Polynomial Acceleration Research Articles

Modified CRI iteration methods for complex symmetric indefinite linear systems

This work investigates the iterative solution of complex symmetric linear systems with indefinite matrix term. Based on a technical equivalent reformulation of the original indefinite systems, an efficient stationary splitting iteration method is established by combining with its real and imaginary parts. In the view of the structure, the new method can be viewed as a modification of the combination method of the real and imaginary parts (CRI) in [Wang & Lu, J Comput Appl Math. 2017;325:188–197], which is originally designed for solving complex symmetric linear systems with positive semi-definite coefficient matrices. Despite that it is devoted to more difficult indefinite problems, the new method keeps the merits of unconditional convergence and parameter free implementation of the original method. Moreover, Chebyshev polynomial acceleration is considered to further improve the convergence rate of the new method. Numerical experiments on a set of model problems are tested to illustrate the efficiency of the proposed methods compared with some existing ones.

A quantum-enhanced solution method for multi classification problems

With the increasing data size of multi classification problems, the running efficiency of classical algorithms is seriously affected. In the paper, in order to improve the implementation efficiency of the algorithm, we propose a quantum-enhanced solution method for multi classification problems. The method mainly introduces the quantum-enhanced technology into the classical algorithm. Aiming at the two steps of solving Euclid distance and kernel function in the classical algorithm, the paper relates the classical inner product principle with the amplitude evolution of quantum states. On the basis of quantizing the sample data, a general quantum circuit that can calculate the inner product is designed and constructed. The circuit can make full use of the advantages of quantum parallel computing to achieve exponential acceleration of computing efficiency. Aiming at solving linear equations in the classical algorithm, a quantum circuit based on the quantum singular value estimation is designed and constructed. The circuit makes use of the acceleration advantage of quantum computing in matrix computing to achieve polynomial acceleration of matrix computing. The experimental results show that the method can not only find the optimal solution for multi classification problems, but also greatly improve the operation efficiency of the algorithm. Compared with the classical methods, the method has at least polynomial improvement in time complexity and spatial complexity.

A Hybrid Quantum-Classical Algorithm for Underwater Target Classification

Offshore channel clearance is an essential underwater task to protect vessels and divers effectively, but current underwater target classification relies heavily on operator identification. Machine learning provides highly accurate methods for image classification as well as detection. In this paper, a new hybrid quantum-classical classification algorithm is proposed. It uses quantum devices to reduce dimension and denoise data sets, greatly reducing the difficulty of classical computer processing data. Using abundant classical classification algorithms, the classification problem of different scenarios can be processed, improving the classification efficiency. Using two kinds of underwater object data sets as examples, the numerical simulation results show that the quantum algorithm can accurately achieve dimensionality reduction. This hybrid algorithm has polynomial acceleration in dimension reduction than classical methods, even considering the classical readout of quantum data. The results also show that the classification accuracy of the training set improves from 0.772 to 0.821 compared to the original dataset. Furthermore, different classical classifiers can be selected in the case of different objects, so this hybrid algorithm has broad application prospects in different fields.

Accelerated capacity model of lithium-ion battery based on non-linear polynomial method with stress coupling analysis under two electrical variables

For lithium-ion batteries, accelerated degradation test (ADT) plays a pivotal role for their long cycle life and high reliability. In some applications, the degradation of batteries is affected by the electrical stresses. Hence, the extended Eyring model applied with the temperature coupling other non-thermal stress, is not appropriate. Motivated by this idea, a non-linear polynomial method with stress coupling analysis (NLP-SCA) model for double electrical variables is proposed as it can extend the application scenarios of current acceleration model. Meanwhile, the NLP-SCA is also developed with better goodness-of-fit compared with the models, by which the stress couplings are ignored. For the life estimation with normal condition, four evaluation criterions (root mean square error, R-Squared, sum of squares due to error, and the mean absolute error) calculated with the proposed model increase by 45.37%, 42.15%, 3.94% and 8.17% separately, compared to the general polynomial acceleration model and exponential acceleration model.

