An accelerated inversion algorithm using the resolvent matrix method

A prescaled multiplicative regularized Gauss-Newton inversion (GNI) algorithm is proposed which utilizes a priori information about the expected ratio between the average magnitude of the real and imaginary parts of the true contrast as well as the expected ratio between the average magnitude of the gradient of the real and imaginary parts of the true contrast. Using both synthetically and experimentally collected data sets, we show that this prescaled inversion algorithm is successful in reconstructing both real and imaginary parts of the contrast when there is a large imbalance between the average magnitude of these two parts where the standard multiplicative regularized Gauss-Newton inversion algorithm fails. We further show that the proposed prescaled inversion algorithm is robust and does not require the a priori information to be exact.

In this article, we propose the Fast Algorithms for Maxwell’s Equations (FAME) package for solving Maxwell’s equations for modeling three-dimensional photonic crystals. FAME combines the null-space free method with fast Fourier transform (FFT)-based matrix-vector multiplications to solve the generalized eigenvalue problems (GEPs) arising from Yee’s discretization. The GEPs are transformed into a null-space free standard eigenvalue problem with a Hermitian positive-definite coefficient matrix. The computation times for FFT-based matrix-vector multiplications with matrices of dimension 7 million are only 0.33 and 3.6 × 10 − 3 seconds using MATLAB with an Intel Xeon CPU and CUDA C++ programming with a single NVIDIA Tesla P100 GPU, respectively. Such multiplications significantly reduce the computational costs of the conjugate gradient method for solving linear systems. We successfully use FAME on a single P100 GPU to solve a set of GEPs with matrices of dimension more than 19 million, in 127 to 191 seconds per problem. These results demonstrate the potential of our proposed package to enable large-scale numerical simulations for novel physical discoveries and engineering applications of photonic crystals.

In this paper we discuss the case in which, using the generalized Cannon's algorithm, it is possible to reduce communications in matrix multiplication. We then apply reduction of communications to the case in which we have to multiply large matrices, in particular rectangular matrices. Two strategies are proposed to solve the problem of multiplying two large squared matrices. For the case in which we have to deal with small matrices, some methods are proposed to use the entire number of processors.

Convolutional neural networks (CNNs) have demonstrated promising results in various applications such as computer vision, speech recognition, and natural language processing. One of the key computations in many CNN applications is matrix multiplication, which accounts for a significant portion of computation. Therefore, hardware accelerators to effectively speed up the computation of matrix multiplication have been proposed, and several studies have attempted to design hardware accelerators to perform better matrix multiplications in terms of both speed and power consumption. Typically, accelerators with either a two-dimensional (2D) systolic array structure or a single instruction multiple data (SIMD) architecture are effective only when the input matrix has shapes that are close to or similar to a square. However, several CNN applications require multiplications of non-squared matrices with various shapes and dimensions, and such irregular shapes lead to poor utilization efficiency of the processing elements (PEs). This study proposes a configurable engine for neural network acceleration, called CONNA, whose computation engine can conduct matrix multiplications with highly utilized computing units, regardless of the access patterns, shapes, and dimensions of the input matrices by changing the shape of matrix multiplication conducted in the physical array. To verify the functionality of the CONNA accelerator, we implemented CONNA as an SoC platform that integrates a RISC-V MCU with CONNA on Xilinx VC707 FPGA. SqueezeNet on CONNA achieved an inference performance of 100 frames per second (FPS) with 2.36 mm2 and 83.55 mW in a 65 nm process, improving efficiency by up to 34.1 times better than existing accelerators in terms of FPS, silicon area, and power consumption.

A hierarchical fast direct solver is proposed for the method of moment (MoM) based on the low-rank compression method. The dense impendent matrix of MoM is decomposed into a hierarchical off-diagonal low-rank (HODLR) system with the rank deficiency of the off-diagonal matrixes for far field interactions. Furthermore, the HODLR matrix can be easily expressed with a multiplication of several diagonal matrixes. Each small diagonal matrix block can be inversed easily. So that, the current solution of the integral equation can be expressed with a series of matrix vector products (MVPs) with given excitation vectors.

We reduce the problem of computing the rank and a null-space basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input n x n matrix of degree, d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension, n and degree, d. If the latter multiplication is done in MM(n,d)= O~(nωd operations, with ω the exponent of matrix multiplication over K, then the algorithm uses O~MM(n,d) operations in, K. For m x n matrices of rank r and degree d, the cost expression is O(nmr ω-2d). The soft-O notation O~ indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of matrix Hensel high-order lifting and matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a K[x]-module.

A new inversion algorithm for simultaneously reconstructing the slip vectors and rupture times of a propagating finite fault is developed based on the recent progress in the nonlinear programming study. To check the validity of this new inversion algorithm, several numerical tests are conducted. The results show that this new source rupture process inversion algorithm is computationally efficient and numerically stable, and depends less on the initial model compared with the two popular inversion methods, i.e. the linear matrix method and the global stochastic search method. Therefore, this new inversion algorithm is expected to be useful in inverting earthquake rupture processes.

In this paper, we advocate the use of the uncompressed form of i-vector and depend on subspace modeling using probabilistic linear discriminant analysis (PLDA) in handling the speaker and session (or channel) variability. An i-vector is a low-dimensional vector containing both speaker and channel information acquired from a speech segment. When PLDA is used on an i-vector, dimension reduction is performed twice: first in the i-vector extraction process and second in the PLDA model. Keeping the full dimensionality of the i-vector in the i-supervector space for PLDA modeling and scoring would avoid unnecessary loss of information. We refer to the uncompressed i-vector as the i-supervector. The drawback in using the i-supervector with PLDA is the inversion of large matrices in the estimation of the full posterior distribution, which we show can be solved rather efficiently by portioning large matrices into smaller blocks. We also introduce the Gaussianized rank-norm, as an alternative to whitening, for feature normalization prior to PLDA modeling. We found that the i-supervector performs better during normalization. A better performance is obtained by combining the i-supervector and i-vector at the score level. Furthermore, we also analyze the computational complexity of the i-supervector system, compared with that of the i-vector, at four different stages of loading matrix estimation, posterior extraction, PLDA modeling, and PLDA scoring.

