Approximate Version Research Articles

Ground-state properties of an electron-phonon coupled quantum wire within the dynamic mean-field approximation

We investigate the ground-state properties of an electron-phonon (e-ph) coupled quantum wire in which the effect of coupling between longitudinal optical (LO) phonons and the wire electrons is considered, in addition to the conventional electron-electron (e-e) interactions. The e-ph coupling is included through the Fröhlich interaction potential and the dynamics of electron correlations using the quantum (dynamic) version of self-consistent mean-field approximation of Singwi et al (the qSTLS approach). Numerical results for static structure factor, pair-correlation function and static density susceptibility are presented over a wide range of electron number density parameter r s . We find that the electrons undergo Wigner crystallization at a critical electron density parameter r s c owing to the dynamics of severely correlated electrons. Notably at a fixed wire width, the inclusion of e-ph coupling tends to increase the value of r s c significantly in comparison to a situation in which only the e-e interactions are considered. Besides this, we compute the plasmon-phonon (pl-ph) coupled modes and the energy contribution of e-ph interactions (E e−ph ) to the ground-state energy of the quantum wire, also known as polaronic energy. We find that pl-ph coupled modes resonantly split into two branches, and ∣E e−ph ∣ monotonically increases with rise in r s then becomes almost constant as . Wherever interesting, we compare our results with the static STLS theory and the lattice regularized diffusion Monte Carlo (LRDMC) simulation data of Casula et al.

Coprime mappings and lonely runners

For x real, let { x } $ \lbrace x \rbrace$ be the fractional part of x (that is, { x } = x − ⌊ x ⌋ $\lbrace x\rbrace = x - \lfloor x \rfloor$ ). The lonely runner conjecture can be stated as follows: for any n positive integers v 1 < v 2 < ⋯ < v n $ v_1 < v_2 < \dots < v_n$ , there exists a real number t such that 1 / ( n + 1 ) ⩽ { v i t } ⩽ n / ( n + 1 ) $ 1/(n+1) \leqslant \lbrace v_i t\rbrace \leqslant n/(n+1)$ for i = 1 , ⋯ , n $ i = 1, \dots , n$ . In this paper, we prove that if ε > 0 $ \epsilon >0$ and n is sufficiently large (relative to ε), then such a t exists for any collection of positive integers v 1 < v 2 < ⋯ < v n $ v_1 < v_2 < \dots < v_n$ such that v n < ( 2 − ε ) n $ v_n < (2-\epsilon )n$ . This is an approximate version of a natural next step for the study of the lonely runner conjecture suggested by Tao. A key ingredient in our proof is a result on coprime mappings. Let A and B be sets of integers. A bijection f : A → B $ f:A \rightarrow B$ is a coprime mapping if a and f ( a ) $f(a)$ are coprime for every a ∈ A $ a \in A$ . We show that if A , B ⊂ [ n ] $A,B \subset [n]$ are intervals of length 2m where m = e Ω ( ( log log n ) 2 ) $ m = e^{ \Omega ({(\log \log n)}^2)}$ then there exists a coprime mapping from A to B.

Covering Convex Bodies and the Closest Vector Problem

We present algorithms for the (1+epsilon )-approximate version of the closest vector problem for certain norms. The currently fastest algorithm (Dadush and Kun 2016) for general norms in dimension n has running time of 2^{O(n)}(1/epsilon )^n. We improve this substantially in the following two cases. First, for ell _p-norms with p>2 (resp. p in [1,2]) fixed, we present an algorithm with a running time of 2^{O(n)}(1+1/epsilon )^{n/2} (resp. 2^{O(n)} (1+1/epsilon )^{n/p}). This result is based on a geometric covering problem, that was introduced in the context of CVP by Eisenbrand et al.: How many convex bodies are needed to cover the ball of the norm such that, if scaled by factor 2 around their centroids, each one is contained in the (1+epsilon )-scaled homothet of the norm ball? We provide upper bounds for this (2,epsilon )-covering number by exploiting the modulus of smoothness of the ell _p-balls. Applying a covering scheme, we can boost any 2-approximation algorithm for CVP to a (1+epsilon )-approximation algorithm with the improved run time, either using a straightforward sampling routine or using the deterministic algorithm of Dadush for the construction of an epsilon net. Second, we consider polyhedral and zonotopal norms. For centrally symmetric polytopes (resp. zonotopes) in {mathbb R}^n with O(n) facets (resp. generated by O(n) line segments), we provide a deterministic O(log _2(2+1/epsilon ))^{O(n)} time algorithm. This generalizes the result of Eisenbrand et al. which applies to the ell _infty -norm. Finally, we establish a connection between the modulus of smoothness and lattice sparsification. As a consequence, using the enumeration and sparsification tools developped by Dadush, Kun, Peikert, and Vempala, we present a simple alternative to the boosting procedure with the same time and space requirement for ell _p norms. This connection might be of independent interest.

