Alternative Approximation Research Articles

Technical design note: differential infrared thermography of methane jets

In this note a novel approach for temperature measurements of methane jets is presented. Differential infrared thermography (DIT) is a contactless, tracer-free temperature determination method for semi-transparent objects, based on an infrared camera. DIT does not rely on a specific a priori value for the emissivity, but typically assumes constant emissivity within the relevant wavelength band. This is reasonable for complex hydrocarbons (i.e. as in liquid fuel sprays) but no longer justified for the discrete absorption spectrum of simple molecules such as methane. An alternative approximation is suggested and discussed, and the feasibility of DIT for the study of supercritical methane jets in a pressure chamber at conditions relevant for internal combustion engines is demonstrated. As DIT also determines the gas emissivity, a combined two-dimensional temperature and projected density visualisation becomes possible and is shown to highlight supersonic structurues such as Mach disks.

Surrogate models for performance evaluation of multi-skill multi-layer overflow loss systems

We consider a model of overflow loss systems in which server groups are arranged into layers, and alternate routing within each layer creates mutual overflow effects, increasing the amount of traffic that can be carried by the system. Such a model has wide applications in communications and service systems. However, the presence of both hierarchical inter-layer overflow and mutual intra-layer overflow makes accurate, robust, yet scalable blocking probability evaluation of such systems a difficult challenge. To address this challenge, we apply and extend the recently developed Information Exchange Surrogate Approximation (IESA) framework to a multi-layer system, adding new surrogate models to the framework and incorporating moment-matching techniques. In contrast to the conventional fixed-point approximation (FPA) approach, which directly decomposes the overflow loss system into independent subsystems with inherent problems of convergence and uniqueness, IESA performs decomposition on a carefully designed surrogate model with guaranteed convergence and uniqueness. Extensive numerical results demonstrate that IESA is consistently more accurate than the conventional FPA approach, showing an improvement in accuracy of several orders of magnitude in many cases. Furthermore, the new extensions to IESA introduced in this paper provide consistent improvements in accuracy relative to the current state-of-the-art of the IESA framework.

The Binomial CEV Model and the Greeks

AbstractThis article compares alternative binomial approximation schemes for computing the option hedge ratios studied by Chung and Shackleton (2002), Chung, Hung, Lee, and Shih (2011), and Pelsser and Vorst (1994) under the lognormal assumption, but now considering the constant elasticity of variance (CEV) process proposed by Cox (1975) and using the continuous‐time analytical Greeks recently offered by Larguinho, Dias, and Braumann (2013) as the benchmarks. Among all the binomial models considered in this study, we conclude that an extended tree binomial CEV model with the smooth and monotonic convergence property is the most efficient method for computing Greeks under the CEV diffusion process because one can apply the two‐point extrapolation formula suggested by Chung et al. (2011). © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:90–104, 2017

Combining existing numerical models with data assimilation using weighted least-squares finite element methods.

A new approach has been developed for combining and enhancing the results from an existing computational fluid dynamics model with experimental data using the weighted least-squares finite element method (WLSFEM). Development of the approach was motivated by the existence of both limited experimental blood velocity in the left ventricle and inexact numerical models of the same flow. Limitations of the experimental data include measurement noise and having data only along a two-dimensional plane. Most numerical modeling approaches do not provide the flexibility to assimilate noisy experimental data. We previously developed an approach that could assimilate experimental data into the process of numerically solving the Navier-Stokes equations, but the approach was limited because it required the use of specific finite element methods for solving all model equations and did not support alternative numerical approximation methods. The new approach presented here allows virtually any numerical method to be used for approximately solving the Navier-Stokes equations, and then the WLSFEM is used to combine the experimental data with the numerical solution of the model equations in a final step. The approach dynamically adjusts the influence of the experimental data on the numerical solution so that more accurate data are more closely matched by the final solution and less accurate data are not closely matched. The new approach is demonstrated on different test problems and provides significantly reduced computational costs compared with many previous methods for data assimilation. Copyright © 2016 John Wiley & Sons, Ltd.

