Vector Multiplication Research Articles

Large-scale focusing joint inversion of gravity and magnetic data with Gramian constraint

SUMMARYA fast algorithm for the large-scale joint inversion of gravity and magnetic data is developed. The algorithm uses a non-linear Gramian constraint to impose correlation between the density and susceptibility of the reconstructed models. The global objective function is formulated in the space of the weighted parameters, but the Gramian constraint is implemented in the original space, and the non-linear constraint is imposed using two separate Lagrange parameters, one for each model domain. It is significant that this combined approach, using the two spaces provides more similarity between the reconstructed models. Moreover, it is shown theoretically that the gradient for the use of the unweighted space is not a scalar multiple of that used for the weighted space, and hence cannot be accounted for by adjusting the Lagrange parameters. It is assumed that the measured data are obtained on a uniform grid and that a consistent regular discretization of the volume domain is imposed. Then, the sensitivity matrices exhibit a block-Toeplitz-Toeplitz-block structure for each depth layer of the model domain, and both forward and transpose operations with the matrices can be implemented efficiently using two dimensional fast Fourier transforms. This makes it feasible to solve for large scale problems with respect to both computational costs and memory demands, and to solve the non-linear problem by applying iterative methods that rely only on matrix–vector multiplications. As such, the use of the regularized reweighted conjugate gradient algorithm, in conjunction with the structure of the sensitivity matrices, leads to a fast methodology for large-scale joint inversion of geophysical data sets. Numerical simulations demonstrate that it is possible to apply a non-linear joint inversion algorithm, with Lp-norm stabilisers, for the reconstruction of large model domains on a standard laptop computer. It is demonstrated, that while the p = 1 choice provides sparse reconstructed solutions with sharp boundaries, it is also possible to use p = 2 in order to provide smooth and blurred models. The methodology is used for inverting gravity and magnetic data obtained over an area in northwest of Mesoproterozoic St Francois Terrane, southeast of Missouri, USA.

HERMES-Core—A 1.59-TOPS/mm2 PCM on 14-nm CMOS In-Memory Compute Core Using 300-ps/LSB Linearized CCO-Based ADCs

We present a 256 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times$ </tex-math></inline-formula> 256 in-memory compute (IMC) core designed and fabricated in 14-nm CMOS technology with backend-integrated multi-level phase change memory (PCM). It comprises 256 linearized current-controlled oscillator (CCO)-based A/D converters (ADCs) at a compact 4- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> pitch and a local digital processing unit (LDPU) performing affine scaling and ReLU operations. A frequency-linearization technique for CCO is introduced, which increases the maximum CCO frequency beyond 3 GHz, while ensuring accurate on-chip matrix–vector multiplications (MVMs). Moreover, the design and functionality of the digital ADC calibration procedure is described in detail and the MVM accuracy is quantified. Finally, the measured classification accuracies of deep learning (DL) inference applications on the MNIST and CIFAR-10 datasets, when two IMC cores are employed, are presented. For a performance density of 1.59 TOPS/mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , a measured energy efficiency of 10.5 TOPS/W, at a main clock frequency of 1 GHz, is achieved.

Performance of an iterative wavelet reconstructor for the Multi-conjugate Adaptive Optics RelaY of the Extremely Large Telescope

The Multi-conjugate Adaptive Optics RelaY (MAORY) is one of the key adaptive optics (AO) systems on the European Southern Observatory’s Extremely Large Telescope. MAORY aims to achieve good wavefront correction over a large field of view, which involves a tomographic estimation of the three-dimensional atmospheric wavefront disturbance. Mathematically, the reconstruction of turbulent layers in the atmosphere is severely ill-posed, hence, limits the achievable reconstruction accuracy. Moreover, the reconstruction has to be performed in real time at a few hundred to one thousand hertz frame rates. Huge amounts of data have to be processed and thousands of actuators of the deformable mirrors have to be controlled by elaborated algorithms. Even with extensive parallelization and pipelining, direct solvers, such as the matrix vector multiplication method, are extremely demanding. Thus, research in recent years shifted into the direction of iterative methods. We focus on the iterative finite-element wavelet hybrid algorithm (FEWHA). The key feature of FEWHA is a matrix-free representation of all operators involved, which makes the algorithm fast and enables on-the-fly system updates whenever parameters at the telescope or in the atmosphere change. We provide a performance analysis of the method regarding quality and run-time for the MAORY instrument using the AO software package COMPASS.

