Number Of Arithmetic Operations Research Articles

Parallel wavelet schemes for images

In this paper, we introduce several new schemes for calculation of discrete wavelet transforms of images. These schemes reduce the number of steps and, as a consequence, allow to reduce the number of synchronizations on parallel architectures. As an additional useful property, the proposed schemes can reduce also the number of arithmetic operations. The schemes are primarily demonstrated on CDF 5/3 and CDF 9/7 wavelets employed in JPEG 2000 image compression standard. However, the presented method is general, and it can be applied on any wavelet transform. As a result, our scheme requires only two memory barriers for 2-D CDF 5/3 transform compared to four barriers in the original separable form or three barriers in the non-separable scheme recently published. Our reasoning is supported by exhaustive experiments on high-end graphics cards.

Sparse high-dimensional FFT based on rank-1 lattice sampling

In this paper, we suggest approximate algorithms for the reconstruction of sparse high-dimensional trigonometric polynomials, where the support in frequency domain is unknown. Based on ideas of constructing rank-1 lattices component-by-component, we adaptively construct the index set of frequencies belonging to the non-zero Fourier coefficients in a dimension incremental way. When we restrict the search space in frequency domain to a full grid [ − N , N ] d ∩ Z d of refinement N ∈ N and assume that the cardinality of the support of the trigonometric polynomial in frequency domain is bounded by the sparsity s ∈ N , our method requires O ( d s 2 N ) samples and O ( d s 3 + d s 2 N log ⁡ ( s N ) ) arithmetic operations in the case N ≲ s ≲ N d . Moreover, we discuss possibilities to reduce the number of samples and arithmetic operations by applying methods from compressed sensing and a version of Prony's method. For the latter, the number of samples is reduced to O ( d s + d N ) and the number of arithmetic operations is O ( d s 3 ) . Various numerical examples demonstrate the efficiency of the suggested method.

Direct Torque Control of Sensorless Induction Machine Drives: A Two-Stage Kalman Filter Approach

Extended Kalman filter (EKF) has been widely applied for sensorless direct torque control (DTC) in induction machines (IMs). One key problem associated with EKF is that the estimator suffers from computational burden and numerical problems resulting from high order mathematical models. To reduce the computational cost, a two-stage extended Kalman filter (TEKF) based solution is presented for closed-loop stator flux, speed, and torque estimation of IM to achieve sensorless DTC-SVM operations in this paper. The novel observer can be similarly derived as the optimal two-stage Kalman filter (TKF) which has been proposed by several researchers. Compared to a straightforward implementation of a conventional EKF, the TEKF estimator can reduce the number of arithmetic operations. Simulation and experimental results verify the performance of the proposed TEKF estimator for DTC of IMs.

Parameter Estimation of Induction Machine at Standstill Using Two-Stage Recursive Least Squares Method

This paper presents a two-stage recursive least squares (TSRLS) algorithm for the electric parameter estimation of the induction machine (IM) at standstill. The basic idea of this novel algorithm is to decouple an identifying system into two subsystems by using decomposition technique and identify the parameters of each subsystem, respectively. The TSRLS is an effective implementation of the recursive least squares (RLS). Compared with the conventional (RLS) algorithm, the TSRLS reduces the number of arithmetic operations. Experimental results verify the effectiveness of the proposed TSRLS algorithm for parameter estimation of IMs.

Study of Adaptive Detection and Channel Estimation for MIMO–OFDM Systems

Requirements for higher data rates and lower power consumption set new challenges for hardware implementation of multiple-input multiple-output orthogonal frequency division multiplexing receivers. Simple detectors have the advantage of low complexity and power consumption, but they cannot offer as good performance as more complex detectors. Therefore, it would be beneficial to be able to adapt the detector algorithm to suit the channel conditions to minimize the receiver processing power consumption while satisfying the quality of service requirements. At low signal-to-noise ratio and/or low rank channel, more power and computation resources could be used for detection in order to guarantee reliable communication, while in good conditions, a simple and less power consuming detector could be used. In this paper, we compare the performance of different detection algorithms. The performance results are based on simulations in a long term evolution system where precoding and hybrid automatic repeat request are used. The effect of channel estimation on the performance is shown. Theoretical complexities of the detectors as numbers of arithmetic operations are presented. Also hardware implementation results based on the existing literature are included in the comparison. We discuss when it would be beneficial to use a complex detector and when a simple one would be sufficient and what kind of impact the channel estimation has on the choice of the detector.

Generalized concatenated system with embedded space-time codes for MIMO systems

Multiple-input multiple-output systems are communication systems employing multiple transmitting and receiving antennas. In the present study, a new generalized concatenated signal-code construction is proposed. Its inner codes are embedded Golden codes, and its outer codes are products of Reed-Solomon codes. New algorithms for decoding inner codes, outer codes, and the generalized signal-code construction itself are proposed. The decoder for space-time Golden codes makes it possible to obtain a measure of reliability for certain symbols without an increase in the number of arithmetic operations. The decoder for outer product codes removes the “floor” on the error-probability curve. In addition, the lower bounds of decoding-error probability for the product-code iterative decoders have been obtained. The error-correcting capacity of the new decoder for the generalized concatenated construction is higher than that of the currently known decoders. A computer simulation has shown the efficiency of the proposed construction and decoding algorithms.

