexible boundary condition treatment, weak continuity conditions imposed on coefficients, and a second order convergence rate in the spatial grids of the approximating CTMC under suitable regularity conditions. Replacing the stochastic variance process with a discrete-state approximation greatly simplifies the direct sampling of the integrated variance, thus enabling a highly efficient simulation scheme. Extensive numerical examples illustrate the accuracy and efficiency of our estimator, which outperforms both biased and unbiased simulation estimators in the literature in terms of root mean squared error (RMSE) and computational time. This paper is focused primarily on the simulation of SDEs which arise in finance, but this new simulation approach has potential for applications in other contextual areas in operations research, such as queuing theory.

High accuracy nonconforming biharmonic element over n‐rectangular meshes

AbstractThis work gives the high accuracy analysis of a rectangular biharmonic element in arbitrarily high‐dimensional cases. Given an n‐rectangle, we construct the nonconforming finite element and show its explicit standard basis representation. We prove that, if the n‐rectangular mesh is uniform, this element can achieve a second order convergence rate in energy norm when applied to biharmonic problems. Numerical examples for n = 3 are also presented.

Discontinuous Galerkin methods using poly-sinc approximation

In this paper, we develop a discontinuous Galerkin method using poly-Sinc approximation, in which non-equidistant points are generated by conformal mappings, such as Sinc points. Sinc points are characterized by their dense distribution near the endpoints of an interval, which render them effective in dealing with jumps introduced by the partitioning process of the interval in a discontinuous Galerkin scheme. Sinc methods are characterized by their fast convergence rate of exponential order and effective handling of singularities. We use equidistant, Chebyshev, and Sinc partitions to handle the singularities. We demonstrate that, using poly-Sinc-based discontinuous Galerkin approximation, Sinc partitions outperforms those with equidistant and Chebyshev partitions as we near the singularity. We also demonstrate that for Sinc partitions, poly-Sinc-based discontinuous Galerkin approximation outperforms those with Chebyshev points. We used the stability condition of the discontinuous Galerkin method to demonstrate the stability of our approach. It was shown that the approximation error between a smooth function and its poly-Sinc approximation over the global partition has a convergence rate of exponential order. For a function with a singularity at an endpoint, we use the weighted L2 norm over the Sinc partitions. Using the weighted L2 norm, we show that, by computing the ℓ2 norm over all Sinc partitions constituting the global partition except the last partition, the approximation error between the exact solution of the ordinary differential equation and its poly-Sinc-based discontinuous Galerkin approximation has a convergence rate of exponential order.