Forecasting with imperfect models, dynamically constrained inverse problems, and gradient descent algorithms

This paper is part of a larger study investigating the meaning of, and appropriate procedures for, forecasting with imperfect models. (In the author’s opinion there is currently no satisfactory general theory and practice for doing so with complex nonlinear systems.) The focus of this paper is on initialisation of the forecast. At the heart of every forecasting scheme there is an inverse problem that translates observations of reality into an initial state, or ensemble of states, of the model. Inverse problems are divided into two classes depending on whether the underlying model of reality is that of a stochastic process or a (deterministic) dynamical process. The two classes have quite different formulations of their inverse problems and consequent solutions methods. This paper considers dynamical process models and their inverse problems, which will be referred to as the dynamically constrained inverse problem (DCIP) line. The interpretation and solutions of the DCIP line are investigated and new algorithms for solving them are presented. The new algorithms are modifications of classical gradient descent algorithms. The new algorithms are applied to a low-dimensional chaotic system and a high-dimensional operational weather forecasting model. Our examination of DCIP shows that gradient descent algorithms are an effective way of solving the inverse problem for complex nonlinear system given an imperfect dynamical model.     

Cooperative communication with imperfect channel information: Performance analysis and optimum power allocation

In this paper, symbol-error-rate (SER) performance analysis is provided for decode-and-forward (DF) and amplify-and-forward (AF) cooperation schemes in wireless networks with imperfect channel information. We derive closed-form SER formulations for a single relay system with square MQAM signals in a flat Rayleigh fading channel. Moreover, closed-form and high SNR tight SER approximations are established to show the asymptotic performance of the cooperation protocols. Simulations and comparisons verify that these approximations lead to similar results to those from the exact SER formulations for different power allocation methods. Furthermore, based on these SER performance analyses, we determine the optimum power allocation for the AF and DF cooperation scenarios.     

Interaction-round-a-face models with fixed boundary conditions: The ABF fusion hierarchy

We use boundary weights and reflection equations to obtain families of commuting double-row transfer matrices for interaction-round-a-face models with fixed boundary conditions. In particular, we consider the fusion hierarchy of the Andrews-Baxter-Forrester (ABF) models, for which we obtain diagonal, elliptic solutions to the reflection equations, and find that the double-row transfer matrices satisfy functional equations with the same form as in the case of periodic boundary conditions.     

Metastates in disordered mean-field models: Random field and hopfield models

We rigorously investigate the size dependence of disordered mean-field models with finite local spin space in terms of metastates. Thereby we provide an illustration of the framework of metastates for systems of randomly competing Gibbs measures. In particular we consider the thermodynamic limit of the empirical metastate , whereμ _n (η) is the Gibbs measure in the finite volume {1,…,n} and the frozen disorder variableη is fixed. We treat explicitly the Hopfield model with finitely many patterns and the Curie-Weiss random field Ising model. In both examples in the phase transition regime the empirical metastate is dispersed for largeN. Moreover, it does not converge for a.e.η, but rather in distribution, for whose limits we given explicit expressions. We also discuss another notion of metastates, due to Aizenman and Wehr.     

The difference of drivers' anticipation behaviors in a new macro model of traffic flow and numerical simulation

A valuable consideration is inserted into car-following theory to investigate the difference of drivers' anticipation. Furthermore, a fresh macro traffic model involving the difference of the drivers' anticipation is deduced from the car-following model by applying transformation relationship from micro to macro variables. And this effect resulted from the difference of the drivers' anticipation is observed via numerical simulation. The results show that the difference of drivers' anticipation has a negligible impact on traffic flow.     

