Adaptive Approximation Algorithm Research Articles

Coordinated Vehicle Platooning with Fixed Routes: Adaptive Time Discretization, Strengthened Formulations and Approximation Algorithms

We consider the coordinated vehicle platooning problem over a road network with time constraints and the routes of vehicles are given. The problem is to coordinate the departure time of each vehicle to enable platoon formation hence maximizing the total fuel saving. We first focus on the case that the routes form a tree. In this case, the flexibility time window of each vehicle admits a unified representation, under which the synchronization of departure time is equivalent to the management of time-window overlapping. Based on this observation, a time-window-overlapping induced mixed-integer linear program formulation has been established which is equivalent to the continuous-time formulation established in the previous work. By imposing an adaptive time-discretization procedure, we can further reformulate the time coordination problem using a vehicle-to-time-bucket assignment induced mixed-integer linear program. Compared to the continuous-time formulation, the assignment formulation is free of big-M coefficients, and it is demonstrated by numerical experiments that the assignment formulation is indeed more effective in computational performance. As an independent interest, three approximation algorithms have been derived with provable competitive ratios under a certain regularity assumptions, which is for the first time on this discrete optimization problem. The solution approach has been extended to general cases in which the graph formed by vehicle routes can have loops, by adding a systematic loop breaking scheme. An insightful discovery is that whether there exists an exact equivalent assignment formulation for a general problem instance has a close connection to the pattern of lattice groups.

Overview of Adaptive and Low-Rank Approximation Algorithms for Modeling of The Influence of Electromagnetic Waves Generated by The Cell Phone Antenna on The Human Head

This paper presents an overview of formulations and algorithms dedicated to modeling the influence of electromagnetic waves on the human head. We start from the three-dimensional MRI scan of the human head. We approximate the MRI scan by the continuous approximation span over three-dimensional h adaptive mesh with quadratic polynomials. Next, we introduce time-harmonic Maxwell equations with a 1.8 GHz cell-phone antenna. We solve the problem of the propagation of electromagnetic waves on the human head. We compute the specific absorption rate used as the heat source for the Pennes bioheat equation. Finally, we introduce the Pennes bio-heat equation modeling the heat generated by the electromagnetic waves propagating through the skull, tissue, and air layers in the human head. We discuss the discretization and time-stepping algorithm for the Pennes equation’s solution over the human head. Namely, we focus on the Crank-Nicolson time integration scheme, to solve the bioheat transfer equations. We employ the hp finite elements with hierarchical shape functions and hp adaptive algorithm in three-dimensions. We propose an adaptive algorithm mixed with time-stepping iterations, where we simultaneously adapt the computational mesh, solve the Maxwell and Pennes equations, and we iterative with time steps. We employ the sparse Gaussian elimination algorithm with low-rank compression of the off-diagonal matrix blocks for the factorization of matrices. We conclude with the statement that 15 minutes of talk with a 1.8 GHz antenna of 1 Wat power results in increased brain tissue temperature up to 38.4 Celsius degree.

An adaptive patch approximation algorithm for bicriteria convex mixed-integer problems

Pareto frontiers of bicriteria continuous convex problems can be efficiently computed and optimal theoretical performance bounds have been established. In the case of bicriteria mixed-integer problems, the approximation of the Pareto frontier becomes, however, significantly harder. In this paper, we propose a new algorithm for approximating the Pareto frontier of bicriteria mixed-integer programs with convex constraints. Such Pareto frontiers are composed of patches of solutions with shared assignments for the discrete variables. By adaptively creating such a patchwork, our algorithm is able to create approximations that converge quickly to the true Pareto frontier. As a quality measure, we use the difference in hypervolume between the approximation and the true Pareto frontier. At least a certain number of patches is required to obtain an approximation with a given quality. This patch complexity gives a lower bound on the number of required computations. We show that our algorithm performs a number of optimization steps that are of a similar order as this lower bound. We provide an efficient MIP-based implementation of this algorithm. The efficiency of our algorithm is illustrated with numerical results showing that our algorithm has a strong theoretical performance guarantee while being competitive with other state-of-the-art approaches in practice.

