Higher-order Approximations Research Articles

Study on reliability of emergency braking performance of high-speed and heavy-load monorail crane

The reliability of monorail crane braking system has an important influence on braking safety, high-speed and heavy-load operation, which poses great challenges to braking safety, and it is necessary to evaluate its braking reliability accurately and efficiently. Firstly, the dynamic performance and thermal-mechanical coupling characteristics of high-speed and heavy-load monorail crane under different braking parameters were analyzed. Secondly, the random response model of braking distance and braking temperature was established by combining the design of experiment method (DoE) and Dendrite Net (DD). Finally, the high-order moment saddlepoint approximation (SPA) method was used to evaluate the emergency braking reliability of the monorail crane. The results can provide a reference for the selection of key parameters and the evaluation of braking safety of the monorail crane braking system under high-speed and heavy-load conditions.

ABOUT THE BOUNDARY CONDITION APPROXIMATION IN THE HIGHER-ORDER GRID-CHARACTERISTIC SCHEMES

In this paper, we consider the problem of constructing a numerical solution to the system of equations of an acoustic medium in a fixed domain with a boundary. Physically, it corresponds to the process of the seismic wave propagation in geological media during the procedure of the seismic exploration of hydrocarbon deposits. The system of partial differential equations under consideration is hyperbolic. To construct its numerical solution, a grid-characteristic method is used on an extended spatial stencil. This approach makes it possible to construct a higher-order approximation scheme at the internal points of the computational domain. However, it requires a careful construction of the numerical solution near the boundaries. In this paper, the approach that preserves the increased approximation order up to the boundary is proposed. The verification numerical simulations were carried out.

Deep Kusuoka Approximation: High-Order Spatial Approximation for Solving High-Dimensional Kolmogorov Equations and Its Application to Finance

Deep Kusuoka Approximation: High-Order Spatial Approximation for Solving High-Dimensional Kolmogorov Equations and Its Application to Finance

A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations

In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner. We first derive continuous energy estimates, and then proceed to the discrete setting. We formulate the discrete approximation using high-order finite difference methods on summation-by-parts form and implement the boundary conditions weakly using the simultaneous approximation term method. By applying the discrete energy method and imitating the continuous analysis, the discrete estimate that resembles the continuous counterpart is obtained proving stability. We also show that these newly derived boundary conditions removes the singularities associated with the null-space of the nonlinear discrete spatial operator. Numerical experiments that verifies the high-order accuracy of the scheme and coincides with the theoretical results are presented. The numerical results are compared with the well-known Blasius similarity solution as well as that resulting from the solution of the incompressible Navier–Stokes equations.

A Novel High-Order Surface Characteristics Scheme for the Neutron Transport Equation in 2D Unstructured Meshes

The development of higher-order method of characteristics (MOC) discretizations has become of great interest to improve the performance of solvers based on the standard Stepwise Constant (SC) MOC approximation. Many codes nowadays implement a Stepwise Linear (SL) volume flux approximation or diamond differencing schemes. In the multigroup lattice solver TDT of the industrial code APOLLO3®, developed at CEA, a Linear Surface (LS) scheme was implemented. In this method, the flux is reconstructed from a linear interpolation made from surface values, therefore ensuring a similar spatial linear development but with a lower computational cost than the volume-based approximations. However, the LS-MOC scheme can conserve only the constant spatial moment of the flux. To overcome this limitation, in this paper we propose an improved version of the LS scheme called LS- xy able to preserve the linear spatial moments of the flux. Compared to the other high-order volume-based approximations, the LS- xy scheme also preserves flux surface moments, which guarantees higher accuracy. Moreover, our scheme has a lower memory footprint because it does not require the storage of response matrices that are dependent on region, group, and anisotropy order. Tests carried out on severe rodded assembly cases show the superior performance of the proposed method with respect to not only the classic SC or LS MOC scheme but also the SL scheme.