A Novel Method for Developing Post-quantum Digital Signature Algorithms on Non-commutative Associative Algebras

Introduction: Development of practical post-quantum signature algorithms is a current challenge in the area of cryptography. Recently, several candidates on post-quantum signature schemes, in which the exponentiation operations in a hidden commutative group contained in a non-commutative algebra is used, were proposed. Search for new mechanisms of using a hidden group, while developing signature schemes resistant to quantum attacks, is of significant practical interest. Purpose: Development of a new method for designing post-quantum signature algorithms on finite non-commutative associative algebras. Results: A novel method for developing digital signature algorithms on non-commutative algebras. A new four-dimensional finite non-commutative associative algebra set over the ground field GF(p) have been proposed as algebraic support of the signature algorithms. To provide a higher performance of the algorithm, in the introduced algebra the vector multiplication is defined by a sparse basis vector multiplication table. Study of the algebra structure has shown that it can be represented as a set of commutative subalgebras of three different types, which intersect exactly in the set of scalar vectors. Using the proposed method and introduced algebra, a new post-quantum signature scheme has been designed. The introduced method is characterized in using one of the elements of the signature (e, S) in form of the four-dimensional vector S that is computed as a masked product of two exponentiated elements G and H of a hidden commutative group: S = B-1GnHmC-1, where non-permutable vectors B and C are masking multipliers; the natural numbers n and m are calculated depending on the signed document M and public key. The pair &lt;G, H&gt; composes a minimum generator systems of the hidden group. The signature verification equation has the form R = (Y1SZ1)e(Y2SZ2)e2, where pairwise non-permutable vectors Y1, Z1, Y2, and Z2 are element of the public key and natural number e that is computed depending on the value M and the vector R. Practical relevance: Due to sufficiently small size of public key and signature and high, the developed digital signature scheme represents interest as a practical post-quantum signature algorithm. The introduced method is very attractive to develop a post-quantum digital signature standard.

Modeling and Compensation of IR Drop in Crosspoint Accelerators of Neural Networks

In-memory computing (IMC) has the potential of accelerating data-intensive computing tasks, such as inference and training of neural networks, by the matrix vector multiplication (MVM) in the crosspoint array. As the array size increases, however, a significant problem arises due to the parasitic IR drop at row and column lines. Understanding and compensating IR drop is essential to enable high-density accelerators of MVM. This work presents an analytical model for the IR drop and a numerical algorithm capable of accelerating the nodal analysis of crosspoint arrays up to five orders of magnitude with respect to SPICE analysis. The numerical algorithm is used to study the impact of array size, data density, device/wire resistance, and nonlinearity on the IR drop and the related loss of accuracy in IMC. We also derive two simple compensation schemes and an architectural solution for the mitigation of IR drop that are able to increase the accuracy of neural networks from 59&#x0025; up to 96.6&#x0025;, compared to an ideal accuracy of 97&#x0025;.

Multiple-precision sparse matrix–vector multiplication on GPUs

Sparse matrix–vector multiplication (SpMV) appears in many application domains, and performance is the key consideration when implementing SpMV kernels. At the same time, accuracy is also important because rounding errors can drastically change the computed result. Multiple-precision arithmetic is a common approach to improve the accuracy of results. In this paper, we implement and evaluate multiple-precision SpMV kernels in the CSR, JAD, ELLPACK, and DIA matrix storage formats for graphics processing units. In the proposed implementation, the matrix is represented in double precision, while the input and output vectors are in multiple precision and internal computations are also performed in multiple precision. Our underlying floating-point arithmetic algorithms are based on the residue number system, which is attractive because of its carry-free nature and provides an arbitrary level of precision determined by the set of moduli that comprise the base of the system. In particular, we apply sets from 8 to 64 moduli, reaching precision values from 106 to 848 bits, and demonstrate how higher precision reduces the rounding errors in the SpMV kernel. We also conduct a thorough analysis of the CSR kernel in terms of roofline performance, occupancy and memory bandwidth, and evaluate the performance and memory consumption for all the proposed kernels. Numerical experiments on matrices from real-world applications show that ELLPACK offers the highest performance in many examples, while JAD is a good trade-off between performance and space requirements. Furthermore, our proposed kernels in general substantially improve the efficiency of sparse matrix–vector multiplication compared to implementations built on top of existing multiple-precision CUDA libraries. Finally, we integrate the multiple-precision SpMV into a preconditioned conjugate gradient linear solver and identify test cases where our implementation exhibits superior convergence and numerical robustness.

