Number Of Auxiliary Variables Research Articles

Mapping dynamical systems with distributed time delays to sets of ordinary differential equations

Real-world dynamical systems with retardation effects are described in general not by a single, precisely defined time delay, but by a range of delay times. An exact mapping onto a set of N + 1 ordinary differential equations exists when the respective delay distribution is given in terms of a gamma distribution with discrete exponents. The number of auxiliary variables one needs to introduce, N, is inversely proportional to the variance of the delay distribution. The case of a single delay is therefore recovered when . Using this approach, denoted here the ‘kernel series framework’, we examine systematically how the bifurcation phase diagram of the Mackey–Glass system changes under the influence of distributed delays. We find that local properties, f.i. the locus of a Hopf bifurcation, are robust against the introduction of broadened memory kernels. Period-doubling transitions and the onset of chaos, which involve non-local properties of the flow, are found in contrast to be more sensitive to distributed delays. In general, the observed effects are found to scale as . Furthermore, we consider time-delayed systems exhibiting chaotic diffusion, which is present in particular for sinusoidal flows. We find that chaotic diffusion is substantially more pronounced for distributed delays. Our results indicate in consequence that modeling approaches of real-world processes should take the effects of distributed delay times into account.

An efficient outer space branch-and-bound algorithm for globally minimizing linear multiplicative problems

<abstract><p>We propose an efficient outer space branch-and-bound algorithm for minimizing linear multiplicative problems (LMP). First, by introducing auxiliary variables, LMP is transformed into an equivalent problem (ELMP), where the number of auxiliary variables is equal to the number of linear functions. Subsequently, based on the properties of exponential and logarithmic functions, further equivalent transformation of ELMP is performed. Next, a novel linear relaxation technique is used to obtain the linear relaxation problem, which provides a reliable lower bound for the global optimal value of LMP. Once more, branching operation takes place in the outer space of the linear function while embedding compression technique to remove infeasible regions to the maximum extent possible, which significantly reduces the computational cost. Therefore, an outer space branch-and-bound algorithm is proposed. In addition, we conduct convergence analysis and complexity proof for the algorithm. Finally, the computational performance of the algorithm is demonstrated based on the experimental results obtained by testing a series of problems.</p></abstract>

Quasi-Markovian property of strong wave turbulence.

This paper is concerned with the reduced-order modeling of the strongly nonlinear wave turbulence system. The motivation for such an attempt comes from the utility of the probabilistic coarse-grained model in facilitating the theoretical and numerical analysis of the true dynamical system model. One typical practice of simplifying the complex physical model is, in the spirit of Brownian motion, to replace the nonlinear interactions by white noise forcing and linear dissipation. For the case of slowly varying longwave, the resulting Markov process is an accurate approximate model. However, this conventional scheme is highly inappropriate for the description of shortwaves because the rapidly varying turbulent signal acquires a significantly non-Markovian character resulting from the poor timescale separation between the relevant mode and the environmental wave field. To resolve the issue, we discuss a simplification technique for which the central concept is the quasi-Markovian property; a non-Markov stochastic process is referred to as quasi-Markovian if it can be represented as a component of Markovian system made by adding a finite number of auxiliary variables. Our contribution in this work is to single out the nontrivial and near resonances from the nonlinear interactions in search of the auxiliary variable. We perform a comparison analysis of the autocorrelation matrices of the true and approximate models, and numerically demonstrate the effectiveness of our Markovian formulation of the inherently non-Markov turbulent signal.