Quantifying Privacy Vulnerability to Socialbot Attacks: An Adaptive Non-Submodular Model

Privacy breaches are one of the biggest concerns on Online Social Networks (OSNs), especially with an introduction of automated attacks by socialbots, which can automatically extract victims' private content by exploiting social behavior to befriend them. The key insight of this attack is that by intelligently sending friend requests to a small subset of users, called the Critical Friending Set (CFS), such a bot can evade current defense mechanisms. We study the vulnerability of OSNs to socialbot attacks. Specifically, we introduce a new optimization problem, Min-Friending, which identifies a minimum CFS to friend in order to obtain at least Q benefit, which quantifies the amount of private information the bot obtains. The two main challenges of this problem are how to cope with incomplete knowledge of network topology and how to model users' responses to friend requests. In this paper, we show that Min-Friending is inapproximable within a factor of (1-o(1)) ln Q and present an adaptive approximation algorithm using adaptive stochastic optimization. The key feature of our solution lies in the adaptive method, where partial network topology is revealed after each successful friend request. Thus the decision of whom to send a friend request to next is made with the outcomes of past decisions taken into account. Traditional tools break down when attempting to place a bound on the performance of this technique with realistic user models. Therefore, we additionally introduce a novel curvature-based technique to construct an approximation ratio of ln Q for a model of user behavior learned from empirical measurements on Facebook.

Stochastic Delay Characterization for Multicoupled RLC Interconnects Under Process Variations

The characterization of interconnect delay metrics in terms of process variations is an important but complicated task for statistical timing analysis in today’s integrated circuits (ICs). This paper presents a stochastic delay characterization framework for multicoupled interconnects in the presence of process variation. The proposed method starts with deriving the stochastic nodal equations for the RLC network that models multicoupled interconnects. By employing polynomial chaos (PC) expansion, a nonsampling-based stochastic prediction method, we further represent the voltage responses of network nodes as a series of orthogonal polynomials of random variables. During the expansion procedure, we use an adaptive approximation algorithm to reduce the number of required sampling points. We then use a stochastic collocation method to estimate the coefficients in the PC expansion model. With the voltage response determined as an expression of a multi-dimensional polynomial of random variables, the stochastic properties of the delay of multicoupled interconnects can be predicted. The proposed method not only takes into account the strong correlations among process variations, but also extracts an explicit delay representation for multicoupled interconnects in terms of process variations. Experimental results demonstrate that the delay characteristics predicted by the proposed method match well with the results by the brute-force Monte-Carlo method. Moreover, a significant speedup over the Monte-Carlo method has been achieved by the proposed delay characterization framework.

Adaptive Caching Networks With Optimality Guarantees

We study the optimal placement of content over a network of caches, a problem naturally arising in several networking applications. Given a demand of content request rates and paths followed, we wish to determine the content placement that maximizes the expected caching gain, i.e., the reduction of routing costs due to intermediate caching. The offline version of this problem is NP-hard and, in general, the demand and topology may be a priori unknown. Hence, a distributed, adaptive approximation algorithm for placing contents into caches is desired. We show that path replication, a simple algorithm frequently encountered in literature, can be arbitrarily suboptimal when combined with traditional eviction policies. We propose a distributed, adaptive algorithm that performs stochastic gradient ascent on a concave relaxation of the expected caching gain, and constructs a probabilistic content placement within a $1-1/e$ factor from the optimal, in expectation. Motivated by our analysis, we also propose a novel greedy eviction policy to be used with path replication, and show through numerical evaluations that both algorithms significantly outperform path replication with traditional eviction policies over a broad array of network topologies.

Adaptive stochastic approximation algorithm

In this paper, stochastic approximation (SA) algorithm with a new adaptive step size scheme is proposed. New adaptive step size scheme uses a fixed number of previous noisy function values to adjust steps at every iteration. The algorithm is formulated for a general descent direction and almost sure convergence is established. The case when negative gradient is chosen as a search direction is also considered. The algorithm is tested on a set of standard test problems. Numerical results show good performance and verify efficiency of the algorithm compared to some of existing algorithms with adaptive step sizes.

An Improved Tone Reservation Scheme in OFDM Systems for Adaptive Amplitude Clipping

Orthogonal frequency division multiplexing(OFDM) suffers from a high peak to average power ratio(PAPR). Tone reservation (TR) technology is considered as one of the most promising methods because of no additional distortion, no side information, and low implementation cost. However, in all PAPR reduction schemes，determining the optimal target clipping level is difficult. In this paper, we consider an adaptive amplitude least squares approximation(AALAS-TR) algorithm. Adjust adaptive optimal convergence factor for initial clipping threshold. That is results an approximately PAPR reduction performance with a low computational complexity. Simulation results show that the proposed algorithm can achieve better PAPR reduction as that different clipping threshold .