A new computational decoding complexity measure of convolutional codes

Abstract This paper presents a computational complexity measure of convolutional codes well suitable for software implementations of the Viterbi algorithm (VA) operating with hard decision. We investigate the number of arithmetic operations performed by the decoding process over the conventional and minimal trellis modules. A relation between the complexity measure defined in this work and the one defined by McEliece and Lin is investigated. We also conduct a refined computer search for good convolutional codes (in terms of distance spectrum) with respect to two minimal trellis complexity measures. Finally, the computational cost of implementation of each arithmetic operation is determined in terms of machine cycles taken by its execution using a typical digital signal processor widely used for low-power telecommunications applications.

A low-complexity ECG processing algorithm based on the Haar wavelet transform for portable health-care devices

This paper presents a simple Electrocardiogram (ECG) processing algorithm for portable healthcare devices. This algorithm consists of the Haar wavelet transform (HWT), the modulus maxima pair detection (MMPD) and the peak position modification (PPM). To lessen the computational complexity, a novel no multiplier structure is introduced to implement HWT. In the MMPD, the HWT coefficient at scale 24 is processed to find candidate peak positions of ECG. The PPM is designed to correct the time shift in digital process and accurately determine the location of peaks. Some new methods are proposed to improve anti-jamming performance in MMPD and PPM. Evaluated by the MIT-BIH arrhythmia database, the sensitivity (Se) of QRS detection is 99.53% and the positive prediction (Pr) of QRS detection is 99.70%. The QT database is chosen to fully validate this algorithm in complete delineation of ECG waveform. The mean µ and standard deviation σ between test results and annotations are calculated. Most of σ satisfies the CSE limits which indicates that the results are stable and reliable. A detailed and rigorous computational complexity analysis is presented in this paper. The number of arithmetic operations in N input samples is chosen as the criterion of complexity. Without any multiplication operations, the number of addition operations is only about 16.33N. This algorithm achieves high detection accuracy and the lower computational complexity.

A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets

Let $$\mathrm {R}$$ be a real closed field and $$\mathrm{D}\subset \mathrm {R}$$ an ordered domain. We present an algorithm that takes as input a polynomial $$Q \in \mathrm{D}[X_{1},\ldots ,X_{k}]$$ and computes a description of a roadmap of the set of zeros, $$\mathrm{Zer}(Q,\,\mathrm {R}^{k}),$$ of Q in $$\mathrm {R}^{k}.$$ The complexity of the algorithm, measured by the number of arithmetic operations in the ordered domain $$\mathrm{D},$$ is bounded by $$d^{O(k \sqrt{k})},$$ where $$d = \deg (Q)\ge 2.$$ As a consequence, there exist algorithms for computing the number of semialgebraically connected components of a real algebraic set, $$\mathrm{Zer}(Q,\,\mathrm {R}^{k}),$$ whose complexity is also bounded by $$d^{O(k \sqrt{k})},$$ where $$d = \deg (Q)\ge 2.$$ The best previously known algorithm for constructing a roadmap of a real algebraic subset of $$\mathrm {R}^{k}$$ defined by a polynomial of degree d has complexity $$d^{O(k^{2})}.$$

A two-grid method for an elliptic equation with boundary layers on a Shishkin mesh

A linear elliptic equation with regular and parabolic boundary layers is considered. It is solved by using an upwind difference scheme on a Shishkin mesh with the property of uniform convergence with respect to a small parameter. The problem of reducing the number of arithmetic operations required to implement the difference scheme on the basis of a two-grid method is investigated. The application of the Richardson extrapolation method to increase the accuracy of the difference scheme is studied. Results of numerical experiments are presented.

Explicit Formulas for Interpolating Splines of Degree 5 on the Triangle

Explicit formulas are derived for 21 Zlamal---Zenisek basis interpolating polynomials of degree 5 in each triangle of the triangulation. Their use significantly reduces the number of arithmetic operations in the FEM because otherwise 21 systems with 21 unknowns should be solved in each triangle to find all the 21 coefficients of each of the basis interpolating polynomials of degree 5. The formulas are also presented for interpolation operators with the use of these basis polynomials and for the integral representation of the remainder term of the approximation of differentiable functions by these operators.

Modified policy iteration algorithms are not strongly polynomial for discounted dynamic programming

This note shows that the number of arithmetic operations required by any member of a broad class of optimistic policy iteration algorithms to solve a deterministic discounted dynamic programming pr...