Proof transfer for fast certification of multiple approximate neural networks

Developers of machine learning applications often apply post-training neural network optimizations, such as quantization and pruning, that approximate a neural network to speed up inference and reduce energy consumption, while maintaining high accuracy and robustness. Despite a recent surge in techniques for the robustness verification of neural networks, a major limitation of almost all state-of-the-art approaches is that the verification needs to be run from scratch every time the network is even slightly modified. Running precise end-to-end verification from scratch for every new network is expensive and impractical in many scenarios that use or compare multiple approximate network versions, and the robustness of all the networks needs to be verified efficiently. We present FANC, the first general technique for transferring proofs between a given network and its multiple approximate versions without compromising verifier precision. To reuse the proofs obtained when verifying the original network, FANC generates a set of templates – connected symbolic shapes at intermediate layers of the original network – that capture the proof of the property to be verified. We present novel algorithms for generating and transforming templates that generalize to a broad range of approximate networks and reduce the verification cost. We present a comprehensive evaluation demonstrating the effectiveness of our approach. We consider a diverse set of networks obtained by applying popular approximation techniques such as quantization and pruning on fully-connected and convolutional architectures and verify their robustness against different adversarial attacks such as adversarial patches, L 0 , rotation and brightening. Our results indicate that FANC can significantly speed up verification with state-of-the-art verifier, DeepZ by up to 4.1x.

Newton’s method for reinforcement learning and model predictive control

The purpose of this paper is to propose and develop a new conceptual framework for approximate Dynamic Programming (DP) and Reinforcement Learning (RL). This framework centers around two algorithms, which are designed largely independently of each other and operate in synergy through the powerful mechanism of Newton’s method. We call these the off-line training and the on-line play algorithms; the names are borrowed from some of the major successes of RL involving games. Primary examples are the recent (2017) AlphaZero program (which plays chess), and the similarly structured and earlier (1990s) TD-Gammon program (which plays backgammon). In these game contexts, the off-line training algorithm is the method used to teach the program how to evaluate positions and to generate good moves at any given position, while the on-line play algorithm is the method used to play in real time against human or computer opponents.Both AlphaZero and TD-Gammon were trained off-line extensively using neural networks and an approximate version of the fundamental DP algorithm of policy iteration. Yet the AlphaZero player that was obtained off-line is not used directly during on-line play (it is too inaccurate due to approximation errors that are inherent in off-line neural network training). Instead a separate on-line player is used to select moves, based on multistep lookahead minimization and a terminal position evaluator that was trained using experience with the off-line player. The on-line player performs a form of policy improvement, which is not degraded by neural network approximations. As a result, it greatly improves the performance of the off-line player.Similarly, TD-Gammon performs on-line a policy improvement step using one-step or two-step lookahead minimization, which is not degraded by neural network approximations. To this end it uses an off-line neural network-trained terminal position evaluator, and importantly it also extends its on-line lookahead by rollout (simulation with the one-step lookahead player that is based on the position evaluator).An important lesson from AlphaZero and TD-Gammon is that the performance of an off-line trained policy can be greatly improved by on-line approximation in value space, with long lookahead (involving minimization or rollout with the off-line policy, or both), and terminal cost approximation that is obtained off-line. This performance enhancement is often dramatic and is due to a simple fact, which is couched on algorithmic mathematics and is the focal point of this work:(a) Approximation in value space with one-step lookahead minimization amounts to a step of Newton’s method for solving Bellman’s equation.(b) The starting point for the Newton step is based on the results of off-line training, and may be enhanced by longer lookahead minimization and on-line rollout. Indeed the major determinant of the quality of the on-line policy is the Newton step that is performed on-line, while off-line training plays a secondary role by comparison.Significantly, the synergy between off-line training and on-line play also underlies Model Predictive Control (MPC), a major control system design methodology that has been extensively developed since the 1980s. This synergy can be understood in terms of abstract models of infinite horizon DP and simple geometrical constructions, and helps to explain the all-important stability issues within the MPC context. In this work we aim to provide insights (often based on visualization), which explain the beneficial effects of on-line decision making on top of off-line training. In the process, we will bring out the strong connections between the artificial intelligence view of RL, and the control theory views of MPC and adaptive control. While we will deemphasize mathematical proofs, there is considerable related analysis, which supports our conclusions and can be found in the author’s recent RL books (Bertsekas, 2019; Bertsekas, 2020), and the abstract DP monograph (Bertsekas, 2022).One of our principal aims is to show, through the algorithmic ideas of Newton’s method and the unifying principles of abstract DP, that the AlphaZero/TD-Gammon methodology of approximation in value space and rollout applies very broadly to deterministic and stochastic optimal control problems, involving both discrete and continuous search spaces, as well as finite and infinite horizon.