Convergence analysis of stochastic higher-order majorization–minimization algorithms

Majorization–minimization schemes are a broad class of iterative methods targeting general optimization problems, including nonconvex, nonsmooth and stochastic. These algorithms minimize successively a sequence of upper bounds of the objective function so that along the iterations the objective value decreases. We present a stochastic higher-order algorithmic framework for minimizing the average of a very large number of sufficiently smooth functions. Our stochastic framework is based on the notion of stochastic higher-order upper bound approximations of the finite-sum objective function and minibatching. We derive convergence results for nonconvex and convex optimization problems when the higher-order approximation of the objective function yields an error that is p times differentiable and has Lipschitz continuous p derivative. More precisely, for general nonconvex problems we present asymptotic stationary point guarantees and under Kurdyka–Lojasiewicz property we derive local convergence rates ranging from sublinear to linear. For convex problems with uniformly convex objective function, we derive local (super)linear convergence results for our algorithm. Compared to existing stochastic (first-order) methods, our algorithm adapts to the problem's curvature and allows using any batch size. Preliminary numerical tests support the effectiveness of our algorithmic framework.

Standard error estimates in hierarchical generalized linear models

Hierarchical generalized linear models are often used to fit random effects models. However, attention is mostly paid to the estimation of fixed unknown parameters and inference for latent random effects. In contrast, standard error estimators receive less attention than they should be. Currently, the standard error estimators are based on various approximations, even when the mean parameters may be estimated from a higher-order approximation of the likelihood and the dispersion parameters are estimated by restricted maximum likelihood. Existing standard error estimation procedures are reviewed. A numerical illustration shows that the current standard errors are not necessarily accurate. Alternative standard errors are also proposed. In particular, a sandwich estimator that accounts for the dependence between the mean parameters and the dispersion parameters greatly improve the current standard errors.

Meshing strategies for 3d geo-electromagnetic modeling in the presence of metallic infrastructure

In 3D geo-electromagnetic modeling, an adequate discretisation of the modeling domain is crucial to obtain accurate forward responses and reliable inversion results while reducing the computational cost. This paper investigates the mesh design for subsurface models, including steel-cased wells, which is relevant for many exploration settings but still remains a numerically challenging task. Applying a goal-oriented mesh refinement technique and subsequent calculations with the high-order edge finite element method, simulations of 3D controlled-source electromagnetic models in the presence of metallic infrastructure are performed. Two test models are considered, each needing a distinct version of approximation methods to incorporate the conductive steel casings of the included wells. The influence of mesh quality, goal-oriented meshing, and high-order approximations on problem sizes, computational cost, and accuracy of electromagnetic responses is investigated. The main insights of our work are: (a) the applied numerical schemes can mitigate the computational burden of geo-electromagnetic modeling in the presence of steel artifacts; (b) investigating the processes driving the meshing of models with embedded metallic infrastructures can lead to adequate strategies to deal with the inversion of such electromagnetic data sets. Based on the modeling results and analyses conducted, general recommendations for modeling strategies are proposed when performing simulations for challenging steel infrastructure scenarios.

Mathematical modeling of the multicomponent flow in porous media using higher-order methods

In this paper, we present a detailed numerical scheme for a single-phase compressible flow without diffusion of a multi-component mixture in porous media with the higher-order approximation in both space and time. The mathematical model consists of Darcy velocity, transport equations for each component of a mixture, pressure equation and associated relations for physical quantities such as viscosity or equation of state. The discrete problem is obtained using a combination of the discontinuous Galerkin method for the transport equations and the mixed-hybrid finite element method for the Darcy velocity and the pressure equation. In both methods the higher-order approximation is used. The resulting nonlinear problem for concentrations is solved with the fully mass-conservative iterative IMPEC method. Experimental order of convergence analysis (EOC) and some numerical experiments of a 2D flow are carried out.

