Capacitance Matrix Research Articles

CAPACITANCE MATRIX REVISITED

The capacitance matrix relates potentials and charges on a system of conductors. We review and rigorously generalize its properties, block-diagonal structure and inequalities, deduced from the geometry of system of conductors and analytic properties of the permittivity tensor. Furthermore, we discuss alternative choices of regularization of the capacitance matrix, which allow us to find the charge exchanged between the conductors having been brought to an equal potential. Finally, we discuss the tacit approximations used in standard treatments of the electric circuits, demonstrating how the formulae for the capacitance of capacitors connected in parallel and series may be recovered from the capacitance matrix.

The improved FASTmrEMMA and GCIM algorithms for genome-wide association and linkage studies in large mapping populations

Owing to high power and accuracy and low false positive rate in our multi-locus approaches for genome-wide association studies and linkage analyses, these approaches have attracted considerable attention in plant and animal genetics. In large mapping population, however, fast multi-locus random-SNP-effect efficient mixed model association (FASTmrEMMA) and genome-wide composite interval mapping (GCIM) run a relatively long time. To address this issue, we proposed the improved FASTmrEMMA and GCIM algorithms in this study. In the new algorithms, some matrix identities, such as the Woodbury matrix identity, were used. In scanning each marker on the entire genome, in other words, the improved algorithms effectively replace the expensive eigenvector solutions in (restricted) maximum likelihood estimations in original algorithms with two (one) updated inner products and one updated vector-matrix-vector multiplication. Simulated and real data analyses showed that their computational efficiencies are increased sharply in large mapping population, although there are no mapping result differences between original and improved algorithms. In addition, the related software packages (mrMLM.GUI and QTL.gCIMapping.GUI) can be downloaded from the R and BioCode websites.

Analytical determination of the turn-to-turn capacitances for the prediction of voltage peaks in a PWM-fed motor winding

The number of inverter-fed motors is increasing due to the good controllability of the motor and the meanwhile low acquisition costs. The steep voltage slopes of the converters lead to an uneven voltage distribution along the winding and thus to voltage peaks between the conductors, which stresses the insulation. The voltage distribution can be predicted by means of equivalent circuit diagrams, which take into account the capacitive coupling between the conductors. This paper presents a novel approach for an analytical determination of the turn-to-turn capacitances, which, in addition to the geometry and the placement of the conductors, considers the influence of materials with different permittivities.The conductors are simulated by means of line charges discretely placed inside the electrodes and receptor points attached to the conductor surfaces. The capacitances are determined by means of the Maxwell capacitance matrix. The method is validated by means of FEM simulations for different geometries and materials.

Low computational complexity methods for decoding of STBC in the uplink of a massive MIMO system

Reducing the computational complexity of the modern wireless communication systems such as massive MIMO configurations is of utmost interest. In this paper, we propose algorithms which can be used to accelerate matrix inversion and reduce the complexity of common spatial multiplexing schemes in massive MIMO systems. Here, we specifically investigate the performance of the proposed methods in systems that utilize STBC (Space-Time Block Code) in the uplink of dynamic massive MIMO systems for different scenarios. A multi-user system in which the base station is equipped with a large number of antennas and each user has two antennas is considered. In addition, users can enter or exit the system dynamically. For a given space-time block coding/decoding scheme, the computational complexity of the receiver will be significantly reduced by employing the proposed methods. The first approach is utilizing Neumann series to approximate the inverse matrix for linear decoders. The second tactic is reducing the computational complexity of the STBC decoders when a user is added to system or removed from it. In the proposed schemes, the matrix inversion for ZF and MMSE decoding is derived from inversing a partitioned matrix and Woodbury matrix identity. Furthermore, the suggested techniques can be utilized when the number of users is fixed but the CSI changes for a particular user. The mathematical equations for both approaches are derived and the complexity of the suggested methods is compared to the direct computation of the inverse matrix. Moreover, the performance of the proposed algorithms is evaluated in terms of the system BER (bit error rate). Evaluations confirm the effectiveness of the proposed approaches.

Generalized Sherman–Morrison–Woodbury formula based algorithm for the inverses of opposite-bordered tridiagonal matrices

Matrix inverse computation is one of the fundamental mathematical problems of linear algebra and has been widely used in many fields of science and engineering. In this paper, we consider the inverse computation of an opposite-bordered tridiagonal matrix which has attracted much attention in recent years. By exploiting the low-rank structure of the matrix, first we show that an explicit formula for the inverse of the opposite-bordered tridiagonal matrix can be obtained based on the combination of a specific matrix splitting and the generalized Sherman–Morrison–Woodbury formula. Accordingly, a numerical algorithm is outlined. Second, we present a breakdown-free symbolic algorithm of $$O(n^2)$$ for computing the inverse of an n-by-n opposite-bordered tridiagonal matrix, which is based on the use of GTINV algorithm and the generalized symbolic Thomas algorithm. Finally, we have provided the results of some numerical experiments for the sake of illustration.

