Parallel Polynomial Research Articles

A Basic Study of the Parallel Polynomial Approximation-based Hologram Computation Method with a GPU

A Basic Study of the Parallel Polynomial Approximation-based Hologram Computation Method with a GPU

Aerodynamic Performance Uncertainty Analysis and Optimization of a Conventional Axisymmetric Vehicle Based on Parallel Polynomial Chaos Expansions

In this study, the aerodynamic uncertainty analysis and optimization of a conventional axisymmetric vehicle with an aerodynamic configuration were investigated. The prediction precision of the typical aerodynamic performance estimating methods, namely, engineering estimation and numerical simulation, was compared using the wind tunnel test data of the vehicle. Then, using a modified missile data compendium (DATCOM) software, a high-efficiency and high-precision method was developed, which was applied to analyze and characterize the aerodynamic parameters of the axisymmetric vehicle. To enhance the robustness and reliability of aerodynamic performance, an uncertainty-based design optimization (UDO) framework was established. The design space was scaled by parameter sensitivity analysis, and improved computational efficiency was achieved by developing parallel polynomial chaos expansions (PCEs). The optimized results show that the modified method exhibits high accuracy in predicting aerodynamic performance. For the same constraints, the results of the deterministic design optimization (DDO) showed that compared with the initial scheme, the probability of the controllability-to-stability ratio satisfying the constraint decreased from 98.8% to 72.4%, and this value increased to 99.9% in the case of UDO. Compared with the results of the initial scheme and DDO, UDO achieved a considerable reduction in mean values and standard deviation of aerodynamic performances, which can ensure a higher probability of constraints meeting the design requirements, thereby, realizing a reliable and robust design.

Low complexity design of bit parallel polynomial basis systolic multiplier using irreducible polynomials

Encryption schemes like AES require finite field modular multiplication. The encryption speed is highly dependent on the performance of the finite field multiplier. Several high-speed systolic bit parallel multipliers with low area complexity have been proposed in the literature. In this paper, a modular multiplication algorithm is used to propose a bit parallel polynomial basis systolic multiplier which has achieved 89% less and 17% less area-delay product than the best existing multipliers. It has been observed that the area complexity of the proposed systolic multiplier for irreducible polynomials matches with the best-reported multiplier with 17% less time complexity. The results are further verified with the help of the FPGA implementation of the proposed multiplier for m = 8,163. Being generic, the proposed multiplier can be optimized further for trinomials and pentanomials.

A low complexity bit parallel polynomial basis systolic multiplier for general irreducible polynomials and trinomials

The bit parallel multiplication scheme is characterized by an important feature called concurrency, which makes its execution faster. The output through this scheme is generated in every clock cycle, after taking ‘m’ clock cycles in the beginning. In this paper, a polynomial basis systolic multiplier over irreducible polynomials (xm+tm−1xm−1+....t2x2+t1x1+t0) in GF(2m) is designed specifically for odd ‘m’, which takes ‘m’ clock cycles and this is further, used to design the low complexity bit-parallel architecture for general irreducible polynomials and trinomials (xm+xk+1). The area complexity of the proposed multiplier for general irreducible polynomials matches with the best existing multiplier with a 17% reduction in time complexity due to a considerable decrease in critical path delay. To the best of our knowledge, the area-delay product of the proposed multipliers is the lowest achieved when compared with the best multipliers available in the literature. FPGA implementation results also show that the proposed multiplier has 59% less space complexity and 47% less time complexity than the best-reported multiplier for m = 233.

Critique of “Planetary Normal Mode Computation: Parallel Algorithms, Performance, and Reproducibility” by SCC Team From ETH Zurich

This report analyzes the reproducibility of the article “Computing Planetary Interior Normal Modes with A Highly Parallel Polynomial Filtering Eigensolver” by Jia Shi et al. (Shi, 2018). To reproduce the results we perform different weak and strong scaling studies using a series of Mars models. All experimental runs were performed during the SC19 Student Cluster Competition on a four node Intel Skylake cluster. We show that the findings of the original article can be reproduced in a different environment.