SparseP

Several manufacturers have already started to commercialize near-bank Processing-In-Memory (PIM) architectures, after decades of research efforts. Near-bank PIM architectures place simple cores close to DRAM banks. Recent research demonstrates that they can yield significant performance and energy improvements in parallel applications by alleviating data access costs. Real PIM systems can provide high levels of parallelism, large aggregate memory bandwidth and low memory access latency, thereby being a good fit to accelerate the Sparse Matrix Vector Multiplication (SpMV) kernel. SpMV has been characterized as one of the most significant and thoroughly studied scientific computation kernels. It is primarily a memory-bound kernel with intensive memory accesses due its algorithmic nature, the compressed matrix format used, and the sparsity patterns of the input matrices given. This paper provides the first comprehensive analysis of SpMV on a real-world PIM architecture, and presents SparseP, the first SpMV library for real PIM architectures. We make three key contributions. First, we implement a wide variety of software strategies on SpMV for a multithreaded PIM core, including (1) various compressed matrix formats, (2) load balancing schemes across parallel threads and (3) synchronization approaches, and characterize the computational limits of a single multithreaded PIM core. Second, we design various load balancing schemes across multiple PIM cores, and two types of data partitioning techniques to execute SpMV on thousands of PIM cores: (1) 1D-partitioned kernels to perform the complete SpMV computation only using PIM cores, and (2) 2D-partitioned kernels to strive a balance between computation and data transfer costs to PIM-enabled memory. Third, we compare SpMV execution on a real-world PIM system with 2528 PIM cores to an Intel Xeon CPU and an NVIDIA Tesla V100 GPU to study the performance and energy efficiency of various devices, i.e., both memory-centric PIM systems and conventional processor-centric CPU/GPU systems, for the SpMV kernel. SparseP software package provides 25 SpMV kernels for real PIM systems supporting the four most widely used compressed matrix formats, i.e., CSR, COO, BCSR and BCOO, and a wide range of data types. SparseP is publicly and freely available at https://github.com/CMU-SAFARI/SparseP. Our extensive evaluation using 26 matrices with various sparsity patterns provides new insights and recommendations for software designers and hardware architects to efficiently accelerate the SpMV kernel on real PIM systems.

Broadband photonic tensor core with integrated ultra-low crosstalk wavelength multiplexers

Abstract The integration of artificial intelligence (AI) systems in the daily life greatly increases the amount of data generated and processed. In addition to the large computational power required, the hardware needs to be compact and energy efficient. One promising approach to fulfill those requirements is phase-change material based photonic neuromorphic computing that enables in-memory computation and a high degree of parallelization. In the following, we present an optimized layout of a photonic tensor core (PTC) which is designed to perform real valued matrix vector multiplications and operates at telecommunication wavelengths. We deploy the well-studied phase-change material Ge2Sb2Te5 (GST) as an optical attenuator to perform single positive valued multiplications. In order to generalize the multiplication to arbitrary real factors, we develop a novel symmetric multiplication unit which directly includes a reference-computation branch. The variable GST attenuator enables a modulation depth of 5 dB over a wavelength range of 100 nm with a wavelength dependency below 0.8 dB. The passive photonic circuit itself ensures equal coupling to the main-computation and reference-computation branch over the complete wavelength range. For the first time, we integrate wavelength multiplexers (MUX) together with a photonic crossbar array on-chip, paving the way towards fully integrated systems. The MUX are crucial for the PTC since they enable multiple computational channels in a single photonic crossbar array. We minimize the crosstalk between the channels by designing Bragg scattering based MUX. By cascading, we achieve an extinction ratio larger than 61 dB while the insertion loss is below 1 dB.