Stirling Refrigerating Machine Modeling Using Schmidt and Finite Physical Dimensions Thermodynamic Models: A Comparison with Experiments

The purpose of the study is to show that two simple models that take into account only the irreversibility due to temperature difference in the heat exchangers and imperfect regeneration are able to indicate refrigerating machine behavior. In the present paper, the finite physical dimensions thermodynamics (FPDT) method and 0-D modeling using the Schmidt model with imperfect regeneration were applied in the study of a β type Stirling refrigeration machine.The 0-D modeling is improved by including the irreversibility caused by imperfect regeneration and the finite temperature difference between the gas and the heat exchangers wall. A flowchart of the Stirling refrigerator exergy balance is presented to show the internal and external irreversibilities. It is found that the irreversibility at the regenerator level is more important than that at the heat exchangers level. The energies exchanged by the working gas are expressed according to the practical parameters, necessary for the engineer during the entire project. The results of the two thermodynamic models are presented in comparison with the experimental results, which leads to validation of the proposed FPDT model for the functional and constructive parameters of the studied refrigerating machine.     

Renormalization group at criticality and complete analyticity of constrained models: A numerical study

We study the majority rule transformation applied to the Gibbs measure for the 2D Ising model at the critical point. The aim is to show that the renormalized Hamiltonian is well defined in the sense that the renormalized measure is Gibbsian. We analyze the validity of Dobrushin-Shlosman uniqueness (DSU) finite-size condition for the constrained models corresponding to different configurations of the image system. It is known that DSU implies, in our 2D case, complete analyticity from which, as recently shown by Haller and Kennedy. Gibbsianness follows. We introduce a Monte Carlo algorithm to compute an upper bound to Vasserstein distance (appearing in DSU) between finite-volume Gibbs measures with different boundary conditions. We get strong numerical evidence that indeed the DSU condition is verified for a large enough volumeV for all constrained models.     

An approach to estimating and extrapolating model error based on inverse problem methods:towards accurate numerical weather prediction

Model error is one of the key factors restricting the accuracy of numerical weather prediction(NWP). Considering the continuous evolution of the atmosphere, the observed data(ignoring the measurement error) can be viewed as a series of solutions of an accurate model governing the actual atmosphere. Model error is represented as an unknown term in the accurate model, thus NWP can be considered as an inverse problem to uncover the unknown error term. The inverse problem models can absorb long periods of observed data to generate model error correction procedures. They thus resolve the deficiency and faultiness of the NWP schemes employing only the initial-time data. In this study we construct two inverse problem models to estimate and extrapolate the time-varying and spatial-varying model errors in both the historical and forecast periods by using recent observations and analogue phenomena of the atmosphere. Numerical experiment on Burgers’ equation has illustrated the substantial forecast improvement using inverse problem algorithms. The proposed inverse problem methods of suppressing NWP errors will be useful in future high accuracy applications of NWP.     

Pulsated free jets with polydisperse spray injection: Experiments and numerical simulations

In the present contribution, we consider acoustically pulsated free jets with a polydisperse spray injection in an axisymmetrical configuration. The acoustic excitation creates periodical large vortical structures, which are representative of the dynamics of gaseous flows in more complex configurations, and a strong interaction with the injected spray. In this context, we provide a series of detailed experimental measurements using laser diagnostics; we analyze droplet size distributions, associated size-conditioned dynamics and evaporation, for three liquids, leading to various droplet preferential concentrations due to the vortices. Besides the achievements in terms of diagnostics, we use the Eulerian multi-fluid model and dedicated robust numerical schemes in order to conduct simulations in the chosen configuration. Detailed comparisons between numerical simulations and experimental measurements are provided in terms of spray velocity and number density. Size-conditioned preferential concentration of both non-evaporating and evaporating sprays are observed and reproduced by the numerical simulations, which eventually yields a validation of the proposed model.     

The kinetic Potts chain and related potts models with competing interactions

We consider a general kinetic model for a chain of three-state Potts spins. From the time-evolution operator we infer points in two-dimensional Potts systems where certain spin correlations have one-dimensional character and the model is exactly solvable. This occurs in square lattice models with different kinds of competing interactions.     