Efficient iterative massive MIMO detection using Chebyshev acceleration

Massive multiple-input multiple-output (MIMO) detection is one of the most important, yet complex parts of the fifth generation (5G) baseband receiver. The linear minimum mean square error (MMSE) signal detection achieves almost optimum efficiency when the number of antennas at the base station is asymptotically large. However, the matrix inversion required for MMSE can also be very complex when the number of users increases. In this paper, a low complexity signal detection algorithm based on modified accelerated overrelaxation (MAOR) method is proposed to iteratively approach the MMSE performance. We calculate optimal values of two key parameters of MAOR and also provide a suitable and less complex initial solution to accelerate the convergence. Furthermore, we adopt the Chebyshev polynomial acceleration technique to present the MAOR method with a new vector combinations, which enhances the performance of the detection algorithm. The spectral radius of MAOR is also calculated to demonstrate its suitability for Chebyshev acceleration. This complete solution is referred to as Chebyshev-MAOR. The results have revealed that the proposed method can achieve faster convergence and better performance than other state-of-the-art detection algorithms. It is also shown that Chebyshev-MAOR reduces computational complexity by an order of magnitude from O(K3) to O(K2), with K denoting the number of transmit antennas. Our performance results show that these complexity gains are achieved with negligible impact on the bit error rate (BER) performance.

Porting an aggregation-based algebraic multigrid method to GPUs

We present a hybrid GPU-CPU version of the AGMG software, a popular algebraic multigrid (AMG) solver which implements an aggregation-based AMG method. With the new implementation, the solution stage runs on a GPU, except operations on the coarsest grid, which are executed on a CPU. To maximize the speedup, two novel %new features are introduced. On the one hand, ℓ1-Jacobi smoothing is combined with polynomial acceleration (or polynomial smoothing), leading to improved performance compared with standard ℓ1-Jacobi smoothing, while not requiring to compute eigenvalue estimates as standard polynomial smoothing does. On the other hand, besides the K-cycle used in standard AGMG, we introduce the relaxed W-cycle, which tends to combine the advantages of the K-cycle and the standard W-cycle. Numerical results show that the new implementation inherits the robustness of the original AGMG software, while bringing significant speedups on GPUs. A comparison with AmgX, a reference AMG solver from NVIDIA, suggests that the presented hybrid GPU-CPU version of AGMG is more robust and often significantly faster in the solution stage.

On relaxed filtered Krylov subspace method for non-symmetric eigenvalue problems

In this paper, by introducing a class of relaxed filtered Krylov subspaces, we propose the relaxed filtered Krylov subspace method for computing the eigenvalues with the largest real parts and the corresponding eigenvectors of non-symmetric matrices. As by-products, the generalizations of the filtered Krylov subspace method and the Chebyshev–Davidson method for solving non-symmetric eigenvalue problems are also presented. We give the convergence analysis of the complex Chebyshev polynomial, which plays a significant role in the polynomial acceleration technique. In addition, numerical experiments are carried out to show the robustness of the relaxed filtered Krylov subspace method and its great superiority over some state-of-the-art iteration methods.

ChASE

Solving dense Hermitian eigenproblems arranged in a sequence with direct solvers fails to take advantage of those spectral properties that are pertinent to the entire sequence and not just to the single problem. When such features take the form of correlations between the eigenvectors of consecutive problems, as is the case in many real-world applications, the potential benefit of exploiting them can be substantial. We present the Chebyshev Accelerated Subspace iteration Eigensolver (ChASE), a modern algorithm and library based on subspace iteration with polynomial acceleration. Novel to ChASE is the computation of the spectral estimates that enter in the filter and an optimization of the polynomial degree that further reduces the necessary floating-point operations. ChASE is written in C++ using the modern software engineering concepts that favor a simple integration in application codes and a straightforward portability over heterogeneous platforms. When solving sequences of Hermitian eigenproblems for a portion of their extremal spectrum, ChASE greatly benefits from the sequence’s spectral properties and outperforms direct solvers in many scenarios. The library ships with two distinct parallelization schemes, supports execution over distributed GPUs, and is easily extensible to other parallel computing architectures.