A refined long time asymptotic bound for 3D axially symmetric Boussinesq system with zero thermal diffusivity

In this paper, we obtain a refined temporal asymptotic upper bound of the global axially symmetric solution to the Boussinesq system with no thermal diffusivity. We show the spacial W1,p-Sobolev (2≤p<∞) norm of the velocity can only grow at most algebraically as t→+∞. Under a signed potential condition imposed on the initial data, we further derive that the aforementioned norm is uniformly bounded at all times. Higher order estimates are also given: We find the H1 norm of the temperature fluctuation grows sub-exponentially as t→+∞. Meanwhile, for any m≥1, we deduce that the Hm-temporal growth of the solution is slower than a double exponential function. As a result, these improve the results in [11] where the authors only provided rough temporal asymptotic upper bounds while proving the global well-posedness.

The dissipative Generalized Hydrodynamic equations and their numerical solution

“Generalized Hydrodynamics” (GHD) stands for a model that describes one-dimensional integrable systems in quantum physics, such as ultra-cold atoms or spin chains. Mathematically, GHD corresponds to nonlinear equations of kinetic type, where the main unknown, a statistical distribution function f(t,z,θ), lives in a phase space which is constituted by a one-dimensional position variable z, and a one-dimensional “kinetic” variable θ, actually a wave-vector, called “rapidity”. Two key features of GHD equations are first a non-local and nonlinear coupling in the advection term, and second an infinite set of conserved quantities, which prevent the system from thermalizing. To go beyond this, we consider the dissipative GHD equations, which are obtained by supplementing the right-hand side of the GHD equations with a non-local and nonlinear diffusion operator or a Boltzmann-type collision integral. In this paper, we deal with new high-order numerical methods to efficiently solve these kinetic equations. In particular, we devise novel backward semi-Lagrangian methods for solving the advective part (the so-called Vlasov equation) by using a high-order time-Taylor series expansion for the advection fields, whose successive time derivatives are obtained by a recursive procedure. This high-order temporal approximation of the advection fields is used to design new implicit/explicit Runge–Kutta semi-Lagrangian methods, which are compared to Adams–Moulton semi-Lagrangian schemes. For solving the source terms, constituted by the diffusion and collision operators, we use and compare different numerical methods of the literature.

Stability and instability in a three-component chemotaxis model for alopecia areata

This paper investigates the stability and instability of a generalized reaction–diffusion–advection system that describes the spatio-temporal dynamics of alopecia areata lesions. Using the spectral analysis, we investigate the effect of chemotactic strengths on the stability and instability of a linearized problem around the positive constant equilibrium. We then extend our analysis to the nonlinear system, utilizing higher-order energy estimates and bootstrap techniques to study its stability and instability based on the linearized results. Additionally, we provide numerical simulations to support our theoretical results.

Analysis on physical-constraint-preserving high-order discontinuous Galerkin method for solving Kapila's five-equation model

This work focuses on the analysis and development of a high-order physical-constraint-preserving discontinuous Galerkin (DG) method for compressible two-fluid flows by solving Kapila's five-equation model with the ideal gas equation of state. The proposed method is proved to possess the following two attractive advantages: first, the high-order DG discretization intrinsically satisfies the mechanical equilibrium criterion around an isolated material interface, and the numerical oscillations due to the use of high-order approximations are well suppressed by using a cell-based characteristic-wise weighted essentially non-oscillatory (WENO) limiter without destroying the mechanical equilibrium criterion; second, it maintains the physical-constraint-preserving property, including the bound-preserving property of volume fractions and positivity-preserving property of partial densities and internal energy. A thorough analysis on the sufficient condition is given to achieve the physical-constraint-preserving property. Several benchmark test cases are examined to validate the accuracy and robustness of the proposed method. Further discussion on the application of the physical-constraint-preserving DG method for solving Kapila's five-equation model with the stiffened gas equation of state is also presented.