Closure Results for Polynomial Factorization

$ \newcommand{\VNP}{\mathsf{VNP}} \newcommand{\VP}{\mathsf{VP}} \newcommand{\poly}{\mathsf{poly}} $ In a sequence of fundamental results in the 1980s, Kaltofen (SICOMP 1985, STOC'86, STOC'87, RANDOM'89) showed that factors of multivariate polynomials with small arithmetic circuits have small arithmetic circuits. In other words, the complexity class $\VP$ is closed under taking factors. A natural question in this context is to understand if other natural classes of multivariate polynomials, for instance, arithmetic formulas, algebraic branching programs, bounded-depth arithmetic circuits or the class $\VNP$, are closed under taking factors. In this paper, we show that all factors of degree at most $\log^a n$ of polynomials with $\poly(n)$-size depth-$k$ circuits have $\poly(n)$-size circuits of depth at most $O(k + a)$. This partially answers a question of Shpilka--Yehudayoff (Found. Trends in TCS, 2010) and has applications to hardness--randomness tradeoffs for bounded-depth arithmetic circuits. As direct applications of our techniques, we also obtain simple proofs of the following results. The complexity class $\VNP$ is closed under taking factors. This confirms Conjecture 2.1 in Bürgisser's monograph (2000) and improves upon a recent result of Dutta, Saxena and Sinhababu (STOC'18) who showed a quasipolynomial upper bound on the number of auxiliary variables and the complexity of the verifier circuit of factors of polynomials in $\VNP$. A factor of degree at most $d$ of a polynomial $P$ which can be computed by an arithmetic formula (or an algebraic branching program) of size $s$ has a formula (an algebraic branching program, resp.) of size at most $\poly(s, d^{\log d}, \deg(P))$. This result was first shown by Dutta et al. (STOC'18) and we obtain a slightly different proof as an easy consequence of our techniques. Our proofs rely on a combination of the ideas, based on Hensel lifting, developed in the polynomial factoring literature, and the depth-reduction results for arithmetic circuits, and hold over fields of characteristic zero or of sufficiently large characteristic. ------------- A preliminary version of this paper, titled ``Hardness vs Randomness for Bounded Depth Arithmetic Circuits,'' appeared in the Proceedings of the 33rd Computational Complexity Conference, 2018.

Simple PML implementation for finite‐difference time‐domain method

Via the auxiliary differential equation method, the authors propose a simple and efficient implementation of the stretched coordinate perfectly matched layer (SC-PML) in this Letter. The proposal can successfully halve the number of auxiliary variables associated with the PML while maintaining the capacity of the original SC-PML in terms of simple implementation. Thus, it can achieve a higher computational efficiency in the finite-difference time-domain simulations. The effectiveness of this new implementation is verified through an example carried out in a 3D Debye dispersive space.

Is there an association between survey characteristics and representativeness? A meta-analysis

How to achieve survey representativeness is a controversially debated issue in the field of survey methodology. Common questions include whether probability-based samples produce more representative data than nonprobability samples, whether the response rate determines the overall degree of survey representativeness, and which survey modes are effective in generating highly representative data. This meta-analysis contributes to this debate by synthesizing and analyzing the literature on two common measures of survey representativeness (R-Indicators and descriptive benchmark comparisons). Our findings indicate that probability-based samples (compared to nonprobability samples), mixed-mode surveys (compared to single-mode surveys), and other-than-Web modes (compared to Web surveys) are more representative, respectively. In addition, we find that there is a positive association between representativeness and the response rate as well as the number of auxiliary variables used in representativeness assessments. Furthermore, we identify significant gaps in the research literature that we hope might encourage further research in this area.

A functional regression model for inventories supported by aerial laser scanner data or photogrammetric point clouds

Abstract Forest inventories, with a probability sampling of a target variable Y and a potentially very large number of auxiliary variables (X) obtained from an aerial laser scanner or photogrammetry, are faced with the issue of model and variable selection when a model for linking Y to X is formulated. To bypass this step we propose a generic functional regression model (FRM) for use in both a design- and a model-based framework of inference. We demonstrate applications of FRM with inventory data from France, Germany, and Norway. The generic FRM achieved results that were comparable to those obtained with more traditional approaches based on model and variable selections. The proposed FRM generates interpretable regression coefficients and enables testing of practically relevant hypotheses regarding estimated models.