Impact of electron-phonon interactions on Wigner crystallization in an electron quantum wire

In this paper, we explore the impact of electron-phonon (e-ph) coupling on the Wigner crystallization occurring due to strong electron-electron (e-e) interactions in a semiconductor-based electron quantum wire. The e-ph coupling is represented by the Fröhlich interaction potential and electron correlations are included within the dynamic version of self-consistent mean-field approximation of Hasegawa and Shimizu. Numerical results are reported for the static structure factor and pair-correlation function for fixed wire width and some selected values of electron number density parameter rs . A strong peak in the static structure factor and pronounced periodic oscillations in pair-correlation function indicate the formation of Wigner crystal in the quantum wire at a critical electron number density rs c . We find that the inclusion of e-ph interactions in addition to the usual e-e interactions significantly enhance rs c , at a fixed wire width.

DeepTEA

In this paper, we study anomalous trajectory detection, which aims to extract abnormal movements of vehicles on the roads. This important problem, which facilitates understanding of traffic behavior and detection of taxi fraud, is challenging due to the varying traffic conditions at different times and locations. To tackle this problem, we propose the &lt;u&gt;deep&lt;/u&gt; -probabilistic-based &lt;u&gt;t&lt;/u&gt;ime-d&lt;u&gt;e&lt;/u&gt;pendent &lt;u&gt;a&lt;/u&gt;nomaly detection algorithm ( DeepTEA ). This method, which employs deep-learning methods to obtain time-dependent outliners from a huge volume of trajectories, can handle complex traffic conditions and detect outliners accurately. We further develop a fast and approximation version of DeepTEA, in order to capture abnormal behaviors in real-time. Compared with state-of-the-art solutions, our method is 17.52% more accurate than seven competitors on average, and can handle millions of trajectories.

A Mixed Finite Differences Scheme for Gradient Approximation

In this paper, we focus on the linear functionals defining an approximate version of the gradient of a function. These functionals are often used when dealing with optimization problems where the computation of the gradient of the objective function is costly or the objective function values are affected by some noise. These functionals have been recently considered to estimate the gradient of the objective function by the expected value of the function variations in the space of directions. The expected value is then approximated by a sample average over a proper (random) choice of sample directions in the domain of integration. In this way, the approximation error is characterized by statistical properties of the sample average estimate, typically its variance. Therefore, while useful and attractive bounds for the error variance can be expressed in terms of the number of function evaluations, nothing can be said on the error of a single experiment that could be quite large. This work instead is aimed at deriving an approximation scheme for linear functionals approximating the gradient, whose error of approximation can be characterized by a deterministic point of view in the case of noise-free data. The previously mentioned linear functionals are no longer considered as expected values over the space of directions, but rather as the filtered derivative of the objective function by a Gaussian kernel. By using this new approach, a gradient estimation based on a suitable linear combination of central finite differences at different step sizes is proposed and deterministic bounds that do not depend on the particular sample of points considered are computed. In the noisy setting, on the other end, the variance of the estimation error of the proposed method is showed to be strictly lower than the one of the estimation error of the Central Finite Difference scheme. Numerical experiments on a set of test functions are encouraging, showing good performances compared to those of some methods commonly used in the literature, also in the noisy setting.