Adaptive Quadrature for Maximum Likelihood Estimation of a Class of Dynamic Latent Variable Models

Maximum likelihood estimation of models based on continuous latent variables generally requires to solve integrals that are not analytically tractable. Numerical approximations represent a possible solution to this problem. We propose to use the adaptive Gaussian---Hermite (AGH) numerical quadrature approximation for a particular class of continuous latent variable models for time-series and longitudinal data. These dynamic models are based on time-varying latent variables that follow an autoregressive process of order 1, AR(1). Two examples are the stochastic volatility models for the analysis of financial time series and the limited dependent variable models for the analysis of panel data. A comparison between the performance of AGH methods and alternative approximation methods proposed in the literature is carried out by simulation. Empirical examples are also used to illustrate the proposed approach.

Approximated prediction of genomic selection accuracy when reference and candidate populations are related.

BackgroundGenomic selection is still to be evaluated and optimized in many species. Mathematical modeling of selection schemes prior to their implementation is a classical and useful tool for that purpose. These models include formalization of a number of entities including the precision of the estimated breeding value. To model genomic selection schemes, equations that predict this reliability as a function of factors such as the size of the reference population, its diversity, its genetic distance from the group of selection candidates genotyped, number of markers and strength of linkage disequilibrium are needed. The present paper aims at exploring new approximations of this reliability.ResultsTwo alternative approximations are proposed for the estimation of the reliability of genomic estimated breeding values (GEBV) in the case of non-independence between candidate and reference populations. Both were derived from the Taylor series heuristic approach suggested by Goddard in 2009. A numerical exploration of their properties showed that the series were not equivalent in terms of convergence to the exact reliability, that the approximations may overestimate the precision of GEBV and that they converged towards their theoretical expectations. Formulae derived for these approximations were simple to handle in the case of independent markers. A few parameters that describe the markers’ genotypic variability (allele frequencies, linkage disequilibrium) can be estimated from genomic data corresponding to the population of interest or after making assumptions about their distribution. When markers are not in linkage equilibrium, replacing the real number of markers and QTL by the “effective number of independent loci”, as proposed earlier is a practical solution. In this paper, we considered an alternative, i.e. an “equivalent number of independent loci” which would give a GEBV reliability for unrelated individuals by considering a sub-set of independent markers that is identical to the reliability obtained by considering the full set of markers.ConclusionsThis paper is a further step towards the development of deterministic models that describe breeding plans based on the use of genomic information. Such deterministic models carry low computational burden, which allows design optimization through intensive numerical exploration.Electronic supplementary materialThe online version of this article (doi:10.1186/s12711-016-0183-3) contains supplementary material, which is available to authorized users.

SOP: parallel surrogate global optimization with Pareto center selection for computationally expensive single objective problems

This paper presents a parallel surrogate-based global optimization method for computationally expensive objective functions that is more effective for larger numbers of processors. To reach this goal, we integrated concepts from multi-objective optimization and tabu search into, single objective, surrogate optimization. Our proposed derivative-free algorithm, called SOP, uses non-dominated sorting of points for which the expensive function has been previously evaluated. The two objectives are the expensive function value of the point and the minimum distance of the point to previously evaluated points. Based on the results of non-dominated sorting, P points from the sorted fronts are selected as centers from which many candidate points are generated by random perturbations. Based on surrogate approximation, the best candidate point is subsequently selected for expensive evaluation for each of the P centers, with simultaneous computation on P processors. Centers that previously did not generate good solutions are tabu with a given tenure. We show almost sure convergence of this algorithm under some conditions. The performance of SOP is compared with two RBF based methods. The test results show that SOP is an efficient method that can reduce time required to find a good near optimal solution. In a number of cases the efficiency of SOP is so good that SOP with 8 processors found an accurate answer in less wall-clock time than the other algorithms did with 32 processors.

On Sample Size and Power Calculation for Variant Set-Based Association Tests.