秋千参数自激振动的数学模型与数值模拟

将秋千视为变长度单摆,基于归一化高斯函数构造了摆球相对摆杆的速度表达式.在此基础上建立了秋千的非线性动力学微分方程,并据此数学模型对秋千运动进行了数值模拟,得到了秋千的运动规律和功能转化的机制.     

The Hanle effect with partial frequency redistribution. Construction of a frequency-dependent polarization matrix and numerical solution by a PALI method

The polarized approximate lambda iteration (PALI) technique developed for the weak field Hanle effect relies on the decomposition of the Stokes parameters I, Q and U into six cylindrically symmetrical components. It has been applied to complete and partial frequency redistribution with redistribution matrices in which frequency redistribution is decoupled from scattering polarization. For partial frequency redistribution, the decoupling is obtained by an adequate decomposition of the frequency space into several domains. By angle-averaging frequency-dependent terms in the exact weak field Hanle redistribution matrix for a normal Zeeman triplet, we construct a redistribution matrix with coupling between frequency redistribution and polarization and no domain decomposition. The coupling is contained in generalized frequency redistribution functions that depend on the magnetic field. The redistribution matrix is expanded in the Landi Degl’Innocenti spherical tensors for polarimetry and the Stokes parameters are decomposed into cylindrically symmetrical components. A PALI method is set up for the calculation of these components. The Stokes parameters are calculated for different simple atmospheric models. The positive Q direction corresponds to the linear polarization perpendicular to the solar limb. It is shown that the frequency space decomposition may induce large errors on Stokes U in the transition region between line core and line wings but can safely be used for Stokes I and Q, the errors staying less than 1% at all the frequencies.     

Potts models with magnetic field: Arithmetic,geometry, and computation

We give a sheaf theoretic interpretation of Potts models with external magnetic field, in terms of constructible sheaves and their Euler characteristics. We show that the polynomial countability question for the hypersurfaces defined by the vanishing of the partition function is affected by changes in the magnetic field: elementary examples suffice to see non-polynomially countable cases that become polynomially countable after a perturbation of the magnetic field. The same recursive formula for the Grothendieck classes, under edge-doubling operations, holds as in the case without magnetic field, but the closed formulae for specific examples like banana graphs differ in the presence of magnetic field. We give examples of computation of the Euler characteristic with compact support, for the set of real zeros, and find a similar exponential growth with the size of the graph. This can be viewed as a measure of topological and algorithmic complexity. We also consider the computational complexity question for evaluations of the polynomial, and show both tractable and NP-hard examples, using dynamic programming.     

One- and two-channel Kondo model with logarithmic Van Hove singularity: A numerical renormalization group solution

Simple scaling consideration and NRG solution of the one- and two-channel Kondo model in the presence of a logarithmic Van Hove singularity at the Fermi level is given. The temperature dependences of local and impurity magnetic susceptibility and impurity entropy are calculated. The low-temperature behavior of the impurity susceptibility and impurity entropy turns out to be non-universal in the Kondo sense and independent of the s–d coupling J. The resonant level model solution in the strong coupling regime confirms the NRG results. In the two-channel case the local susceptibility demonstrates a non-Fermi-liquid power-law behavior.     

The effects of aircraft on climate and pollution. Part I: Numerical methods for treating the subgrid evolution of discrete size- and composition-resolved contrails from all commercial flights worldwide

This paper provides and evaluates mass conservative, positive-definite, unconditionally-stable, and non-iterative numerical techniques for simulating the evolution of discrete, size- and composition-resolved aerosol and contrail particles in individual aircraft exhaust plumes in a global or regional 3-D atmospheric model and coupling the subgrid exhaust plume information to the grid scale. Such treatment represents a new method of simulating the effects of aircraft on climate, contrails, and atmospheric composition. Microphysical processes solved within each plume include size-resolved coagulation among and between aerosol and contrail particles and their inclusions, aerosol-to-hydrometeor particle ice and liquid nucleation, deposition/sublimation, and condensation/evaporation. Each plume has its own emission and supersaturation, and the spreading and shearing of each plume’s cross-section are calculated as a function of time. Aerosol- and contrail-particle core compositions are tracked for each size and affect optical properties in each plume. When line contrails sublimate/evaporate, their size- and composition-resolved aerosol cores and water vapor are added to the grid scale where they affect large-scale clouds. Algorithm properties are analyzed, and the end-result model is evaluated against in situ and satellite data.     