Information efficient learning of complexly structured preferences: Elicitation procedures and their application to decision making under uncertainty

In this paper we propose efficient methods for elicitation of complexly structured preferences and utilize these in problems of decision making under (severe) uncertainty. Based on the general framework introduced in Jansen et al. (2018) [37], we now design elicitation procedures and algorithms that enable decision makers to reveal their underlying preference system (i.e. two relations, one encoding the ordinal, the other the cardinal part of the preferences) while having to answer as few as possible simple ranking questions. Here, two different approaches are followed. The first approach directly utilizes the collected ranking data for obtaining the ordinal part of the preferences, while their cardinal part is constructed implicitly by measuring the decision maker's consideration times. In contrast, the second approach explicitly elicits also the cardinal part of the decision maker's preference system, however, only an approximate version of it. This approximation is obtained by additionally collecting labels of preference strength during the elicitation procedure. For both approaches, we give conditions under which they produce the decision maker's true preference system and investigate how their efficiency can be improved. For the latter purpose, besides data-free approaches, we also discuss ways for statistically guiding the elicitation procedure if data from elicitations of previous decision makers is available. Finally, we demonstrate how the proposed elicitation methods can be utilized in problems of decision under (severe) uncertainty. Precisely, we show that under certain conditions optimal decisions can be found without fully specifying the preference system.

Efficient Beampattern Synthesis for Sparse Frequency Diverse Array via Matrix Pencil Method.

Due to the introduction of frequency offsets, the pattern synthesis problem of sparse Frequency diverse array (FDA) becomes more complicated than that of the phased array. A typical way to solve this problem is to use a global optimization algorithm, but this is usually time-consuming. In this paper, we propose an efficient non-iterative beampattern synthesis approach for sparse FDA. For a given reference pattern, which can be generated by other synthesis methods, we first sample it uniformly and construct the Hankel matrix with the sampled data. By low-rank processing, a low-rank approximation version of the Hankel matrix can then be obtained. Finally, the matrix enhancement and matrix pencil (MEMP) and matrix pencil (MP) methods are applied to estimate the antenna positions, frequency offsets, and excitations of the obtained array from the approximated matrix. Besides this, two typical FDA frameworks including multi-carrier FDA (MCFDA) and standard FDA (SFDA) are considered. Numerical simulation results prove that the proposed method outperforms the existing methods in terms of synthesis error, average runtime, and percentage of saving elements.

Exploring smoking-gun signals of the Schwinger mechanism in QCD

In Quantum Chromodynamics, the Schwinger mechanism endows the gluons with an effective mass through the dynamical formation of massless bound-state poles that are longitudinally coupled. The presence of these poles affects profoundly the infrared properties of the interaction vertices, inducing crucial modifications to their fundamental Ward identities. Within this general framework, we present a detailed derivation of the non-Abelian Ward identity obeyed by the pole-free part of the three-gluon vertex in the soft-gluon limit, and determine the smoking-gun displacement that the onset of the Schwinger mechanism produces to the standard result. Quite importantly, the quantity that describes this distinctive feature coincides formally with the bound-state wave function that controls the massless pole formation. Consequently, this signal may be computed in two independent ways: by solving an approximate version of the pertinent Bethe-Salpeter integral equation, or by appropriately combining the elements that enter in the aforementioned Ward identity. For the implementation of both methods we employ two- and three-point correlation functions obtained from recent lattice simulations, and a partial derivative of the ghost-gluon kernel, which is computed from the corresponding Schwinger-Dyson equation. Our analysis reveals an excellent coincidence between the results obtained through either method, providing a highly nontrivial self-consistency check for the entire approach. When compared to the null hypothesis, where the Schwinger mechanism is assumed to be inactive, the statistical significance of the resulting signal is estimated to be 3 standard deviations.