Recurrent Neural Networks With Column-Wise Matrix–Vector Multiplication on FPGAs

This article presents a reconfigurable accelerator for REcurrent Neural networks with fine-grained cOlumn-Wise matrix–vector multiplicatioN (RENOWN). We propose a novel latency-hiding architecture for recurrent neural network (RNN) acceleration using column-wise matrix–vector multiplication (MVM) instead of the state-of-the-art row-wise operation. This hardware (HW) architecture can eliminate data dependencies to improve the throughput of RNN inference systems. Besides, we introduce a configurable checkerboard tiling strategy which allows large weight matrices, while incorporating various configurations of element-based parallelism (EP) and vector-based parallelism (VP). These optimizations improve the exploitation of parallelism to increase HW utilization and enhance system throughput. Evaluation results show that our design can achieve over 29.6 tera operations per second (TOPS) which would be among the highest for field-programmable gate array (FPGA)-based RNN designs. Compared to state-of-the-art accelerators on FPGAs, our design achieves 3.7–14.8 times better performance and has the highest HW utilization.

Accelerating an iterative eigensolver for nuclear structure configuration interaction calculations on GPUs using OpenACC

To accelerate the solution of large eigenvalue problems arising from many-body calculations in nuclear physics on distributed-memory parallel systems equipped with general-purpose Graphic Processing Units (GPUs), we modified a previously developed hybrid MPI/OpenMP implementation of an eigensolver written in Fortran 90 by using an OpenACC directives based programming model. Such an approach requires making minimal changes to the original code and enables a smooth migration of large-scale nuclear structure simulations from a distributed-memory many-core CPU system to a distributed GPU system. However, in order to make the OpenACC based eigensolver run efficiently on GPUs, we need to take into account the architectural differences between a many-core CPU and a GPU device. Consequently, the optimal way to insert OpenACC directives may be different from the original way of inserting OpenMP directives. We point out these differences in the implementation of sparse matrix–vector multiplications with multiple vectors, which constitutes the main cost of the eigensolver, as well as other differences in the preconditioning step and dense linear algebra operations. We compare the performance of the OpenACC based implementation executed on multiple GPUs with the performance on distributed-memory many-core CPUs, and demonstrate significant speedup achieved on GPUs compared to the on-node performance of a many-core CPU. We also show that the overall performance improvement of the eigensolver on multiple GPUs is more modest due to the communication overhead among different MPI ranks.

A low-latency LSTM accelerator using balanced sparsity based on FPGA

Long Short-Term Memory (LSTM) has been widely used in the fields of Automatic Speech Recognition (ASR) and Natural Language Processing (NLP). In order to accelerate LSTM inference, previous works have proposed various compression methods for weight pruning. Bank-Balanced Sparsity (BBS) is an efficient compression method that has a balanced distribution of non-zero elements and negligible precision degradation. However, it costs considerable additional memory overhead to store the indices, which limits the compression ratio and poses a challenge for FPGAs with limited on-chip resources.A Shared Index Bank-Balanced Sparsity (SIBBS) compression method is presented in this paper. The rows of a weight matrix are divided into multiple bank clusters to balance the non-zero weight distribution. The banks in one cluster share the indices. The overall cost of indices is reduced by 2x–8x compared with BBS. A coarse-grained inputs similarity skipping scheme is proposed at the same time to utilize the SIBBS pruning balance, which achieves 10% LSTM operation reduction with little accuracy degradation and negligible overhead. A sparse matrix and vector multiplication architecture for the SIBBS method is proposed and the customized accelerator is implemented on the Xilinx XCKU115 FPGA. Compared with the state-of-the-art LSTM accelerators based on FPGA, our accelerator achieved a 1.47x–79.5x reduction in latency with little degradation in accuracy.

