Higher-order Approximations Research Articles

Higher-order bias corrections for kernel type density estimators on the unit or semi-infinite interval

For the data of size n from the unit or semi-infinite interval, several asymmetric kernel density estimators, having the mean integrated squared errors of order or , are available in the literature. We develop more higher-order bias-corrected estimators, achieving the order , where is a given integer. We illustrate the finite sample performance of the estimators through the simulations.

Improving pairwise approximations for network models with susceptible-infected-susceptible dynamics

Network models of disease spread play an important role in elucidating the impact of long-lasting infectious contacts on the dynamics of epidemics. Moment-closure approximation is a common method of generating low-dimensional deterministic models of epidemics on networks, which has found particular success for diseases with susceptible-infected-recovered (SIR) dynamics. However, the effect of network structure is arguably more important for sexually transmitted infections, where epidemiologically relevant contacts are comparatively rare and longstanding, and which are in general modelled via the susceptible-infected-susceptible (SIS)-paradigm. In this paper, we introduce an improvement to the standard pairwise approximation for network models with SIS-dynamics for two different network structures: the isolated open triple (three connected individuals in a line) and the k-regular network. This improvement is achieved by tracking the rate of change of errors between triple values and their standard pairwise approximation. For the isolated open triple, this improved pairwise model is exact, while for k-regular networks a closure is made at the level of triples to obtain a closed set of equations. This improved pairwise approximation provides an insight into the errors introduced by the standard pairwise approximation, and more closely matches both higher-order moment-closure approximations and explicit stochastic simulations with only a modest increase in dimensionality to the standard pairwise approximation.

Study of a Flame Kernel Evolution in a Turbulent Mixing Layer Using LES with a Laminar Chemistry Model

Temporal evolution of an ignition kernel in a spark ignited turbulent hydrogen–air mixing layer is studied using the large eddy simulation approach with an implicit treatment of the reaction source terms. The applied numerical code is based on a high-order compact difference approximation combined with a weighted essentially non-oscillatory scheme, which provide an accurate resolution of the small scale phenomena. The spark is modelled by an energy deposition model coupled with the enthalpy equation. Since the fuel and oxidiser streams move in the opposite directions the flow is dominated by shear stresses and vortical structures. The ignition follows three different scenarios depending on the spark location. An interaction between the developing flame kernel and large turbulent structure is analysed starting from the energy transfer up to a fully reacting state. The size and shape of the flames are correlated with the initial ignition scenario. A 3D visualisation of the transient position of the flame kernel shows its different spatio-temporal behaviour. It is observed that the flame can stay close to the initial position and spread equally in all directions or it can move far from the initial location and follow the evolving flow field. In the latter scenario two separate sub-stages are convincingly identified. Concerning the ignition probability (P_{ign}), when the spark is located in the immediate vicinity of the mixing layer, the P_{ign} on the fuel-rich side is higher. On the other hand, when the ignition occurs far from the mixing layer, the P_{ign} is significantly larger on the lean side. Unlike the local composition the impact of the strain rate of the velocity field was found to have very limited impact on the P_{ign}.

English

Survival analysis is applied in a wide range of sectors (medicine, economy, etc.), and its main idea is based on evaluating the time until the occurrence of an event of interest. The effect of some particular covariates on survival time is usually described by the Cox proportional hazards model and the statistical significance of the impact of covariates is verified by the likelihood ratio test, the Wald test, or the score test. In addition to standard tests, appropriate higher-order approximations based on Barndorff-Nielsen and Lugannani-Rice formulas are used for more accurate approximations. In this paper, comparison of these tests’ size and power for small sample sizes is performed on simulated datasets with various proportions of right-censored data, distributions of baseline hazard functions and different types of covariate—continuous or discrete.

Higher-Order Proximity-Based MiRNA-Disease Associations Prediction.