Compact Difference Schemes for the Diffusion and Schröodinger Equations Approximation, Stability, Convergence, Effectiveness, Monotony

Various compact difference schemes (both old and new, explicit and implicit, one-level and two-level), which approximate the diffusion equation and Schrodinger equation with periodical boundary conditions are constructed by means of the general approach. The results of numerical experiments for various initial data and right hand side are presented. We evaluate the real order of their convergence, as well as their stability, effectiveness, and various kinds of monotony. The optimal Courant number depends on the number of grid knots and on the smoothness of solutions. The competition of various schemes should be organized for the fixed number of arithmetic operations, which are necessary for numerical integration of a given Cauchy problem. This approach to the construction of compact schemes can be developed for numerical solution of various problems of mathematical physics.

A FAST FRACTIONAL DIFFERENCE ALGORITHM

AbstractWe provide a fast algorithm for calculating the fractional difference of a time series. In standard implementations, the calculation speed (number of arithmetic operations) is of order T2, where T is the length of the time series. Our algorithm allows calculation speed of order TlogT. For moderate and large sample sizes, the difference in computation time is substantial. Copyright © 2014 John Wiley & Sons, Ltd.

Communication lower bounds and optimal algorithms for numerical linear algebra

The traditional metric for the efficiency of a numerical algorithm has been the number of arithmetic operations it performs. Technological trends have long been reducing the time to perform an arithmetic operation, so it is no longer the bottleneck in many algorithms; rather, communication, or moving data, is the bottleneck. This motivates us to seek algorithms that move as little data as possible, either between levels of a memory hierarchy or between parallel processors over a network. In this paper we summarize recent progress in three aspects of this problem. First we describe lower bounds on communication. Some of these generalize known lower bounds for dense classical (O(n3)) matrix multiplication to all direct methods of linear algebra, to sequential and parallel algorithms, and to dense and sparse matrices. We also present lower bounds for Strassen-like algorithms, and for iterative methods, in particular Krylov subspace methods applied to sparse matrices. Second, we compare these lower bounds to widely used versions of these algorithms, and note that these widely used algorithms usually communicate asymptotically more than is necessary. Third, we identify or invent new algorithms for most linear algebra problems that do attain these lower bounds, and demonstrate large speed-ups in theory and practice.

Computational complexity of local posteriori inference algorithms in algebraic Bayesian networks.

The paper presents algorithmical complexity estimates for local posteriori inference in algebraic Bayesian networks. We consider the ways of implementing the inference for three types of evidence (deterministic, stochastic, and imprecise). If we need to solve linear programming tasks for inference, the comlexity estimations are given in numbers of such tasks and numbers of variables and constraints in each task. In other cases, complexity estimates are given in numbers of arithmetic operations.

Solving second-order conic systems with variable precision

We describe and analyze an interior-point method to decide feasibility problems of second-order conic systems. A main feature of our algorithm is that arithmetic operations are performed with finite precision. Bounds for both the number of arithmetic operations and the finest precision required are exhibited.

Application of Java-Called DLL in Crop Breeding Data Analysis System

The collected massive data during the breeding process usually cost plenty of human power. Therefore, it is urgent to create highly-efficient specific breeding data analysis software to assist breeders to screen fine varieties. For this reason, this paper developed a breeding data management analysis system by taking MyEclipse as development tool, which covers various statistical analysis techniques. In the development process of each algorithm module, JNI technology is adopted to realize Java-called DLL and complete algorithm call, improving the execution efficiency of a large number of arithmetical operations. This paper introduced Java-called DLL by aid of JNI technology in One-Way ANOVA in details and the parameter passing between Java and C++ and how to realize Java-called DLL in the operating system without C++ operating environment. Through the above-mentioned call, the advantages of C++ and Java are comprehensively considered, which not only lightens the burden of Java virtual machine and avoids the repeatability, but also improves the efficiency of arithmetical operation and the utilization rate of codes.

Non-Maximally Decimated Analysis/Synthesis Filter Banks: Applications in Wideband Digital Filtering

We present a new class of highly effective and low complexity digital filters for processing very wideband signals. Digital filtering for wideband signals is often limited by the number of arithmetic operations that has to be performed per input sampling interval. We will show the new architecture permits filtering to be performed on partitioned spectral segments of the input signal at significantly reduced sample rate. The digital filtering will be shown to include various tasks such as linear/non-linear phase finite impulse response (FIR) filtering, fractional delay filter among others. The proposed technique utilizes the framework of non-maximally decimated filter banks (NMDFBs) with perfect reconstruction (PR) property, which makes the filter bank design simpler and more flexible. Compact representation for the generalized DFT based NMDFBs as well as its efficient polyphase implementation will be provided. The digital filtering is made possible by incorporating the desired intermediate processing elements in between the analysis and synthesis filter banks. We will show this embedded intermediate processing elements can achieve spectral shaping/signal manipulation task, which is a generalization of the concept of manipulating a digital filter in the frequency domain. We also analyze both analytically and experimentally, the spectral shaping accuracy based on different design strategies.

Method of optimization in flight dynamics

Consideration was given to the problem of suboptimal control of the flight vehicle in atmosphere which was solved numerically. The results presented corroborate the efficiency of this procedure. The present paper is distinguished for the fact that the complex nonlinear boundary problem was solved using a finite number of arithmetic operations.