Quadratic reformulations of nonlinear binary optimization problems

Very large nonlinear unconstrained binary optimization problems arise in a broad array of applications. Several exact or heuristic techniques have proved quite successful for solving many of these problems when the objective function is a quadratic polynomial. However, no similarly efficient methods are available for the higher degree case. Since high degree objectives are becoming increasingly important in certain application areas, such as computer vision, various techniques have been recently developed to reduce the general case to the quadratic one, at the cost of increasing the number of variables by introducing additional auxiliary variables. In this paper we initiate a systematic study of these quadratization approaches. We provide tight lower and upper bounds on the number of auxiliary variables needed in the worst-case for general objective functions, for bounded-degree functions, and for a restricted class of quadratizations. Our upper bounds are constructive, thus yielding new quadratization procedures. Finally, we completely characterize all minimal quadratizations of negative monomials.

Formal verification of logical descriptions with functional uncertainty based on logarithmic encoding of conditions

We consider the implementability testing problem for a description with functional uncertainty represented with a system of partial Boolean functions (PBF) given by a combinatorial circuit. Implementability testing is based on a formal approach in which we construct a conjunctive normal form (CNF) based on encoding multi-output intervals from the domain of this system. We propose a method for CNF construction related to a system of PBFs which is based on logarithmic encoding of multi-output intervals from the domain of this system and lets one reduce the number of auxiliary variables introduced in the resulting CNF.

A GENERAL CLASS OF ESTIMATORS UNDER MULTI PHASE SAMPLING

This paper derives the general estimators for finite population mean using multivariate auxiliary information under multiphase sampling. Here a number of auxiliary variables are considered in each phase under general sampling design. The properties of these estimators are studied and the results are presented for simple random sampling without replacement (SRSWOR) scheme. Using a modified cost function the optimum sample sizes are also derived.

Coloured thermal noise-driven dynamical system: upper bound of time derivative of information entropy

In this paper we have studied the upper bound of the time derivative of information entropy for non-Markovian and thermodynamically closed system(s) using reduced model theory (RMT). The upper bound is calculated on the basis of the Fokker–Planck equation and the Schwartz inequality principle. Our calculation shows that the upper bound exhibits extremal nature in the variation of system parameters such as noise correlation time, dissipation strength. The present calculation also considers how the upper bound does change if we increase the number of auxiliary variables involved in the RMT.

Estimating Function Jackknife Variance Estimators Under Stratified Multistage Sampling

Generalized regression (GREG) uses auxiliary variables with known population totals to improve efficiency of estimators and to ensure consistency with the known totals. Variance estimation for the GREG estimator of a total under stratified multistage sampling is considered. Customary resampling methods (jackknife, balanced repeated replication and bootstrap) for estimating the variance of a GREG estimator require the inversion of a P × P matrix for each resample, where P is the number of auxiliary variables with known population totals. This could lead to illconditioned matrices for some of the resamples. We apply the estimating function (EF) resampling method of Hu and Kalbfleisch [Hu, F., Kalbfleisch, J. D. (2000). The estimating function bootstrap (with discussion). Can. J. Statist. 28:449–499] to obtain variance estimators, using jackknife resampling. This method avoids repeated inverses. We extend the results to cover parameters defined as solutions of census estimating equations. The proposed method can be implemented from micro data files containing the GREG weights and the associated EF jackknife weights.

Efficient balanced sampling: The cube method

A balanced sampling design is defined by the property that the Horvitz-Thompson estimators of the population totals of a set of auxiliary variables equal the known totals of these variables. Therefore the variances of estimators of totals of all the variables of interest are reduced, depending on the correlations of these variables with the controlled variables. In this paper, we develop a general method, called the cube method, for selecting approximately balanced samples with equal or unequal inclusion probabilities and any number of auxiliary variables.

Multivariate regression estimates for finite populations

The theory on regression estimate based on one auxiliary variable has been extended to that for more than one auxiliary variable. It has been found that the multivariate regression estimate (MRE) is not unbiased in general. The form of the approximate standard error of MRE is the same as that of simple regression estimate based on one auxiliary variable with the exception that the multiple correlation coefficient replaces the total correlation coefficient in the expression. It has also been found that the precision of MRE is non-decreasing, rather usually increasing as the number of auxiliary variables correlated with the dependent variable increases, assuming sample size to be large compared to the number of auxiliary variables.