MiRNA-disease association prediction plays an important role in identifying human disease-related miRNAs. This approach is helpful not only to formulate individualized diagnosis schemes, but also to understand the pathogenesis of diseases. Many studies have focused on enhancing the prediction performance using explicit side information, such as miRNA functional similarity and disease semantic similarity. The existing approaches, however, often ignore the higher-order implicit proximity among miRNAs and diseases. To this end, in this paper, we first propose a novel approach HOP_MDA (Higher-Order Proximity based MiRNA and Disease Association Prediction) for predicting potential association between miRNA and disease. Both explicit interaction information and implicit higher-order proximity information between miRNA and disease are encoded with different order proximity matrices which are weightily combined into a parameterized prediction matrix. A supervised learning approach based on the known miRNAs-disease associations is proposed to determine the optimal weight parameters. The prediction matrix is then used to achieve effective prediction. Additionally, a higher-order proximity approximation technique (HOPA_MDA) is presented to make more efficient predictions. 5-fold cross validation is used to evaluate the performance of our proposed method. The average AUC values of HOPA_MDA for two real datasets are 0.921+/-0.002 and 0.944+/-0.0015, respectively. Our method can also predict potential miRNAs specific to new diseases with no known related miRNAs.

Application of New Iterative Method to Time Fractional Whitham–Broer–Kaup Equations

This article presents the Fractional Laplace transform with the help of new iterative method for finding the approximate solutions of coupled system of fractional order partial differential equations. The time fractional Whitham-Broer-Kaup system is taken as test example The fractional derivatives are described in the Caputo sense. Numerical results obtained by the proposed method are compared with that of Adomian Decomposition Method (ADM), Variational Iteration Method (VIM) and Optimal Homotopy Asymptotic Method (OHAM).Numerical results show that the proposed method is reliable and efficient for solution of fractional order coupled system partial differential equations. The accuracy of the method increases by taking higher order approximations.

Langevin Equations in the Small-Mass Limit: Higher-Order Approximations

We study the small-mass (overdamped) limit of Langevin equations for a particle in a potential and/or magnetic field with matrix-valued and state-dependent drift and diffusion. We utilize a bootstrapping argument to derive a hierarchy of approximate equations for the position degrees of freedom that are able to achieve accuracy of order $$m^{\ell /2}$$ over compact time intervals for any $$\ell \in \mathbb {Z}^+$$. This generalizes prior derivations of the homogenized equation for the position degrees of freedom in the $$m\rightarrow 0$$ limit, which result in order $$m^{1/2}$$ approximations. Our results cover bounded forces, for which we prove convergence in $$L^p$$ norms and unbounded forces, in which case we prove convergence in probability.

Ferromagnetic 2/1 quasicrystal approximants

The existence of various magnetic orders has recently been established in the Tsai-type 1/1 approximant Au-Al-Gd by variation of the electron-per-atom (e/a) ratio [Ishikawa et al. Phys. Rev. B 98, 220403(R) (2018)]. Here, we report ferromagnetic (FM) 2/1 quasicrystal approximants and show that the magnetic order of higher-order approximants can be tailored starting from known magnetic 1/1 approximants. The (Au,Cu)-(Al,In)-$R\phantom{\rule{4pt}{0ex}}(R=\mathrm{Gd},\mathrm{Tb})2$/1 approximants are synthesized by the simultaneous substitution of isovalent elements Cu and In for Au and Al, respectively, to the FM Au-Al-$R\phantom{\rule{4pt}{0ex}}(R=\mathrm{Gd},\mathrm{Tb})1$/1 approximants with $e/a=1.74$. Both the (Au,Cu)-(Al,In)-$R\phantom{\rule{4pt}{0ex}}(R=\mathrm{Gd},\mathrm{Tb})2$/1 approximants exhibit a FM transition at significantly high Curie temperatures of ${T}_{\mathrm{C}}=30.0$ and 15.3 K, respectively. Therefore, this method of isovalent substitution will also enable the realization of a variety of magnetic orders for the 2/1 approximant by variation of the e/a ratio. Such isovalent substitution may also stabilize magnetic quasicrystals for a given e/a ratio, which would lead to the realization of the long-range magnetic order in quasicrystals.