Experimental observations of 1D quantum walks in a limited region

The quantum walk (QW) is the quantum counterpart of the classical walk (CW) and is widely used in universal quantum computations. QWs provide exponential acceleration in hitting times and polynomial acceleration in searching times. Taking advantage of the integration and stability of waveguide structures, large-scale QWs can be implemented on chips. Here, we simulate both CWs and QWs in limited regions to qualitatively analyze the boundary conditions. Subsequently, we used a silicon chip to achieve 21-step QWs in a limited region and experimentally measured the photon number distributions by utilizing the fiber-coupling platform. Then we simulated the theoretical result by using the splitting ratio of the directional couplers which was experimentally obtained by measuring an individual directional coupler and the phase difference of the boundary waveguides which was fit from the experimental results. We also simulated the photon number distributions that are caused by the parameters of the waveguides, such as the splitting ratio, the phase difference, and the size of the walking region.

Data fusion based multi-rate Kalman filtering with unknown input for on-line estimation of dynamic displacements

Structural dynamic displacement is one of the most important measurands that describe the dynamic characteristics of a structure. However, accurate measurement of dynamic displacements of a civil infrastructure is still a challenging task. To solve the difficulties and drawbacks of direct dynamic displacement measurement, the approach of multi-rate Kalman filtering for the data fusion of displacement and acceleration measurement was developed. Recently, an improved technique for dynamic displacement estimation by fusing biased high-sampling rate acceleration and low-sampling rate displacement measurements has been proposed. However, this technique can only take constant acceleration bias into account. In this paper, based on the algorithm of Kalman filter with unknown input recently developed by the authors, dynamic displacement is on line estimated based on multi-rate data fusion of high-sampling rate acceleration with time-varying bias and low-sampling rate displacement measurements. The time history of time-varying acceleration bias is treated as “unknown input” in the algorithm of Kalman filter with unknown input to overcome the limitations of the previous technique. Some numerical examples with linear or polynomial acceleration bias are used to demonstrate the effectiveness of the proposed approach for on line estimation of structural dynamic displacement.

Polynomial Accelerated Solutions to a Large Gaussian Model for Imaging Biofilms: In Theory and Finite Precision

ABSTRACTThree-dimensional confocal scanning laser microscope images offer dramatic visualizations of living biofilms before and after interventions. Here, we use confocal microscopy to study the effect of a treatment over time that causes a biofilm to swell and contract due to osmotic pressure changes. From these data (the video is provided in the supplementary materials), our goal is to reconstruct biofilm surfaces, to estimate the effect of the treatment on the biofilm’s volume, and to quantify the related uncertainties. We formulate the associated massive linear Bayesian inverse problem and then solve it using iterative samplers from large multivariate Gaussians that exploit well-established polynomial acceleration techniques from numerical linear algebra. Because of a general equivalence with linear solvers, these polynomial accelerated iterative samplers have known convergence rates, stopping criteria, and perform well in finite precision. An explicit algorithm is provided, for the first time, for an iterative sampler that is accelerated by the synergistic implementation of preconditioned conjugate gradient and Chebyshev polynomials. Supplementary materials for this article are available online.

Accelerated Gibbs sampling of normal distributions using matrix splittings and polynomials

Standard Gibbs sampling applied to a multivariate normal distribution with a specified precision matrix is equivalent in fundamental ways to the Gauss–Seidel iterative solution of linear equations in the precision matrix. Specifically, the iteration operators, the conditions under which convergence occurs, and geometric convergence factors (and rates) are identical. These results hold for arbitrary matrix splittings from classical iterative methods in numerical linear algebra giving easy access to mature results in that field, including existing convergence results for antithetic-variable Gibbs sampling, REGS sampling, and generalizations. Hence, efficient deterministic stationary relaxation schemes lead to efficient generalizations of Gibbs sampling. The technique of polynomial acceleration that significantly improves the convergence rate of an iterative solver derived from a symmetric matrix splitting may be applied to accelerate the equivalent generalized Gibbs sampler. Identicality of error polynomials guarantees convergence of the inhomogeneous Markov chain, while equality of convergence factors ensures that the optimal solver leads to the optimal sampler. Numerical examples are presented, including a Chebyshev accelerated SSOR Gibbs sampler applied to a stylized demonstration of low-level Bayesian image reconstruction in a large 3-dimensional linear inverse problem.