Sample size and power calculations are an important part of designing new sequence-based association studies. The recently developed SEQPower and SPS programs adopted computationally intensive Monte Carlo simulations to empirically estimate power for a series of variant set association (VSA) test methods including the sequence kernel association test (SKAT). It is desirable to develop methods that can quickly and accurately compute power without intensive Monte Carlo simulations. We will show that the computed power for SKAT based on the existing analytical approach could be inflated especially for small significance levels, which are often of primary interest for large-scale whole genome and exome sequencing projects. We propose a new χ(2) -approximation-based approach to accurately and efficiently compute sample size and power. In addition, we propose and implement a more accurate "exact" method to compute power, which is more efficient than the Monte Carlo approach though generally involves more computations than the χ(2) approximation method. The exact approach could produce very accurate results and be used to verify alternative approximation approaches. We implement the proposed methods in publicly available R programs that can be readily adapted when planning sequencing projects.

On polynomial chaos expansion via gradient-enhanced ℓ1-minimization

Gradient-enhanced Uncertainty Quantification (UQ) has received recent attention, in which the derivatives of a Quantity of Interest (QoI) with respect to the uncertain parameters are utilized to improve the surrogate approximation. Polynomial chaos expansions (PCEs) are often employed in UQ, and when the QoI can be represented by a sparse PCE, $\ell_1$-minimization can identify the PCE coefficients with a relatively small number of samples. In this work, we investigate a gradient-enhanced $\ell_1$-minimization, where derivative information is computed to accelerate the identification of the PCE coefficients. For this approach, stability and convergence analysis are lacking, and thus we address these here with a probabilistic result. In particular, with an appropriate normalization, we show the inclusion of derivative information will almost-surely lead to improved conditions, e.g. related to the null-space and coherence of the measurement matrix, for a successful solution recovery. Further, we demonstrate our analysis empirically via three numerical examples: a manufactured PCE, an elliptic partial differential equation with random inputs, and a plane Poiseuille flow with random boundaries. These examples all suggest that including derivative information admits solution recovery at reduced computational cost.

Approximating the Conway-Maxwell-Poisson normalizing constant

The Conway-Maxwell-Poisson is a two-parameter family of distributions on the nonnegative integers. Its parameters ? and ? model the intensity and the dispersion, respectively. Its normalizing constant is not always easy to compute, so good approximations are needed along with an assessment of their error. Shmueli, et al. [11] derived an approximation assuming that ? is an integer, and gave an estimate of the relative error. Their numerical work showed that their approximation performs well in some parameter ranges but not in others. Our aims are to show that this approximation applies to all real ? > 0; to provide correction terms to this approximation; and to give different approximations for ? very small and very large. We then investigate the error terms numerically to assess our approximations. In parameter ranges for which Shmueli?s approximation does poorly we show that our correction terms or alternative approximations give considerable improvement.

Spread of pedigree versus genetic ancestry in spatially distributed populations

Ancestral processes are fundamental to modern population genetics and spatial structure has been the subject of intense interest for many years. Despite this interest, almost nothing is known about the distribution of the locations of pedigree or genetic ancestors. Using both spatially continuous and stepping-stone models, we show that the distribution of pedigree ancestors approaches a travelling wave, for which we develop two alternative approximations. The speed and width of the wave are sensitive to the local details of the model. After a short time, genetic ancestors spread far more slowly than pedigree ancestors, ultimately diffusing out with radius ∼t rather than spreading at constant speed. In contrast to the wave of pedigree ancestors, the spread of genetic ancestry is insensitive to the local details of the models.

Measurement of the Self-Oscillating Vortex Rope Dynamics for Hydroacoustic Stability Analysis

Flow instabilities in hydraulic machines often feature oscillating cavitation volumes, which locally introduce compliance and mass flow gain effects. These unsteady characteristics play a crucial role in one-dimensional stability models and can be determined through the definition of transfer functions for the state variables, where the cavitation volume is commonly estimated from the discharge difference between two points located upstream and downstream of the cavity. This approach is demonstrated on a test rig with a microturbine, featuring a self-oscillating vortex rope in its conical draft tube. The fluctuating discharges at the turbine inlet and the draft tube outlet are determined with the pressure–time method using differential pressure transducers. The cavitation volume is then calculated by integrating the corresponding discharge difference over time. In order to validate the results, an alternative volume approximation method is presented, based on the image processing of a high-speed flow visualization. In this procedure, the edges of the vortex rope are detected to calculate the local cross section areas of the cavity. It is shown that the cavitation volumes obtained by the two methods are in good agreement. Thus, the fluctuating part of the cavitation volume oscillation can be accurately estimated by integrating the difference between the volumetric upstream and downstream discharges. Finally, the volume and discharge fluctuations from the pressure–time method are averaged over one mean period of the pressure oscillation. This enables an analysis of the key physical flow parameters’ behavior over one characteristic period of the instability and a discussion of its sustaining mechanisms.