Direct reconstruction method for discontinuous Galerkin methods on higher-order mixed-curved meshes II. Surface integration

As the second extension of the direct reconstruction method (DRM), DRM for the surface integration of the discontinuous Galerkin (DG) weak formulation is presented on multi-dimensional high-order mixed meshes. From the previous study, DRM was successfully extended to the volume integration of the DG method, and produced a significant reduction in both the computational cost and memory overhead, particularly for high-order solution approximations on 3-D high-order meshes. DRM has the potential to reduce the computational cost and memory overhead further by treating the surface integration term consistently. To realize this, we design a linear dependency detector and formulate an adaptive orthonormalization process from the modified Gram-Schmidt (MGS) process. DRM is then applied to the restricted function space of the surface integration in an adaptive and optimal manner. The resulting methods, namely the DRM surface integration with the brute force points (BFP) and shape function points (SFP) methods, perform very well on high-order mixed-curved meshes. In addition, the adaptive MGS process is robust and accurate in extracting linear faces/edges from a given set of mixed-curved meshes. Various benchmark tests from the compressible Euler and Navier-Stokes equations verify the impact of DRM for 3-D large-scale computations on mixed-curved meshes. As an example, the DRM framework applied to the 3-D surface and volume integrations with DG-P3 approximation on P3-mesh may yield O(10)× speed-up, compared to conventional quadrature-based methods.

Mathematical Modeling of Spot Dynamics in a Stratified Medium

The investigation of dynamics of mixed fluid spots in a stratified environment is of interest both for the study of the ocean fine structure and for the study of wake dynamics behind moving underwater objects. The paper is devoted to the construction of a physical and mathematical model for this problem. Salinity is used as the stratifying component. This model is described by the Navier–Stokes equations in the Boussinesq approximation. The problem is solved using a recent version of the splitting method by physical factors the finite difference scheme of which has a high approximation order, minimum scheme viscosity and dispersion, and, which is especially important for problems with large gradients of hydrophysical parameters, problems with a free surface and internal waves, the monotonicity property. Numerous test computations for the study of the influence of grid parameters on the results are performed. The results of comparison with analytical estimates, experimental data, and computations of other researchers are presented. By way of example, the dynamics of the salinity perturbation is discussed, which corresponds to the isophase lines describing the behavior of internal waves during the collapse of spots.

Gedanken experiments at high-order approximation: nearly extremal Reissner-Nordstr\xf6m black holes cannot be overcharged

The new version of the gedanken experiment proposed by Sorce and Wald has been used to examine the weak cosmic censorship conjecture (WCCC) for black holes at the second-order approximation of the matter fields perturbation. However, only considering the perturbation until the second-order approximation is incomplete because there is an optimal option such that the existing condition of the event horizon vanishes at second- order. For this circumstance, we cannot judge whether the WCCC is satisfied at this order. In our investigation, the kth-order perturbation inequality is generally derived. Using the inequalities, we examine the WCCC for nearly extremal Reissner-Nordstöm black holes at higher-order approximation. It is shown that the WCCC cannot be violated yet after the perturbation. From this result, it can be indicated that the WCCC is strictly satisfied at the perturbation level for nearly extremal RN black holes.

Reaction Rates Important for Fusion Targets

To compute controlled fusion problems, four nuclear reactions are conventionally used. In the paper, cross sections of many thermonuclear reactions between the lightest elements are compared. We show that, along with the traditional four reactions, a tangible contribution can be made by the reaction $${\text{T}} + {\text{T}} \to 2n + \,{}^{{\text{4}}}{\text{He}}$$. At low temperatures, the reactions $${\text{D}} + p \to \gamma + {}^{3}{\text{He}}$$ and $${\text{T}} + p \to \gamma + {}^{4}{\text{He}}$$ make a considerable contribution and it appears that the rest of the reactions can be neglected. For the outlined reactions, we perform a more thorough processing of the experimental data and construct high-order accurate approximations. The accuracy of finding the S-factor is 2–6% and that of the reaction rate is 3–4% for the new reactions.