An Improved Tone Reservation Scheme Based Nearly Optimal Peak Reduction for Adaptive Amplitude Clipping

Orthogonal frequency division multiplexing (OFDM) suffers from a high peak to average power ratio (PAPR). Tone reservation (TR) technology is considered as one of the most promising methods because of no additional distortion, no side information, and low implementation cost. However, in all PAPR reduction schemes, determining the optimal target clipping level is difficult. In this paper, we consider an adaptive amplitude least squares approximation (AALAS-TR) algorithm. Adjust adaptive optimal convergence factor for initial clipping threshold. That is results an approximately PAPR reduction performance with a low computational complexity. Simulation results show that the proposed algorithm can achieve better PAPR reduction as that different clipping threshold

Overlays with Preferences: Distributed, Adaptive Approximation Algorithms for Matching with Preference Lists

A key property of overlay networks is the overlay nodes’ ability to establish connections (or be matched) to other nodes by preference, based on some suitability metric related to, e.g., the node’s distance, interests, recommendations, transaction history or available resources. When there are no preference cycles among the nodes, a stable matching exists in which nodes have maximized individual satisfaction, due to their choices, however no such guarantees are currently being given in the generic case. In this work, we employ the notion of node satisfaction to suggest a novel modeling for matching problems, suitable for overlay networks. We start by presenting a simple, yet powerful, distributed algorithm that solves the many-to-many matching problem with preferences. It achieves that by using local information and aggregate satisfaction as an optimization metric, while providing a guaranteed convergence and approximation ratio. Subsequently, we show how to extend the algorithm in order to support and adapt to changes in the nodes’ connectivity and preferences. In addition, we provide a detailed experimental study that focuses on the levels of achieved satisfaction, as well as convergence and reconvergence speed.

Indexable online time series segmentation with error bound guarantee

The volume of time series stream data grows rapidly in various applications. To reduce the storage, transmission and processing costs of time series data, segmentation and approximation is a common approach. In this paper, we propose a novel online segmentation algorithm that approximates time series by a set of different types of candidate functions (polynomials of different orders, exponential functions, etc.) and adaptively chooses the most compact one as the pattern of the time series changes. We call this algorithm the Adaptive Approximation (AA) algorithm. The AA algorithm incrementally narrows the feasible coefficient spaces (FCS) of candidate functions in coefficient coordinate systems to make each segment as long as possible given an error bound on each data point. We propose an algorithm called the FCS algorithm for the incremental computation of the feasible coefficient spaces. We further propose a mapping based index for similarity searches on the approximated time series. Experimental results show that our AA algorithm generates more compact approximations of the time series with lower average errors than the state-of-the-art algorithm, and our indexing method processes similarity searches on the approximated time series efficiently.

Adaptive Galerkin approximation algorithms for Kolmogorov equations in infinite dimensions

Space-time variational formulations and adaptive Wiener–Hermite polynomial chaos Galerkin discretizations of Kolmogorov equations in infinite dimensions, such as Fokker–Planck and Ornstein–Uhlenbeck equations for functions defined on an infinite-dimensional separable Hilbert space $$H$$ , are developed. The well-posedness of these equations in the Hilbert space $$\mathrm{L}^2(H,\mu )$$ of functions on the infinite-dimensional domain $$H$$ , which are square-integrable with respect to a Gaussian measure $$\mu $$ with trace class covariance operator $$Q$$ on $$H$$ , is proved. Specifically, for the infinite-dimensional Fokker–Planck equation, adaptive space-time Galerkin discretizations, based on a wavelet polynomial chaos Riesz basis obtained by tensorization of biorthogonal piecewise polynomial wavelet bases in time with a spatial Wiener–Hermite polynomial chaos arising from the Wiener–Itô decomposition of $$\mathrm{L}^2(H,\mu )$$ , are introduced. The resulting space-time adaptive Wiener–Hermite polynomial Galerkin discretization algorithms of the infinite-dimensional PDE are proved to converge quasioptimally in the sense that they produce sequences of finite-dimensional approximations that attain the best possible convergence rates afforded by best $$N$$ -term approximations of the solution from tensor-products of multiresolution (wavelet) time-discretizations and the Wiener–Hermite polynomial chaos in $$\mathrm{L}^2(H,\mu )$$ . As a consequence, the proposed adaptive Galerkin solution algorithms exhibit dimension-independent performance, which is optimal with respect to the algebraic best $$N$$ -term rate afforded by the solution and the polynomial degree and regularity of the multiresolution (wavelet) time-discretizations in the finite-dimensional case, in particular. All constants in our error and complexity bounds are shown to be independent of the number of “active” coordinates identified by the proposed adaptive Galerkin approximation algorithms. The computational work and memory required by the proposed algorithms scale linearly with the support size of the coefficient vectors that arise in the approximations, with dimension-independent constants.