Universal spatial correlations in the anisotropic Kondo screening cloud: Analytical insights and numerically exact results from a coherent state expansion

We analyze the spatial correlations in the spin density of an electron gas in the vicinity of a Kondo impurity. Our analysis extends to the spin-anisotropic regime, which was not investigated in the literature. We use an original and numerically exact method, based on a systematic coherent-state expansion of the ground state of the underlying spin-boson Hamiltonian, which we apply to the computation of observables that are specific to the fermionic Kondo model. We also present an important technical improvement to the method, that obviates the need to discretize modes of the Fermi sea, and allows one to tackle the problem in the thermodynamic limit. One can thus obtain excellent spatial resolution over arbitrary length scales, for a relatively low computational cost, a feature that gives the method an advantage over popular techniques such as NRG and DMRG. We find that the anisotropic Kondo model shows rich universal scaling behavior in the spatial structure of the entanglement cloud. First, SU(2) spin-symmetry is dynamically restored in a finite domain in parameter space in vicinity of the isotropic line, as expected from poor man's scaling. We are also able to obtain in closed analytical form a set of different, yet universal, scaling curves for strong exchange asymmetry, which are parametrized by the longitudinal exchange coupling. Deep inside the cloud, i.e. for distances smaller than the Kondo length, the correlation between the electron spin density and the impurity spin oscillates between ferromagnetic and antiferromagnetic values at the scale of the Fermi wavelength, an effect that is drastically enhanced at strongly anisotropic couplings. Our results also provide further numerical checks and alternative analytical approximations for the recently computed Kondo overlaps [PRL 114, 080601 (2015)].

About the validity of complex mass scheme for the Δ resonance and higher energy region approaches

It is a well known fact that when a resonance (R) is produced and decays, its unperturbed propagator will be singular at when it appears in an s-type Feynman pole graph. Nevertheless this is not a problem, since it must be dressed by the successive re-interactions of the resonance with itself to all orders through the non-pole T-matrix, which we include in this work for the self-energy one-loop pion–nucleon contributions. Recently, we have shown that within the one-loop approximation for the absorptive part it is possible to justify the use of the complex-mass description over the unperturbed propagator, adopted in previous calculations to treat pion–nucleon elastic and radiative scattering, and photo and weak pion production in the resonance region. Here we discuss the limits of the validity of this approximation, since in actual calculations we must go over this range. We show that, when the resonance energy is increased above , this approach becomes invalid and an alternative approximation should be considered. In addition, we discuss the usual energy dependent width approximation, present in several actual isobar models used to analyze the last neutrino scattering experiments, and show that also it could be unappropriate.

Explicit [formula omitted] approximation of conic sections using Bézier curves of arbitrary degree

In this paper, we propose an explicit and effective method for G1 approximation of conic sections with arbitrary degree polynomial curves. The geometric constraints can be expressed by two variables. Then we construct an objective function as the alternative approximation error function in the L2-norm, which is a quadratic function in these two variables. To minimize the objective function is equivalent to solving a system of linear equations with two variables. Since its coefficient matrix is invertible, we can explicitly obtain the unique solution and then derive the explicit G1 approximation of conic sections by Bézier curves of arbitrary degree. The proposed method has an optimal approximation in the L2-norm, i.e., the L2-distance error reaches minimum, and also can process the conic section with center angle larger than π without subdivision scheme. Finally, numerical examples demonstrate the effectiveness of our method.