An Efficient Scheme for Acoustic Echo Canceller Implementation Using Offset Binary Coding

This article presents an efficient design and implementation scheme for a low-area and low-power acoustic echo canceller. The design employs the block least mean square algorithm-based adaptive filter (ADF) using offset binary coding. The proposed approach first formulates the ADF by splitting the matrix&#x2013;vector multiplication into smaller ones. Each of them is then realized with lookup tables and shift accumulate units with offset terms. An efficient scheme is suggested to update the offset terms from the corresponding lookup tables. In addition, a novel optimization scheme is proposed based on the grouping of partial products (PPs) and moving windows. The PPs are generated in two parallel styles using adders, multiplexers, and registers. The optimized architecture is shared to compute both the filter output and coefficient increment terms in every iteration. The fixed-point quantization model for the architecture is also discussed. Accuracy measure is defined to characterize the proposed design and compare it with the Cramer&#x2013;Rao lower bound. Simulations are carried out to evaluate the performance of the proposed design. Field-programmable gate array implementation results and application-specific integrated circuit synthesis show that the proposed design outperforms the state-of-the-art architectures.

Next-generation reservoir computing based on memristor array

As a kind of brain-inspired computing, reservoir computing (RC) has great potential applications in time sequence signal processing and chaotic dynamics system prediction due to its simple structure and few training parameters. Since in the RC randomly initialized network weights are used, it requires abundant data and calculation time for warm-up and parameter optimization. Recent research results show that an RC with linear activation nodes, combined with a feature vector, is mathematically equivalent to a nonlinear vector autoregression (NVAR) machine, which is named next-generation reservoir computing (NGRC). Although the NGRC can effectively alleviate the problems which traditional RC has, it still needs vast computing resources for multiplication operations. In the present work, a hardware implementation method of using computing-in memory paradigm for NGRC is proposed for the first time. We use memristor array to perform the matrix vector multiplication involved in the nonlinear vector autoregressive process for the improvement of the energy efficiency. The Lorenz63 time series prediction task is performed by simulation experiments with the memristor array, demonstrating the feasibility and robustness of this method, and the influence of the weight precision of the memristor devices on the prediction results is discussed. These results provide a promising way of implementing the hardware NGRC.

Real-Time Nonlinear Behavioral Electrothermal Device-Level Emulation of IGBT on Heterogeneous Adaptive Compute Acceleration Platform

Power converter design evaluation by means of real-time simulation techniques is prevalent, although it is mostly restricted to simple power semiconductor switch models that exclude device-level physical details. In this work, the nonlinear high-order electro-thermal model of the Insulated-gate bipolar transistor (IGBT) is developed and then deployed onto the heterogeneous digital hardware for real-time implementation. As the complexity of the nonlinear behavioral model (NBM) of the IGBT poses a significant computational burden on real-time hardware emulation, machine learning (ML) methodology is utilized so that the trained model can reproduce the characteristics of its original counterpart as much as possible and then it is implemented on the Adaptive Compute Acceleration Platform (ACAP), which composes of the processing system (PS), programmable logic (PL), and Artificial Intelligent Engine (AIE). The vector multiplication feature of the AIE caters to mathematical operations of the ML-based model particularly well and consequently enables it to be executed in real-time with remarkable speedup over the original model with which matrix inversion is otherwise mandatory. Finally, the validation for real-time device-level results and system-level results of a multi-converter system is provided by SaberRD® and MATLAB/Simulink®.