An improved SPSA algorithm for system identification using fuzzy rules for training neural networks

Simultaneous perturbation stochastic approximation (SPSA) belongs to the class of gradient-free optimization methods that extract gradient information from successive objective function evaluation. This paper describes an improved SPSA algorithm, which entails fuzzy adaptive gain sequences, gradient smoothing, and a step rejection procedure to enhance convergence and stability. The proposed fuzzy adaptive simultaneous perturbation approximation (FASPA) algorithm is particularly well suited to problems involving a large number of parameters such as those encountered in nonlinear system identification using neural networks (NNs). Accordingly, a multilayer perceptron (MLP) network with popular training algorithms was used to predicate the system response. We found that an MLP trained by FASPSA had the desired accuracy that was comparable to results obtained by traditional system identification algorithms. Simulation results for typical nonlinear systems demonstrate that the proposed NN architecture trained with FASPSA yields improved system identification as measured by reduced time of convergence and a smaller identification error.

Efficient Batch and Adaptive Approximation Algorithms for Joint Multicast Beamforming and Admission Control

Wireless multicasting is becoming increasingly important for efficient distribution of streaming media and location-aware services to mobile and hand-held devices, network management, and software updates over cellular (UMTS-LTE) and indoor/outdoor wireless networks (e.g., 802.11/16). Multicast beamforming was recently proposed as a means of exploiting the broadcast nature of the wireless medium to boost spectral efficiency and meet Quality of Service (QoS) requirements. Infeasibility is a key issue in this context, due to power or mutual interference limitations. We therefore consider the joint multicast beamforming and admission control problem for one or more co-channel multicast groups, with the objective of maximizing the number of subscribers served and minimizing the power required to serve them. The problem is NP-hard even for an isolated multicast group and no admission control; but drawing upon our earlier work for the multiuser SDMA downlink, we develop an efficient approximation algorithm that yields good solutions at affordable worst-case complexity. For the special case of an isolated multicast, Lozano proposed a particularly simple adaptive algorithm for implementation in UMTS-LTE. We identify strengths and drawbacks of Lozano's algorithm, and propose two simple but worthwhile improvements. All algorithms are carefully tested on publicly available indoor/outdoor measured channel data.

Use of the matching pursuit algorithm with a dictionary of asymmetric waveforms in the analysis of transient evoked otoacoustic emissions

Transiently evoked otoacoustic emissions (TEOAEs) are normally modeled as the sum of asymmetric waveforms. However, some previous studies of TEOAEs used time-frequency (TF) methods to decompose the signals into symmetric waveforms. This approach was justified mainly as a means to reduce the complexity of the calculations. The present study extended the dictionary of numeric functions to incorporate asymmetric waveforms into the analysis. The necessary calculations were carried out using an adaptive approximation algorithm based on the matching pursuit (MP) numerical technique. The classic MP dictionary uses Gabor functions and consists of waveforms described by five parameters, namely, frequency, latency, time span, amplitude, and phase. In the present investigation, a sixth parameter, the degree of asymmetry, was added in order to enhance the flexibility of this approach. The effects of expanding the available functions were evaluated by means of both simulations using synthetic signals and authentic TEOAEs. The resulting analyses showed that the contributions of asymmetric components in the OAE signal are appreciable. In short, the expanded analysis method brought about important improvements in identifying TEOAE components including the correct detection of components with long decays, which are often related to spontaneous OAE activity, the elimination of a "dark energy" effect in TF distributions, and more reliable estimates of latency-frequency relationships. The latter feature is especially important for correct estimation of latency-frequency data, which is a crucial factor in investigations of OAE-generation mechanisms.

An Adaptive Linear Approximation Algorithm for Copositive Programs

We study linear optimization problems over the cone of copositive matrices. These problems appear in nonconvex quadratic and binary optimization; for instance, the maximum clique problem and other combinatorial problems can be reformulated as such problems. We present new polyhedral inner and outer approximations of the copositive cone which we show to be exact in the limit. In contrast to previous approximation schemes, our approximation is not necessarily uniform for the whole cone but can be guided adaptively through the objective function, yielding a good approximation in those parts of the cone that are relevant for the optimization and only a coarse approximation in those parts that are not. Using these approximations, we derive an adaptive linear approximation algorithm for copositive programs. Numerical experiments show that our algorithm gives very good results for certain nonconvex quadratic problems.