Nonlinear Vibrations of Thermo-Hyperelastic Moderately Thick Cylindrical Shells with 2:1 Internal Resonance

The nonlinear vibration of a hyperelastic moderately thick cylindrical shell with 2:1 internal resonance in a temperature field is investigated based on the third-order shear deformation theory. A radial harmonic excitation is applied to the shell. First, by employing the higher-order approximation for the curvature-related expansion, the displacement field of the moderately thick cylindrical shell with improved coefficients is derived. For the temperature field with gradient, the shear modulus of the shell is thickness dependent because of the temperature-dependent nature of the hyperelastic materials used. The graphical results manifest that the temperature gradient has a significant impact on the nonlinear vibration of the shell. In addition, the separation of the resonance peak caused by the variations of the structural parameter and temperature will result in a bubble shaped response curve for the shell.

Switching Volatility in a Nonlinear Open Economy

Uncertainty about an economy’s regime can change drastically around a crisis. An imported crisis such as the global financial crisis in the euro area highlights the effect of foreign shocks. Estimating an open-economy nonlinear dynamic stochastic general equilibrium model for the euro area and the United States including Markov-switching volatility shocks, we show that these shocks were significant during the global financial crisis compared with periods of calm. We describe how U.S. shocks from both the real economy and financial markets affected the euro area economy and how bond reallocation occurred between short- and long-term maturities during the global financial crisis. Importantly, the estimated nonlinearities when domestic and foreign financial markets influence the economy should not be neglected. The nonlinear behavior of market-related variables highlights the importance of higher-order estimation for providing additional interpretations to policymakers.

A meshfree collocation method based on moving Taylor polynomial approximation for high order partial differential equations

This paper presents a meshfree collocation method for solving high order partial differential equations (PDEs). The leading numerical difficulty is the approximation of high order derivatives. To make the approximation simple and efficient, a moving Taylor polynomial (MTP) approximation is presented by using movable expansion point for each sub-domain. Derivatives can be derived straightforward from the corresponding Taylor coefficients, which are determined by solving a weighted least squares problem. A distinct feature of the method is its ability to give the derivatives along with the shape function itself without further cost. To ensure the accuracy of high order approximation, stability of the weighted least squares problems for determining the Taylor coefficients is another issue should be addressed. For this purpose, the basis functions are rescaled by the size of window functions, and QR decomposition is adopted to solve the weighted least squares problems. The collocation method based on this MTP approximation does not require any grid or background cell, so it is a truly meshfree method. When solving the linear algebraic system generated by the MTP collocation method, a preconditioned sparse biconjugate gradients stabilized (BICGSTAB) solver is used to accelerate the computation speed. Numerical tests show that the proposed method is much accurate and efficient for high order PDEs.

Poisson–Nernst–Planck equations with high-order steric effects

The Poisson–Nernst–Planck–Lennard-Jones (PNP–LJ) model is a mathematical model for ionic solutions with Lennard-Jones interactions between the ions. Due to the singular nature of the Lennard-Jones interaction kernel, however, the PNP–LJ model gives rise to an intractable analytic problem and a highly demanding computational problem. Previous works tackled this problem by replacing the LJ potential with a leading order approximation of the LJ potential giving rise to the steric PNP model. The steric PNP proved to be a successful model for the transport of ions in biological ionic channels. However, in other parameter regimes related to high concentrations, it is ill-posed. Presumably, since the steric PNP model is derived using only a leading-order approximation of the LJ potential. In this study, we go beyond the leading order approximation of the Lennard-Jones (LJ) interaction kernel and develop a new class of steric PNP equations that rely on a high-order approximation of the LJ potential. Surprisingly, we show that the introduction of high-order terms does not regularize the steric PNP model. Namely, at the limit of vanishing ionic sizes, high-order steric PNP equations, at all orders including the PNP–LJ model, are ill-defined at certain parameters regimes. Further analysis shows that this is an inherent property of PNP equations that account for steric effects and is related to pattern formation in these systems.