Алгебраические алгоритмы ЭЦП, основанные на трудности решения систем уравнений

Purpose of work is the development of post-quantum digital signature algorithms with comparatively small sizes of the public and secret keys and the signature. Research method is the use of a new concept for constructing signature algorithms on finite non-commutative associative algebras, which is distinguished by the multiple occurrences of the signature S in the power verification equation. A public key is generated in the form of a set of vectors every of which is calculated as the product of triples of secret vectors. With a special choice of these triples, it is possible to calculate a signature that satisfies the verification equation. Results of the study are two developed algebraic post-quantum digital signature algorithms of a new type, security of which is based on the computational difficulty of solving systems of many quadratic equations with many unknowns. The difference from the public-key algorithms of multivariate cryptography is that the system of quadratic equations is derived from the formulas for generating the public-key elements in the form of a set of vectors of m-dimensional finite non-commutative algebra with an associative vector multiplication operation. The said formulas define the system of n quadratic vector equations, which reduces to the system of mn quadratic equations over a finite field. Thanks to the “natural” mechanism for the occurrence of the specified system, it is set above the field, the order of which has a large size (97 and 129 bits). The used procedures for generating the public key and signature include the exponentiation operations to the degree of a large size (96 and 128 bits), which are performed over the elements of the secret (hidden) commutative group contained in the algebra. The signature is formed in the form of two elements: a randomizing natural number e and a “fitting” vector S. The signature authentication equation includes a multiple occurrence of the S element and every entry of the vector S is associated with the formation of a product that is exponentiated to a degree dependent on the value of the e element. A significant reduction in the size of public and secret keys and signatures has been achieved, as well as an increase in performance compared to foreign analogues, considered currently as basic algorithms for the adoption of post-quantum digital signature standards.

An asynchronous and parallel row-wise compressed SpMV kernel on heterogeneous CPU-GPU architectures

Sparse matrix vector multiplication (SpMV) is a fundamental and important algebra used extensively in various fields, such as machine learning, data mining and numerical simulation. Accelerating SpMV benefits the corresponding real-world applications. Meanwhile, heterogeneous CPU-GPU architectures are inevitable to realise high-performance computing. Therefore, this paper designs an asynchronous parallel row-wise compressed SpMV kernel by utilising the heterogeneous CPU-GPU architectures to accelerate applications that use SpMV. First, according to the memory access and control flow of SpMV and the architectural features of the heterogeneous CPU-GPU systems, the heterogeneous parallelisation of the row-wise compressed SpMV based on CSR format is designed. Next, an asynchronous method is designed for the parallel compressed SpMV kernel to create an additional level of parallelism in SpMV execution. The proposed SpMV kernel can obtain up to 8.97% improvement from the asynchronous method on the heterogeneous CPU-GPU architecture and perform the nearly linear speedup across GPU threads.

Electromagnetic Scattering From Multiple Moving Targets Above a Rough Surface

Aimed at solving composite scattering from multiple moving targets above a rough surface, a hybrid method combining the multilevel fast multipole algorithm (MLFMA) and Kirchhoff approximation (KA) is presented. The technology combining the MLFMA with the ray-propagation fast multipole algorithm (RPFMA) with the fast far-field approximation (FAFFA) is applied to accelerate coupling matrix filling. There are two main bottlenecks in this problem. The first is that with moving targets; specifically, we need to refill the impedance matrix and recompute composite scattering, and the second is solving the tremendous coupling matrix between multiple targets and a rough surface. To solve the first bottleneck, the hybrid MLFMA-KA based on the domain decomposition method (DDM) is used, with the target region solved by the MLFMA and the surface region solved by the KA. Additionally, the impedance matrix on each target is divided into the near-field matrix and far-field matrix on the same target, the far-field matrix between different targets and between targets with a rough surface. With moving targets, the impedance matrix on the same target remains unchanged and only needs to be solved once, and the far-field matrix between different targets and between targets with a rough surface needs to be recomputed. To solve the second bottleneck, the MLFMA-RPFMA-FAFFA is used to accelerate matrix vector multiplication between the targets and rough surfaces. Compared with the MLFMA, the proposed method is proven to be efficient and accurate. In addition, electromagnetic scattering from multiple moving targets above the surface are analyzed.

RMIST-Net: Joint Range Migration and Sparse Reconstruction Network for 3-D mmW Imaging