High-order half-step compact numerical approximation for fourth-order parabolic PDEs

High-order half-step compact numerical approximation for fourth-order parabolic PDEs

Higher-order error estimates for physics-informed neural networks approximating the primitive equations

Large-scale dynamics of the oceans and the atmosphere are governed by primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is generally challenging. Neural networks have been shown to be a promising machine learning tool to tackle this challenge. In this work, we employ physics-informed neural networks (PINNs) to approximate the solutions to the PEs and study the error estimates. We first establish the higher-order regularity for the global solutions to the PEs with either full viscosity and diffusivity, or with only the horizontal ones. Such a result for the case with only the horizontal ones is new and required in the analysis under the PINNs framework. Then we prove the existence of two-layer tanh PINNs of which the corresponding training error can be arbitrarily small by taking the width of PINNs to be sufficiently wide, and the error between the true solution and its approximation can be arbitrarily small provided that the training error is small enough and the sample set is large enough. In particular, all the estimates are a priori, and our analysis includes higher-order (in spatial Sobolev norm) error estimates. Numerical results on prototype systems are presented to further illustrate the advantage of using the H^s norm during the training.

High order approximations of solutions to initial value problems for linear fractional integro-differential equations

High order approximations of solutions to initial value problems for linear fractional integro-differential equations

High-order asymptotic approximations for improved inference under exceptionally low false positive error rates

High-order asymptotic approximations for improved inference under exceptionally low false positive error rates

Pricing ratchet equity index annuity with mortality risk by complex Fourier series method

In this paper, we consider the pricing of the ratchet equity-indexed annuity with the incorporation of mortality risk by an efficient and accurate method under the general time-changed Lévy processes. The pricing of the contract is a two-dimensional problem: we apply the complex Fourier series method to the log return dimension of the asset and high-order quadrature approximation to the other dimensions. Depending on the subordinating dynamic of the volatility, we use the time-changed Lévy process to model stochastic volatility via the random time change in the Lévy process, which compensates for limitations in the common Lévy process. By combining the recursive algorithm and complex Fourier series method, we obtain the approximate price of the contract. Error analysis proves that the total error can achieve an exponential convergence rate, and numerical examples demonstrate the accuracy and efficiency of the complex Fourier series expansion method.

Kenward-Roger-type corrections for inference methods of network meta-analysis and meta-regression.

Network meta-analysis has been an essential methodology of systematic reviews for comparative effectiveness research. The restricted maximum likelihood (REML) method is one of the current standard inference methods for multivariate, contrast-based meta-analysis models, but recent studies have revealed the resultant confidence intervals of average treatment effect parameters in random-effects models can seriously underestimate statistical errors; that is, the actual coverage probability of a true parameter cannot retain the nominal level (e.g., 95%). In this article, we provided improved inference methods for the network meta-analysis and meta-regression models using higher-order asymptotic approximations based on the approach of Kenward and Roger (Biometrics 1997;53:983-997). We provided two corrected covariance matrix estimators for the REML estimator and improved approximations for its sample distribution using a t-distribution with adequate degrees of freedom. All of the proposed procedures can be implemented using only simple matrix calculations. In simulation studies under various settings, the REML-based Wald-type confidence intervals seriously underestimated the statistical errors, especially in cases of small numbers of trials meta-analyzed. By contrast, the proposed Kenward-Roger-type inference methods consistently showed accurate coverage properties under all the settings considered in our experiments. We also illustrated the effectiveness of the proposed methods through applications to two real network meta-analysis datasets.

A kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator with higher approximation order

In this paper, a kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator is studied by combining the known multiquadric quasi-interpolation operator with the generalized Taylor polynomial as the expansion in the bivariate Bernoulli polynomials. Some error bounds and convergence rates of the combined operators are studied. A selection of numerical examples is presented to compare the performances of the obtained scheme. Furthermore, our method can be applied to time-dependent differential equations. Its advantage is that the algorithm is very simple and easy to implement.