Fitting models to correlated data III: A comparison between residual analysis and other methods

Applications of the χ² test, the F test, the Durbin-Watson d test, and the f (or Sign) test, to examples of correlated data treatment, show important drawbacks with the d test and (apparently) with the f test. An analytical approach based on residual analysis suggests an improvement in their use that leads to better results at lowest order; it also points out a distinction between goodness-of-fit tests, as the f test, and goodness-of-modeling tests, as the χ² and F tests. The residual analysis method is applied to the same examples; it looks faster, simpler, and often more accurate than the classical ones.     

The 1/D expansion for classical magnets: Low-dimensional models with magnetic field

The field-dependent magnetizationm(H, T) of one- and two-dimensional classical magnets described by theD-component vector model is calculated analytically in the whole range of temperature and magnetic fields with the help of the 1/D expansion. In the first order in 1/D the theory reproduces with a good accuracy the temperature dependence of the zero-field susceptibility of antiferromagnets with maximum atT|J ₀|/D (J ₀ is the Fourier component of the exchange interaction) and describes for the first time the singular behavior of (H, T) at small temperatures and magnetic fields: lim _T0 lim _H0 (H, T)=1/(2|J ₀|)(1–1/D) and lim _H0 lim _T0 (H, T)=1/(2|J ₀|).     

Detailed and simplified kinetic models of n-dodecane oxidation: The role of fuel cracking in aliphatic hydrocarbon combustion

A detailed kinetic model is proposed for the combustion of normal alkanes up to n-dodecane above 850 K. The model was validated against experimental data, including fuel pyrolysis in plug flow and jet-stirred reactors, laminar flame speeds, and ignition delay times behind reflected shock waves, with n-dodecane being the emphasis. Analysis of the computational results reveal that for a wide range of combustion conditions, the kinetics of fuel cracking to form smaller molecular fragments is fast and may be decoupled from the oxidation kinetics of the fragments. Subsequently, a simplified model containing a minimal set of 4 species and 20 reaction steps was developed to predict the fuel pyrolysis rate and product distribution. Combined with the base C₁-C₄ model, the simplified model predicts fuel pyrolysis rate and product distribution, laminar flame speeds, and ignition delays as close as the detailed reaction model.     

Molecular basis of structural make-up of feeds in relation to nutrient absorption in ruminants,revealed with advanced molecular spectroscopy: A review on techniques and models

Progress in ruminant feed research is no more feasible only based on wet chemical analysis, which is merely able to provide information on chemical composition of feeds regardless of their digestive features and nutritive value in ruminants. Studying internal structural make-up of functional groups/feed nutrients is often vital for understanding the digestive behaviors and nutritive values of feeds in ruminant because the intrinsic structure of feed nutrients is more related to its overall absorption. In this article, the detail information on the recent developments in molecular spectroscopic techniques to reveal microstructural information of feed nutrients and the use of nutrition models in regards to ruminant feed research was reviewed. The emphasis of this review was on (1) the technological progress in the use of molecular spectroscopic techniques in ruminant feed research; (2) revealing spectral analysis of functional groups of biomolecules/feed nutrients; (3) the use of advanced nutrition models for better prediction of nutrient availability in ruminant systems; and (4) the application of these molecular techniques and combination of nutrient models in cereals, co-products and pulse crop research. The information described in this article will promote better insight in the progress of research on molecular structural make-up of feed nutrients in ruminants.