Compressed sensing (CS) demonstrates significant potential to improve image quality in 3-D millimeter-wave imaging compared with conventional matched filtering (MF). However, existing sparsity-driven 3-D imaging algorithms always suffer from large-scale storage, excessive computational cost, and nontrivial tuning of parameters due to the huge-dimensional matrix–vector multiplication in complicated iterative optimization steps. In this article, we present a novel range migration (RM) kernel-based iterative-shrinkage thresholding network, dubbed as RMIST-Net, by combining the traditional model-based CS method and data-driven deep learning method for near-field 3-D millimeter-wave (mmW) sparse imaging. First, the measurement matrices in ISTA optimization steps are replaced by RM kernels, by which matrix–vector multiplication is converted to the Hadamard product. Then, the modified ISTA optimization is unrolled into a deep hierarchical architecture, in which all parameters are learned automatically instead of manually tuned. Subsequently, 1000 pairs of oracle images with randomly distributed targets and their corresponding echoes are simulated to train the network. A well-trained RMIST-Net produces high-quality 3-D images from range-focused echoes. Finally, we experimentally prove that RMIST-Net is capable process <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$512 \times 512$ </tex-math></inline-formula> large-scale imaging tasks within 1 s. Besides, we compare RMIST-Net with other state-of-the-art methods in near-field 3-D imaging applications. Both simulations and real-measured experiments demonstrate that RMIST-Net produces impressive reconstruction performance while maintaining high computational speed compared with conventional and sparse imaging algorithms.

НОВАЯ КОНЦЕПЦИЯ РАЗРАБОТКИ ПОСТКВАНТОВЫХ АЛГОРИТМОВ ЦИФРОВОЙ ПОДПИСИ НА НЕКОММУТАТИВНЫХ АЛГЕБРАХ

Purpose of work is the development of a new approach to designing post-quantum digital signature algorithms that are free from the shortcomings of known analogs – large sizes of the signature and public key. Research method is the use of power vector equations with multiple occurrences of the signature S as a signature verification equation. The computational difficulty of solving equations of the said type relatively the unknown value of S ensures the resistance of the signature scheme to attacks using S as a fitting parameter. The possibility of calculating the value of S by the secret key is provided by using the public key in the form of a set of secret elements of the hidden group, masked by performing left and right multiplications by matched invertible vectors. Results of the study include a new proposed concept for the development of post-quantum digital signature algorithms on non-commutative algebras, which use a hidden commutative group. One of its main differences is the use of a secret key in the form of a set of vectors, the knowledge of which makes it possible to calculate the correct signature value for the random powers present in the verification equation. The form of the latter defines a system of quadratic vector equations connecting the public key with the secret, which is reduced to a system of many quadratic equations with many unknowns, given over a finite field. The computational difficulty of finding a solution to the latter system determines the security of the algorithms developed within the framework of the proposed concept. A quantum computer is ineffective for solving this problem, therefore, the said algorithms are post-quantum. As analogs in construction, digital signature algorithms based on the computational difficulty of the hidden discrete logarithm problem are considered, however, the use of a hidden group and exponentiation operations represent only a general technique for ensuring the correctness of the signature schemes developed within the framework of the concept, and not for specifying a basic computationally difficult problem. To improve the performance of the signature generation and verifications procedures, the four-dimensional algebras defined by sparse basis vector multiplication tables are used as an algebraic support. The proposed concept is confirmed by the development of a specific post-quantum algorithm that provides a significant reduction in the size of the public key and signature in comparison with the finalists of the NIST global competition in the nomination of post-quantum digital signature algorithms. Practical relevance: The developed new concept for constructing post-quantum digital signature algorithms expands the areas of their application in conditions of limited computing resources

Full waveform inversion through double-sweeping solver

An efficient method is proposed to accurately approximate the gradient and the Hessian operator for the full-waveform inversion (FWI) in large-scale problems. The key idea is an approximate solver called double-sweeping solver, which divides the domain into smaller slabs and sequentially solves the wavefields through a downward and an upward sweeping. The sequential solution is facilitated by approximating the continuity conditions that suppress the multiples, thus relaxing long-range coupling between the subdomains. The double-sweeping solver is incorporated into an inexact Gauss-Newton approach to perform FWI, where the gradient and the Hessian vector multiplication are computed more efficiently. Through numerical experiments, we show that the convergence of FWI with respect to the number of iterations does not degrade when the double-sweeping approximation is used. Given that the double-sweeping solver is computationally cheaper than full-wave simulation, the proposed method is more efficient than the standard FWI. This paper contains the complete formulation of the proposed methodology as well as an illustration of its effectiveness to problems of varying complexity including the inversion of the Marmousi model from the Geophysics community.