Thus far, all results in this book have been concerned with the limiting behavior of eigenvalues of large dimensional random matrices. As mentioned in the introduction, the development of RMT has been attributed to the investigation of the energy level distribution of a large number of particles in QM; in other words, the original interests of RMT were confined to eigenvalue distributions of large dimensional random matrices. In the beginning, most of the important results in RMT were related to a certain deterministic behavior, to the extent that their empirical distributions tend toward nonrandom ones as the dimension tends to infinity. Moreover, this behavior is invariant under the distribution of the variables making up the matrix.

The magnetotelluric (MT) method is highly advantageous in geothermal resource exploration because of its deep probing capability, low cost, and sensitivity to key geothermal system elements, such as heat sources, reservoirs, and caprocks. To address the challenge of balancing the MT inversion resolution and computational efficiency, we develop a Gauss-Newton optimization inversion algorithm based on the solution space dimensionality reduction (SSDR) technique. This algorithm uses SSDR to establish algebraic mapping relationships between the edges of grid cells, effectively reducing the degrees of freedom in the linear equation system required for forward and pseudoforward modeling, thereby saving computational costs. In addition, during the inversion process, a multiple right-hand direct-iterative hybrid solver is used to solve the dimensionally reduced linear equations, further enhancing the computational efficiency of the inversion. We then validate and analyze the accuracy and efficiency of the forward and inversion algorithms using various synthetic models. Finally, we apply this inversion strategy to field MT data from the Dabie Mountain geothermal area in eastern China. On the basis of the inversion results, we then analyze the thermal genesis model of the region, providing technical support for the exploration and development of deep geothermal resources.

Preconditioned iterative methods for a class of nonlinear eigenvalue problems

This paper presents a distributed Kalman filter to estimate the state of a sparsely connected, large-scale, n -dimensional, dynamical system monitored by a network of N sensors. Local Kalman filters are implemented on n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> -dimensional subsystems, n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> Lt n, obtained by spatially decomposing the large-scale system. The distributed Kalman filter is optimal under an Lth order Gauss-Markov approximation to the centralized filter. We quantify the information loss due to this Lth-order approximation by the divergence, which decreases as L increases. The order of the approximation L leads to a bound on the dimension of the subsystems, hence, providing a criterion for subsystem selection. The (approximated) centralized Riccati and Lyapunov equations are computed iteratively with only local communication and low-order computation by a distributed iterate collapse inversion (DICI) algorithm. We fuse the observations that are common among the local Kalman filters using bipartite fusion graphs and consensus averaging algorithms. The proposed algorithm achieves full distribution of the Kalman filter. Nowhere in the network, storage, communication, or computation of n-dimensional vectors and matrices is required; only n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> Lt n dimensional vectors and matrices are communicated or used in the local computations at the sensors. In other words, knowledge of the state is itself distributed.

Large numeric matrices and multidimensional data arrays appear in many science domains, as well as in applications of financial and business warehousing. Common applications include eigenvalue determination of large matrices, which decompose into a set of linear algebra operations. With the rise of in-memory databases it is now feasible to execute these complex analytical queries directly in a relational database system without the need of transfering data out of the system and being restricted by hard disc latencies for random accesses. In this paper, we present a way to integrate linear algebra operations and large matrices as first class citizens into an in-memory database following a two-layered architectural model. The architecture consists of a logical component receiving manipulation statements and linear algebra expressions, and of a physical layer, which autonomously administrates multiple matrix storage representations. A cost-based hybrid storage representation is presented and an experimental implementation is evaluated for matrix-vector multiplications.

The interpolative decomposition (ID) is combined with the multilevel fast multipole algorithm (MLFMA), denoted by ID‐MLFMA, to handle multiscale problems. The ID‐MLFMA first generates ID levels by recursively dividing the boxes at the finest MLFMA level into smaller boxes. It is specifically shown that near‐field interactions with respect to the MLFMA, in the form of the matrix vector multiplication (MVM), are efficiently approximated at the ID levels. Meanwhile, computations on far‐field interactions at the MLFMA levels remain unchanged. Only a small portion of matrix entries are required to approximate coupling among well‐separated boxes at the ID levels, and these submatrices can be filled without computing the complete original coupling matrix. It follows that the matrix filling in the ID‐MLFMA becomes much less expensive. The memory consumed is thus greatly reduced and the MVM is accelerated as well. Several factors that may influence the accuracy, efficiency and reliability of the proposed ID‐MLFMA are investigated by numerical experiments. Complex targets are calculated to demonstrate the capability of the ID‐MLFMA algorithm.

General matrices can be split uniquely into Frobenius-orthogonal components: a constant row and column sum (type S) part, a vertex cross sum (type V) part and a weight part. We show that for square matrices, the type S part can be expressed as a sum of squares of type V matrices. We investigate the properties of such decomposition under matrix multiplication, in particular how the pseudoinverses of a matrix relates to the pseudoinverses of its component parts. For invertible matrices, this yields an expression for the inverse where only the type S part needs to be (pseudo)inverted; in the example of the Wilson matrix, this component is considerably better conditioned than the whole matrix. We also show a relation between matrix determinants and the weight of their matrix inverses and give a simple proof for Frobenius-optimal approximations with the constant row and column sum and the vertex cross sum properties, respectively, to a given matrix.