Convergence in Variance of Chebyshev Accelerated Gibbs Samplers

A stochastic version of a stationary linear iterative solver may be designed to converge in distribution to a probability distribution with a specified mean $\mu$ and covariance matrix $A^{-1}$. A common example is Gibbs sampling applied to a multivariate Gaussian distribution which is a stochastic version of the Gauss--Seidel linear solver. The iteration operator that acts on the error in mean and covariance in the stochastic iteration is the same iteration operator that acts on the solution error in the linear solver, and thus both the stationary sampler and the stationary solver have the same error polynomial and geometric convergence rate. The polynomial acceleration techniques that are well known in numerical analysis for accelerating the linear solver may also be used to accelerate the stochastic iteration. We derive first-order and second-order Chebyshev polynomial acceleration for the stochastic iteration to accelerate convergence in the mean and covariance by mimicking the derivation for the linear solver. In particular, we show that the error polynomials are identical and hence so are the convergence rates. Thus, optimality of the Chebyshev accelerated solver implies optimality of the Chebyshev accelerated sampler. We give an algorithm for the stochastic version of the second-order Chebyshev accelerated SSOR (symmetric successive overrelaxation) iteration and provide numerical examples of sampling from multivariate Gaussian distributions to confirm that the desired convergence properties are achieved in finite precision.

The Applied Research of Four Polynomial Acceleration and Deceleration Control Method in Multi-Axis DSP Motion Controller

In traditional NC systems, the linear, index and S curve acceleration and deceleration control methods exist lots of problems that jerk is discontinuous or algorithm is too complex, seeking a more superior acceleration and deceleration control method to make up these shortages is needed. In this paper, the Four polynomial acceleration and deceleration control method is researched further, some contrast experiments have been done on NC terrace which is based on 32 bit floating point multi-axis DSP motion controller, and the superiority of this method and the practicability in NC field are analysed by processing two hemispheres with H600 CNC engraving and milling machine.

Chebyshev polynomial acceleration for block SOR methods for solving the rank-deficient least-squares problem

In this article, we give the acceleration of the block successive overrelaxation (SOR) method for solving the rank-deficient least-squares problem. Santos and Silva proposed the two-block SOR method and the three-block SOR method. Here, we consider the acceleration of the two-block SOR method and the three-block SOR method using the Chebyshev polynomial and derive what we term the C-2-block SOR method and the C-3-block SOR method. The advantage of our methods is that we can get good results with very small iteration number. The comparison between the C-2-block method and the C-3-block method is presented. Finally, numerical examples are given.

A new type of quadratic acceleration method

The polynomial acceleration method for time history analysis is presented, in which accelerations between several equal neighboring time steps were assumed to be a polynomial function of time interval. With a higher order polynomial used, a higher accuracy can be obtained, but the stability field of the method becomes narrower. When stability field and computational accuracy are taken into account at the same time, the quadratic acceleration method is superior to linear and cubic acceleration methods in choosing the maximum acceptable time step size. It is also shown that the quadratic acceleration method has desirable arithmetic damp, amplitude decay rate and period elongation rate, though its conditional stability restricts its application in stiff structures.