Critique of “Planetary Normal Mode Computation: Parallel Algorithms, Performance, and Reproducibility” by SCC Team From University of Washington

One of the tasks for the SC19 Student Cluster Competition is to reproduce the results in the reproducibility challenge article “Computing Planetary Interior Normal Modes with a Highly Parallel Polynomial Filtering Eigensolver”, by J. Shi et al., which describes a highly parallel algorithm for computing planetary normal modes. In running experiments from the article, we study the weak and strong scalability of the algorithm, as well as the relationship between model size, degree of polynomial filter, and execution time. We investigate these findings on a two-node, 64-core Intel Skylake-based Xeon cluster. Unfortunately, we are able to confirm some, but not all, of the original findings, with discrepancies possibly due to a low number of experimental runs due to competition time limits as well as nonuniform scaling of compute resources.

Critique of “Planetary Normal Mode Computation: Parallel Algorithms, Performance, and Reproducibility” by SCC Team From University of Warsaw

The Supercomputing conference holds a student cluster competition - A competition that aims to introduce undergraduate students to the world of High Performance Computing. One of the tasks at the SC19 conference was to reproduce the scaling results obtained in the article titled Computing Planetary Interior Normal Modes with a Highly Parallel Polynomial Filtering Eigensolver by Jia Shi et al. They introduced a new approach to the problem of calculating planetary normal modes. The developed method is a highly parallel algorithm that approximates the results via the mixed finite element method on unstructured tetrahedral meshes. The cluster used consisted of five 40-core Intel Xeon Cascade Lake nodes, equipped with 384 GB of RAM each. The original study was demonstrated on Stampede2 system on partitions ranging from 2 to 256 nodes with 48-core Intel Xeon Skylake and 192 GB of RAM each. Our cluster was equipped with Infiniband interconnect while Stampede2 used Intel Omni-Path network. We decided to work with HPC-X MPI library in place of Intel MPI used the reference study. The design - sizes and schedule - of experimental runs was the most challenging part of our study. We run both strong and weak scalability experiments in a one set of runs in order to fit in limited computing capacity and very tight time schedule. Due to limitations of our hardware, we had to split the weak scaling study into two separate smaller studies. We didn't manage to reproduce the exact results, however, we achieved similar scalability trend.

Planetary Normal Mode Computation: Parallel Algorithms, Performance, and Reproducibility

This article is an extension of work entitled “Computing planetary interior normal modes with a highly parallel polynomial filtering eigensolver.” by Shi <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> , <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> originally presented at the SC18 conference. A highly parallel polynomial filtered eigensolver was developed and exploited to calculate the planetary normal modes. The proposed method is ideally suited for computing interior eigenpairs for large-scale eigenvalue problems as it greatly enhances memory and computational efficiency. In this article, the second-order finite element method is used to further improve the accuracy as only the first-order finite element method was deployed in the previous work. The parallel algorithm, its parallel performance up to 20k processors, and the great computational accuracy are illustrated. The reproducibility of the previous work was successfully performed on the Student Cluster Competition at the SC19 conference by several participant teams using a completely different Mars-model dataset on different clusters. Both weak and strong scaling performances of the reproducibility by the participant teams were impressive and encouraging. The analysis and reflection of their results are demonstrated and future direction is discussed.

Critique of “Planetary Normal Mode Computation: Parallel Algorithms, Performance, and Reproducibility” by SCC Team From Tsinghua University

In this article we present our results from the SC19 Student Cluster Competition Reproducibility Challenge. The challenge entails reproducing the article entitled “Computing Planetary Interior Normal Modes with A Highly Parallel Polynomial Filtering Eigensolver” presented at SC'18, which proposes a parallel polynomial filtered Lanczos algorithm to directly calculate the planetary normal modes of heterogeneous planets. The proposed algorithm showed excellent performance with relatively low memory consumption and high parallel efficiency. In this work, we reproduce the scaling tests in that article on a cluster using Intel Cascade Lake architecture and use the proposed algorithm to illustrate specific normal modes of Mars. We compare the results obtained on our cluster with those in the original article. We also design a new metric to better analyze the results. In addition, we use the profiling tool Intel VTune Amplifier to explain our discoveries. Our results demonstrate that the given models show great scalability, which is similar to the original article. The required normal modes of Mars are also successfully calculated and visualized.