Brill–Noether existence on graphs via $${\mathbb {R}}$$-divisors, polytopes and lattices

We study Brill-Noether existence on a finite graph using methods from polyhedral geometry and lattices. We start by formulating analogues of the Brill-Noether conjectures (both the existence and non-existence parts) for $\mathbb{R}$-divisors, i.e. divisors with real coefficients, on a graph. We then reformulate the Brill-Noether existence conjecture for $\mathbb{R}$-divisors on a graph in geometric terms, that we refer to as the covering radius conjecture and we show a weak version, in support of it. Using this, we show an approximate version of the Brill-Noether existence conjecture for divisors on a graph. As applications, we derive upper bounds on the gonality of a graph and its $\mathbb{R}$-divisor analogue.

EM Scattering by Core-Shell Gyroelectric-Isotropic and Isotropic-Gyroelectric BoRs Using the EBCM

We employ the extended boundary condition method (EBCM) and construct a solution for the problem of electromagnetic (EM) scattering by anisotropic core-shell bodies of revolution (BoRs). In particular, two different core-shell configurations are examined: the gyroelectric-isotropic and the isotropic-gyroelectric setup. To construct the solution, we employ two groups of integral representations (IRs)&#x2014;one group for each configuration solved&#x2014;in conjunction with the discrete eigenfunction (DE) expansion of the fields in terms of spherical vector wave functions (SVWFs) for the gyroelectric regions. We demonstrate the validity and the computational performance of the method by comparisons with the HFSS commercial software for various core-shell setups such as spheroidal, cylindrical, and combined spherical-cylindrical BoRs. We also employ ADDA, a particular version of the discrete dipole approximation (DDA) method, to trace the boundaries of validity of the EBCM. Finally, we present an application of the method to the study of magnetically-tunable spheroidal THz antennas. The method can be used in a variety of potential EM applications including microwaves, functional photonics structures, as well as nanoantenna engineering.

Mathematical assessment of the dynamics of the tobacco smoking model: An application of fractional theory

&lt;abstract&gt;&lt;p&gt;In this paper we consider fractional-order mathematical model describing the spread of the smoking model in the sense of Caputo operator with tobacco in the form of snuffing. The threshold quantity $ \mathcal{R}_0 $ and equilibria of the model are determined. We prove the existence of the solution via fixed-point theory and further examine the uniqueness of of the solution of the considered model. The new version of numerical approximation's framework for the approximation of Caputo operator is used. Finally, the numerical results are presented to justify the significance of the arbitrary fractional order derivative. The analysis shows fractional-order model of tobacco smoking in Caputo sense gives useful information as compared to the classical integer order tobacco smoking model.&lt;/p&gt;&lt;/abstract&gt;

A consistent approximation of the total perimeter functional for topology optimization algorithms

This article revolves around the total perimeter functional, one particular version of the perimeter of a shape Ω contained in a fixed computational domainDmeasuring the total area of its boundary∂Ω, as opposed to its relative perimeter, which only takes into account the regions of∂Ω strictly insideD. We construct and analyze approximate versions of the total perimeter which make sense for general “density functions”u, as generalized characteristic functions of shapes. Their use in the context of density-based topology optimization is particularly convenient insofar as they do not involve the gradient of the optimized functionu. Two different constructions are proposed: while the first one involves the convolution of the functionuwith a smooth mollifier, the second one is based on the resolution of an elliptic boundary-value problem featuring Robin boundary conditions. The “consistency” of these approximations with the original notion of total perimeter is appraised from various points of view. At first, we prove the pointwise convergence of our approximate functionals, then the convergence of their derivatives, as the level of smoothing tends to 0, when the considered density functionuis the characteristic function of a “regular enough” shape Ω ⊂D. Then, we focus on the Γ-convergence of the second type of approximate total perimeter functional, that based on elliptic regularization. Several numerical examples are eventually presented in two and three space dimensions to validate our theoretical findings and demonstrate the efficiency of the proposed functionals in the context of structural optimization.