SOMS: SurrOgate MultiStart algorithm for use with nonlinear programming for global optimization

AbstractSOMS is a general surrogate‐based multistart algorithm, which is used in combination with any local optimizer to find global optima for computationally expensive functions with multiple local minima. SOMS differs from previous multistart methods in that a surrogate approximation is used by the multistart algorithm to help reduce the number of function evaluations necessary to identify the most promising points from which to start each nonlinear programming local search. SOMS's numerical results are compared with four well‐known methods, namely, Multi‐Level Single Linkage (MLSL), MATLAB's MultiStart, MATLAB's GlobalSearch, and GLOBAL. In addition, we propose a class of wavy test functions that mimic the wavy nature of objective functions arising in many black‐box simulations. Extensive comparisons of algorithms on the wavy test functions and on earlier standard global‐optimization test functions are done for a total of 19 different test problems. The numerical results indicate that SOMS performs favorably in comparison to alternative methods and does especially well on wavy functions when the number of function evaluations allowed is limited.

Estimating excitonic effects in the absorption spectra of solids: problems and insight from a guided iteration scheme.

A major obstacle for computing optical spectra of solids is the lack of reliable approximations for capturing excitonic effects within time-dependent density functional theory. We show that the accurate prediction of strongly bound electron-hole pairs within this framework using simple approximations is still a challenge and that available promising results have to be revisited. Deriving a set of analytical formulas we analyze and explain the difficulties. We deduce an alternative approximation from an iterative scheme guided by previously available knowledge, significantly improving the description of exciton binding energies. Finally, we show how one can "read" exciton binding energies from spectra determined in the random phase approximation, without any further calculation.

Variational Inference for Nonparametric Bayesian Quantile Regression

Quantile regression deals with the problem of computing robust estimators when the conditional mean and standard deviation of the predicted function are inadequate to capture its variability. The technique has an extensive list of applications, including health sciences, ecology and finance. In this work we present a non-parametric method of inferring quantiles and derive a novel Variational Bayesian (VB) approximation to the marginal likelihood, leading to an elegant Expectation Maximisation algorithm for learning the model. Our method is nonparametric, has strong convergence guarantees, and can deal with nonsymmetric quantiles seamlessly. We compare the method to other parametric and non-parametric Bayesian techniques, and alternative approximations based on expectation propagation demonstrating the benefits of our framework in toy problems and real datasets.

Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability

This paper investigates structural wave propagation in waveguides with randomly varying material and geometrical properties along the axis of propagation. More specifically, it is assumed that the properties vary slowly enough such that there is no or negligible backscattering due to any changes in the propagation medium. This variability plays a significant role in the so called mid-frequency region for dynamics and vibration, but wave-based methods are typically only applicable to homogeneous and uniform waveguides. The WKB approximation is used to find a suitable generalization of the wave solutions for finite waveguides undergoing longitudinal and flexural motion. An alternative wave formulation approximation with piecewise constant properties is also derived and included, so that the internal reflections are taken into account, but this requires a discretization of the waveguide. Moreover, a Fourier like series, the Karhunen–Loeve expansion, is used to represent homogeneous and spatially correlated randomness and subsequently the wave propagation approach allows the statistics of the natural frequencies and the forced response to be derived. Experimental validation is presented using a cantilever beam whose mass per unit length is randomized by adding small discrete masses to an otherwise uniform beam. It is shown how the correlation length of the random material properties affects the natural frequency statistics and comparison with the predictions using the WKB approach is given.

Alternative fluid approximation approach for the steady-state distribution of the two-sided reflected Markov modulated Brownian motion and its computation

The steady-state distribution of the two-sided reflected Markov Modulated Brownian motion is derived through an alternative fluid approximation approach. In this paper, we have shown that the distribution can be obtained by taking limit on the steady-state distribution of a weakly convergent Markov modulated fluid model. In addition, we present how the duality theorem developed by Ahn and Ramaswami is applied to get the limit result. For computation, we introduce a quadratically convergent algorithm and its related theories which rely on the fluid approximation approach. Through numerical examples, we finally illustrate behaviors of the steady-state distribution and also some performances of the algorithm.