Critique of “Planetary Normal Mode Computation: Parallel Algorithms, Performance, and Reproducibility” by SCC Team From Peking University

Shi et al. (2018) proposed a highly parallel polynomial filtering eigensolver for the computation of planetary normal modes. As a challenge at the Student Cluster Competition in The International Conference for High Performance Computing, Networking, Storage and Analysis (SC19), we reproduce the computational efficiency of the polynomial filtering eigensolver on our Intel Xeon machine. We present the weak scalability, scaling of runtime with model size (in a fixed interval) and the strong scalability results in this report.

An Ultra-Highly Parallel Polynomial Multiplier for the Bootstrapping Algorithm in a Fully Homomorphic Encryption Scheme

Fully homomorphic encryption (FHE) is a post-quantum secure cryptographic technology that enables privacy-preserving computing on an untrusted platform without divulging any secret or sensitive information. The core of FHE is the bootstrapping algorithm, which is the intermediate refreshing procedure of a processed ciphertext. However, this step has been the computational bottleneck that prevents real-world deployments among various FHE schemes. This paper, to the best of our knowledge, for the first time, presents a scalable and ultra-highly parallel design for the number theoretic transform (NTT)-based polynomial multiplier with a variable number of reconfigurable processing elements (PEs). Hence, the highest degree of acceleration can be achieved for any targeted hardware platform by implementing as many PEs as possible under the resource constraint. The corresponding addressing and scheduling schemes are also proposed to avoid memory access conflict for the PEs, which yields an extremely high utilization ratio of 99.18% on average. In addition, the latency of the proposed design with the general negative wrapped convolution algorithm is reduced by 59.20% compared to prior works.

High-speed Instruction-set Coprocessor for Lattice-based Key Encapsulation Mechanism: Saber in Hardware

In this paper, we present an instruction set coprocessor architecture for lattice-based cryptography and implement the module lattice-based post-quantum key encapsulation mechanism (KEM) Saber as a case study. To achieve fast computation time, the architecture is fully implemented in hardware, including CCA transformations. Since polynomial multiplication plays a performance-critical role in the module and ideal lattice-based public-key cryptography, a parallel polynomial multiplier architecture is proposed that overcomes memory access bottlenecks and results in a highly parallel yet simple and easy-to-scale design. Such multipliers can compute a full multiplication in 256 cycles, but are designed to target any area/performance trade-offs. Besides optimizing polynomial multiplication, we make important design decisions and perform architectural optimizations to reduce the overall cycle counts as well as improve resource utilization. For the module dimension 3 (security comparable to AES-192), the coprocessor computes CCA key generation, encapsulation, and decapsulation in only 5,453, 6,618 and 8,034 cycles respectively, making it the fastest hardware implementation of Saber to our knowledge. On a Xilinx UltraScale+ XCZU9EG-2FFVB1156 FPGA, the entire instruction set coprocessor architecture runs at 250 MHz clock frequency and consumes 23,686 LUTs, 9,805 FFs, and 2 BRAM tiles (including 5,113 LUTs and 3,068 FFs for the Keccak core).

Predicting the Kentucky Derby Winner! Sort of

Through a series of explorations, this article will demonstrate how the Kentucky Derby winning times dataset provides various opportunities for introductory and advanced topics, from data processing to model building. Although the final goal may be a prediction interval, the dataset is rich enough for it to appear in several places in an introductory or second course in statistics. After adjusting for the change in track length and track condition, winning speed has an interesting nonlinear trend, with one notable outlier. Student investigations can range from validating the phrase “the most exciting two minutes in sports” to predicting the winning speed of next year’s race using parallel polynomial models.

New Block Recombination for Subquadratic Space Complexity Polynomial Multiplication Based on Overlap-Free Approach

In this paper, we present new parallel polynomial multiplication formulas which result in subquadratic space complexity. The schemes are based on a recently proposed block recombination of polynomial multiplication formula. The proposed two-way, three-way, and four-way split polynomial multiplication formulas achieve the smallest space complexities. Moreover, by providing area-time tradeoff method, the proposed formulas enable one to choose a parallel formula for polynomial multiplication which is suited for a design environment.