Adaptive Prospective Interpolation Method for High Speed Machining of Micro Line Blocks Based on the Cubic Polynomial Model

An adaptive prospective interpolation method for high speed machining of micro line blocks, including interpolation preprocess and real time parametric interpolation, is proposed, which is based on the cubic polynomial model. During interpolation preprocess, the high speed adaptive prospective control strategy is established by using path transfer point maximum velocity confirming, adaptive predetermination of deceleration point position and stride segment transfer point speed checking. During real time interpolation, stride multi-path parametric interpolation algorithm is set up based on the cubic polynomial acceleration and deceleration control model. The results demonstrate the proposed method realizes high speed machining of consecutive micro line blocks and determination of deceleration point position in advance. The method achieves high speed machining and the productivity is improved significantly. It has been applied in multi-axis high speed micro machining system successfully.

FPGA implementation of higher degree polynomial acceleration profiles for peak jerk reduction in servomotors

Acceleration profile generation for jerk limitation is a major issue in automated industrial applications like computer numerical control (CNC) machinery and robotics. The automation machinery dynamics should be kept as smooth as possible with suitable controllers where trajectory precision ensures quality while smoothness decreases machinery stress. During the operation of commercially available CNC and robotics controllers, small discontinuities on the dynamics are generated due to the controller position profiler which is generally based on a trapezoidal velocity profile. These discontinuities can produce undesirable high-frequency harmonics on the position reference which consequentially can excite the natural frequencies of the mechanical structure and servomotors. Previous works have developed jerk limited trajectories with higher degree polynomial-based profiles, but lack one or both of computer efficiency for on-line operation and low-cost hardware implementation. The present work shows a low cost, computationally efficient, on-line hardware implementation of a high-degree polynomial-based profile generator with limited jerk dynamics for CNC machines and robotics applications to improve the machining process. The novelty of the paper is the development of a multiplier-free recursive algorithm for computationally efficient polynomial evaluation in profile generation and a low-cost implementation of the digital structure in field programmable gate array (FPGA). Two experimental setups were prepared in order to test the polynomial profile generator: the first one with the servomotor at no load and the second one for the servomotor driving a CNC milling machine axis. From experimental results it is shown that higher degree polynomial profiles, compared to the standard trapezoidal speed profile improve the system dynamics by reducing peak jerk in more than one order of magnitude while precision is maintained the same and on-line operation is guaranteed.

Research on cubic polynomial acceleration and deceleration control model for high speed NC machining

To satisfy the need of high speed NC (numerical control) machining, an acceleration and deceleration (acc/dec) control model is proposed, and the speed curve is also constructed by the cubic polynomial. The proposed control model provides continuity of acceleration, which avoids the intense vibration in high speed NC machining. Based on the discrete characteristic of the data sampling interpolation, the acc/dec control discrete mathematical model is also set up and the discrete expression of the theoretical deceleration length is obtained furthermore. Aiming at the question of hardly predetermining the deceleration point in acc/dec control before interpolation, the adaptive acc/dec control algorithm is deduced from the expressions of the theoretical deceleration length. The experimental result proves that the acc/dec control model has the characteristic of easy implementation, stable movement and low impact. The model has been applied in multi-axes high speed micro fabrication machining successfully.

A strategy for detecting extreme eigenvalues bounding gaps in the discrete spectrum of self-adjoint operators

For a self-adjoint linear operator with a discrete spectrum or a Hermitian matrix, the “extreme” eigenvalues define the boundaries of clusters in the spectrum of real eigenvalues. The outer extreme ones are the largest and the smallest eigenvalues. If there are extended intervals in the spectrum in which no eigenvalues are present, the eigenvalues bounding these gaps are the inner extreme eigenvalues. We will describe a procedure for detecting the extreme eigenvalues that relies on the relationship between the acceleration rate of polynomial acceleration iteration and the norm of the matrix via the spectral theorem, applicable to normal matrices. The strategy makes use of the fast growth rate of Chebyshev polynomials to distinguish ranges in the spectrum of the matrix which are devoid of eigenvalues. The method is numerically stable with regard to the dimension of the matrix problem and is thus capable of handling matrices of large dimension. The overall computational cost is quadratic in the size of a dense matrix; linear in the size of a sparse matrix. We verify computationally that the algorithm is accurate and efficient, even on large matrices.