Preconditioned Splitting Series Approximation for 2D Rough Surface Scattering

ABSTRACTScattering sensitivity of waves to surface roughness has been widely observed in many fields. Some series approximations for rough surface scattering, such as the general Born, the splitting, and the preconditioned splitting series, are presented for a numerical description of rough surface scattering by multiscale surfaces. In fact, the splitting series approximation is a specific form of the preconditioned splitting series. Numerical tests with several benchmark models are compared with the full-waveform numerical solution and the general Born series approximation to investigate the range of validity of the splitting and the preconditioned splitting series approximations. The splitting and the preconditioned splitting series approximations to multiscale surfaces are not subject to the strict limit applied to the general Born series approximation. Each order of the splitting series represents an increase of multiple scatterings between surface points. Therefore, higher-order splitting series approximation accounts for stronger surface scattering. A suitable preconditioned splitting and the splitting series approximations improve the general Born series approximation for the convergence of high-incident angle scattering, and, therefore, become realistic methods for multiscale surfaces with infinite gradients and extremely large surface heights. This series approximation mathematically provides a unified framework for rough surface scattering, which contains Born or Rytov series approximation as specific cases.

Efficient discontinuous Galerkin implementations and preconditioners for implicit unsteady compressible flow simulations

This work presents and compares efficient implementations of high-order discontinuous Galerkin methods: a modal matrix-free discontinuous Galerkin (DG) method, a hybridizable discontinuous Galerkin (HDG) method, and a primal formulation of HDG, applied to the implicit solution of unsteady compressible flows. The matrix-free implementation allows for a reduction of the memory footprint of the solver when dealing with implicit time-accurate discretizations. HDG reduces the number of globally-coupled degrees of freedom relative to DG, at high order, by statically condensing element-interior degrees of freedom from the system in favor of face unknowns. The primal formulation further reduces the element-interior degrees of freedom by eliminating the gradient as a separate unknown. This paper introduces a p-multigrid preconditioner implementation for these discretizations and presents results for various flow problems. Benefits of the p-multigrid strategy relative to simpler, less expensive, preconditioners are observed for stiff systems, such as those arising from low-Mach number flows at high-order approximation. The p-multigrid preconditioner also shows excellent scalability for parallel computations. Additional savings in both speed and memory occur with a matrix-free/reduced version of the preconditioner.

A new regularization of the D-bar method with complex conductivity

The aim of the D-bar method is to solve inverse conductivity problems. For complex conductivity, Hamilton et al. presented a reconstruction method which consists of six steps. In this article, we present a new regularization denoted by based on an iterative algorithm which allows us to reconstruct an approximation of the conductivity of the domain. The advantage of this new regularization is the performance of high-order error estimates.

Analysis of the Thermomechanical Stresses in Double-Wall Effusion Cooled Systems

Abstract In this study, thermal stresses in a double-wall cooling system are analyzed. We consider an infinite flat double-wall geometry and assume that it can be represented by an axisymmetric unit cell, wherein the thermal loadings and deformation at the boundaries are determined by periodicity conditions. A thermal model is initially developed to obtain the thermal fields using a combination of empirical correlations and computational fluid dynamics (CFD) analysis. The thermal fields are then analyzed such that both the first-order and higher order approximations are determined. A theoretical solution is derived assuming that the temperature gradient takes place only across the outer skin using the first-order approximations. The solution yields an equibiaxial stress state in the skins, which are driven by the thermal curvature of the outer skin. To investigate other geometrical features and higher order approximations, a finite element model is used to solve Fourier’s law of heat conduction and mechanical equilibrium equations. The numerical and theoretical results are found to be in excellent agreement. We determine that the increase of distance between pedestals reduces the stresses. Furthermore, the stress concentration factor at the fillet increases with the increase of both the radius and pedestal diameter and the decrease of the skin thickness. Increasing the number of film holes limits stresses to the external surface of the outer skin. The increase of the Reynolds number in the impingement hole increases the Biot number in the outer skin, which increases the stresses. The higher order approximations of the heat transfer coefficients play a minor role.