Review of Parallel Polynomial Multiplier based on FFT using Indian Vedic Mathematics

most of the operations performed by any complex system need a multiplier. Hence, multiplier based on FFT is the desired aim. In this paper, we have presented a review of parallel polynomial multiplier based on FFT using Indian Vedic mathematics. Parallel polynomial multipliers were optimized for throughput and area resources, respectively. These multipliers are used for multiplication of different polynomial numbers based on exponential type, power type, etc. FFT system is used for multiplication so complex multiplier is the main part of this design. The coding of the design can be done in VHDL. For synthesis and simulation of the design Xilinx ISE EDA tool can be used.

Castelnuovo–Mumford Regularity and Computing the de Rham Cohomology of Smooth Projective Varieties

We describe a parallel polynomial time algorithm for computing the topological Betti numbers of a smooth complex projective variety X. It is the first single exponential time algorithm for computing the Betti numbers of a significant class of complex varieties of arbitrary dimension. Our main theoretical result is that the Castelnuovo–Mumford regularity of the sheaf of differential p-forms on X is bounded by p(em+1)D, where e, m, and D are the maximal codimension, dimension, and degree, respectively, of all irreducible components of X. It follows that, for a union V of generic hyperplane sections in X, the algebraic de Rham cohomology of X∖V is described by differential forms with poles along V of single exponential order. By covering X with sets of this type and using a Čech process, we obtain a similar description of the de Rham cohomology of X, which allows its efficient computation. Furthermore, we give a parallel polynomial time algorithm for testing whether a projective variety is smooth.

A new parallel polynomial division by a separable polynomial via hermite interpolation with applications, pp. 770-781.

A new parallel division of polynomials by a common separable divisor over a perfect field is presented and this is done by expressing the remainders as derivatives of a unique polynomial. In order to get this result, a novel variant expression of the classical Lagrange–Sylvester Hermite interpolating polynomial has been utilised, although any known variant may be used. The above findings are utilized to obtain a number of new identities involving polynomial derivatives, including a closed formula for the semi–simple part of the Jordan decomposition of a matrix.

A fuzzy ensemble of parallel polynomial neural networks with information granules formed by fuzzy clustering

In this paper, we introduce a new category of fuzzy models called a fuzzy ensemble of parallel polynomial neural network (FEP 2N 2), which consist of a series of polynomial neural networks weighted by activation levels of information granules formed with the use of fuzzy clustering. The two underlying design mechanisms of the proposed networks rely on information granules resulting from the use of fuzzy C-means clustering (FCM) and take advantage of polynomial neural networks (PNNs). The resulting model comes in the form of parallel polynomial neural networks. In the design procedure, in order to estimate the optimal values of the coefficients of polynomial neural networks we use a weighted least square estimation algorithm. We incorporate various types of structures as the consequent part of the fuzzy model when using the learning algorithm. Among the diverse structures being available, we consider polynomial neural networks, which exhibit highly nonlinear characteristics when being viewed as local learning models. We use FCM to form information granules and to overcome the high dimensionality problem. We adopt PNNs to find the optimal local models, which can describe the relationship between the input variables and output variable within some local region of the input space. We show that the generalization capabilities as well as the approximation abilities of the proposed model are improved as a result of using polynomial neural networks. The performance of the network is quantified through experimentation in which we use a number of benchmarks already exploited within the realm of fuzzy or neurofuzzy modeling.

Parallel Time and Quantifier Prefixes

We characterize the amount of alternation between blocks of Boolean quantifiers (having both existential and universal), blocks of real existential quantifiers, and blocks of real universal quantifiers that can be decided in parallel polynomial time over the reals. We do so under the assumption that blocks have a uniform bound on their size, both for the case of this bound to be polynomial and constant. On the way towards this characterization we prove a real version of Savitch’s Theorem.

VPSPACE and a Transfer Theorem over the Reals

We introduce a new class VPSPACE of families of polynomials. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main theorem is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class $$\sf {PAR}_{\mathbb {R}}$$of decision problems that can be solved in parallel polynomial time over the real numbers collapses to $$\sf{P}_{\mathbb {R}}$$. As a result, one must first be able to show that there are VPSPACE families which are hard to evaluate in order to separate $$\sf{P}_{\mathbb {R}}$$from $$\sf{NP}_{\mathbb {R}}$$, or even from $$\sf{PAR}_{\mathbb {R}}$$.