Flexible motion-adaptive video coding with redundant expansions

This paper presents a highly flexible video coding scheme, based on the use of a redundant dictionary of spatio-temporal three-dimensional (3-D) functions. Directionality and anisotropic scaling are key ingredients to the spatial components, which form a rich collection of two-dimensional (2-D) visual primitives. The temporal component is tuned to capture most of the energy in the temporal signal evolution, along motion trajectories in the video sequence. The video coding scheme (MP3D) first computes motion trajectories that are eventually entropy coded and sent as side information to the decoder. It then applies a spatio-temporal decomposition along motion trajectories, using an adaptive approximation algorithm based on matching pursuit (MP). Quantized coefficients and basis function parameters are entropy-coded in a embedded stream that is constructed to respect multiple rate constraints. The geometric properties of the 2-D primitive dictionary allow for flexible spatial resolution adaptation, so that the flexible MP3D stream enables decoding at different spatio-temporal resolutions, and multiple rates. The MP3D scheme is shown to provide rate-distortion performance that are comparable with state-of-the-art schemes, such as H.264, MPEG-4, at low and medium bit rate. However, the use of a redundant dictionary is penalizing at high coding rate, which makes the MP3D algorithm mostly interesting for low rate applications, or as a flexible base layer in hierarchical coding schemes.

Adaptive multivariate three-timescale stochastic approximation algorithms for simulation based optimization

We develop in this article, four adaptive three-timescale stochastic approximation algorithms for simulation optimization that estimate both the gradient and Hessian of average cost at each update epoch. These algorithms use four, three, two, and one simulation(s), respectively, and update the values of the decision variable and Hessian matrix components simultaneously, with estimates based on the simultaneous perturbation methodology. Our algorithms use coupled stochastic recursions that proceed using three different timescales or step-size schedules. We present a detailed convergence analysis of the algorithms and show numerical experiments using all the developed algorithms on a two-node network of M/G/1 queues with feedback for a 50-dimensional parameter vector. We provide comparisons of the performance of these algorithms with two recently developed two-timescale steepest descent simultaneous perturbation analogs that use randomized and deterministic perturbation sequences, respectively. We also present experiments to explore the sensitivity of the algorithms to their associated parameters. The algorithms that use four and three simulations, respectively, perform significantly better than the rest of the algorithms.

Electrostatic fields without singularities

The following problems that arise in the computation of electrostatic forces and in the Boundary Element Method are considered. Given two convex interior-disjoint polyhedra in 3-space endowed with a volume charge density which is a polynomial in the Cartesian coordinates of R 3 , compute the Coulomb force acting on them. Given two interior-disjoint polygons in 3-space endowed with a surface charge density which is polynomial in the Cartesian coordinates of R 3 , compute the normal component of the Coulomb force acting on them. For both problems adaptive Gaussian approximation algorithms are given, which, for n Gaussian points, in time O( n ), achieve absolute error O(c -√n ) for a constant c > 1. Such a result improves upon previously known best asymptotic bounds. This result is achieved by blending techniques from integral geometry, computational geometry and numerical analysis. In particular, integral geometry is used in order to represent the forces as integrals whose kernal is free from singularities.

An adaptive stochastic approximation algorithm for simultaneous diagonalization of matrix sequences with applications

We describe an adaptive algorithm based on stochastic approximation theory for the simultaneous diagonalization of the expectations of two random matrix sequences. Although there are several conventional approaches to solving this problem, there are many applications in pattern analysis and signal detection that require an online (i.e., real-time) procedure for this computation. In these applications, we are given two sequences of random matrices {A/sub k/} and {B/sub k/} as online observations, with lim/sub k/spl rarr//spl infin//E[A/sub k/]=A and lim/sub k/spl rarr//spl infin//E[B/sub k/]=B, where A and B are real, symmetric and positive definite. For every sample (A/sub k/, B/sub k/), we need the current estimates /spl Phi//sub k/ and /spl Lambda//sub k/, respectively of the eigenvectors /spl Phi/ and eigenvalues /spl Lambda/ of A/sup -1/ B. We have described such an algorithm where /spl Phi//sub k/ and /spl Lambda//sub k/ converge provably with probability one to /spl Phi/ and /spl Lambda/ respectively. A novel computational procedure used in the algorithm is the adaptive computation of A/sup - 1/2 /. Besides its use in the generalized eigen-decomposition problem, this procedure can be used on its own in several feature extraction problems. The performance of the algorithm is demonstrated with an example of detecting a high-dimensional signal in the presence of interference and noise, in a digital mobile communications problem. Experiments comparing computational complexity and performance demonstrate the effectiveness of the algorithm in this real-time application.