Approximate Square Root Circuits with Low Latency and Power Dissipation

This paper proposes a series of approximate square root circuit designs with high accuracy, low latency, low area, and low power dissipation requirements. The proposed designs are constructed using an array of controlled add–subtract cell elements with both exact and approximate versions. The utility of the proposed designs are evaluated by utilizing them in an example image contrast enhancement application with demonstrably satisfactory results and large peak signal-to-noise ratios and structural similarity values. The accuracy and hardware characteristics of the proposed square root designs are also analyzed and compared with previously proposed state-of-the-art approximate square root designs. When applied to a 16-bit radicand (the number under the square root symbol), the proposed designs have the lowest error rates, normalized mean error distances, and mean relative error distances by at least 1.8x when compared to all previous methods using the same number of approximate cells. When the designs were synthesized using Synopsys Design Compiler with a 28 nm bulk CMOS process, the delay, area, power, and power-delay-product characteristics outperform all previous designs in all but a few cases. These results demonstrate that the proposed designs permit the use of a flexible range of approximate designs with varying accuracy and hardware overhead characteristics, and a suitable design can be selected based on the user design requirements.

The recursive variational Gaussian approximation (R-VGA)

We consider the problem of computing a Gaussian approximation to the posterior distribution of a parameter given N observations and a Gaussian prior. Owing to the need of processing large sample sizes N, a variety of approximate tractable methods revolving around online learning have flourished over the past decades. In the present work, we propose to use variational inference to compute a Gaussian approximation to the posterior through a single pass over the data. Our algorithm is a recursive version of variational Gaussian approximation we have called recursive variational Gaussian approximation. We start from the prior, and for each observation, we compute the nearest Gaussian approximation in the sense of Kullback–Leibler divergence to the posterior given this observation. In turn, this approximation is considered as the new prior when incorporating the next observation. This recursive version based on a sequence of optimal Gaussian approximations leads to a novel implicit update scheme which resembles the online Newton algorithm and which is shown to boil down to the Kalman filter for Bayesian linear regression. In the context of Bayesian logistic regression, the implicit scheme may be solved, and the algorithm is shown to perform better than the extended Kalman filter, while being less computationally demanding than its sampling counterparts.

Analysis of an Extremum Seeking Controller Under Bounded Disturbance.

One of the applications of Extremum Seeking (ES) is to localize the source of a scalar field by using a mobile agent that can measure this field at its current location. While the scientific literature has presented many approaches to this problem, a formal analysis of the behavior of ES controllers for source seeking in the presence of disturbances is still lacking. This paper aims to fill this gap by analyzing a specific version of an ES control algorithm in the presence of source movement and measurement disturbances. We define an approximate version of this controller that captures the main features but allows for a simplified analysis and then formally characterize the convergence properties of this approximation. Through simulations and physical experiments, we compare the theoretically-predicted regions of attraction of the simplified system with the behavior of the full system and show that the simplified version is a good predictor of the behavior of the initial ES controller.

Generalization of Vaidya’s Metric for A Radiating Star to Rotating Star

Schwarzschild's external solution of Einstein’s gravitational field equations in the general theory of relativity for a static star has been generalized by Vaidya [1], taking into account the radiation of the star. Here, we generalize Vaidya’s metric to a star that is rotating and radiating. Although, there is a famous Kerr solution [2] for a rotating star, but here is a simple solution for a rotating star which may be termed as a zero approximate version of the Kerr solution. Results are discussed.

ON APPROXIMATION OF PERIODIC ATEB-FUNCTIONS

Two versions of approximation formulae for periodic Ateb-sine and Ateb-cosine in the first quarter of their common period are proposed. The first version is a Pade type approximation derived when constructing analytical solution of corresponding integral equation by iteration method with transforming the power series into a closed sum by Shanks’ formula. Two iteration approximations are considered. The first one is more concise but of worse approximation accuracy which deteriorates with increasing the argument value. To improve the approximation accuracy a hybrid approximation is proposed when the values of the Ateb-functions in the beginning (for the cosine) and in the end (for the sine) of the quarter period are computed by a separate formula obtained a priory by the asymptotic method. The comparison analysis of the approximate and exact values of the special functions indicates the error of the approximation proposed to be less than one per cent. The second variant of approximation is by replacing the periodic Ateb-functions by trigonometric functions of specific argument. The arguments are chosen so that the values of the special functions are exact at specific points of the quarter period. Five such collocation points are introduced in the paper. To implement this version of approximation a separate table of the values of the periodic Ateb-functions at the collocation points is compiled. The computational examples presented in the paper show the approximate values of the special functions obtained by the second version of approximation to have a good accuracy.