A Novel Linking-Domain Extraction Decomposition Method for Parallel Electromagnetic Transient Simulation of Large-Scale AC/DC Networks

Domain decomposition of the network conductance matrix is one of the efficient approaches to solve large-scale networks in parallel, wherein the most commonly-used non-iterative method is the Schur complement (SC) method. However, the SC method could not obtain the network conductance matrix inversion directly, and the computational cost will increase fast when the overlapping domain expands. In this work, a novel Linking-Domain Extraction (LDE) based decomposition method is proposed, in which the network matrix is expressed as the sum of a linking-domain matrix (LDM) and a diagonal block matrix (DBM) composed of multiple block matrices in diagonal. Through mathematical analysis over LDM, one lemma about the nature of LDM and its proof are proposed. Based on this lemma, the general formulation of the inverse matrix of the sum of LDM and DBM can be found using the Woodbury matrix identity, and based on the formulation the network matrix inversion can be directly computed in parallel to significantly accelerate the matrix inversion process. Test systems were implemented on both the FPGA and GPU parallel architectures, and the simulation results and speed-ups over the SC method and Gauss-Jordan elimination demonstrate the validity and efficiency of the proposed LDE method.

Fast algorithms for sparse portfolio selection considering industries and investment styles

In this paper, we consider a large scale portfolio selection problem with and without a sparsity constraint. Neutral constraints on industries are included as well as investment styles. To develop fast algorithms for the use in the real financial market, we shall expose the special structure of the problem, whose Hessian is the summation of a diagonal matrix and a low rank modification. Specifically, an interior point algorithm taking use of the Sherman–Morrison–Woodbury formula is designed to solve the problem without any sparsity constraint. The complexity in each iteration of the proposed algorithm is shown to be linear with the problem dimension. In the occurrence of a sparsity constraint, we propose an efficient three-block alternating direction method of multipliers, whose subproblems are easy to solve. Extensive numerical experiments are conducted, which demonstrate the efficiency of the proposed algorithms compared with some state-of-the-art solvers.

Exuber: Recursive Right-Tailed Unit Root Testing with R

This paper introduces the R package exuber for testing and date-stamping periods of mildly explosive dynamics (exuberance) in time series. The package computes test statistics for the supremum ADF test (SADF) of Phillips, Wu, and Yu (2011), the generalized SADF (GSADF) of Phillips, Shi, and Yu (2015a,b), and the panel GSADF proposed by Pavlidis, Yusupova, Paya, Peel, Martinez-Garcia, Mack, and Grossman (2016); generates finite-sample critical values based on Monte Carlo and bootstrap methods; and implements the corresponding date-stamping procedures. The recursive least-squares algorithm that we introduce in our implementation of these techniques utilizes the matrix inversion lemma and in that way achieves significant speed improvements. We illustrate the speed gains in a simulation experiment, and provide illustrations of the package using artificial series and a panel on international house prices.

Enhanced PEEC Model Based on Automatic Voronoi Decomposition of Triangular Meshes

We present an enhanced partial element equivalent circuit (PEEC) model based on Voronoi decomposition of triangular meshes. A complex geometry is accurately represented by many surface triangles, while keeping the number of elements in the corresponding equivalent circuit far less than the number of triangles. To do this, the triangular geometry is automatically decomposed into surface partitions using Voronoi's algorithm. These partitions describe capacitance cells, while inductance elements are set between all neighboring partition pairs. The inductance ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> ) and capacitance ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ) matrix calculations are done independently using solvers based on quasi-static electric and magnetic full-wave method of moments (MoM). To get the current and charge distribution over the triangles, the MoM results during <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LC</i> evaluation are combined with the results of modified nodal analysis. We validated our method against the standard PEEC and MoM results as well as measured data. Then, we used our method to simulate conducted emission for a real automotive LED brake-light module.

On the Approximation of the Full Mass Matrix in the Rotational-Coordinate-Based Beam Formulation

Abstract A novel approach is developed to approximate the full mass matrix in the rotational-coordinate-based beam formulation, which can improve the efficiency of calculating its inverse in dynamic analyses. While the rotational-coordinate-based beam formulation can reduce numbers of elements and generalized coordinates, its mass matrix is a full matrix, such that corresponding Jacobian matrix is also full, and it is time-consuming to calculate its inverse. To increase efficiency of calculating its inverse, the full mass matrix is approximated in this work. Two approximations are adopted: (1) a double integral is approximated by a single integral; and (2) a full matrix is approximated by a sum of several rank-one matrices. Through this way, the approximate mass matrix can be decomposed as a band-diagonal sparse matrix and multiplication of low-rank matrices, and its inverse can be efficiently calculated using Sherman–Woodbury formula. Through this way, the approximate mass matrix can be efficiently calculated. Several numerical examples are presented to demonstrate the performance of the current approach, and its accuracy and efficiency are analyzed.

Two solution strategies to improve the computational performance of sequentially linear analysis for quasi‐brittle structures

SummarySequentially linear analysis (SLA), an event‐by‐event procedure for finite element (FE) simulation of quasi‐brittle materials, is based on sequentially identifying a critical integration point in the FE model, to reduce its strength and stiffness, and the corresponding critical load multiplier (λcrit), to scale the linear analysis results. In this article, two strategies are proposed to efficiently reuse previous stiffness matrix factorisations and their corresponding solutions in subsequent linear analyses, since the global system of linear equations representing the FE model changes only locally. The first is based on a direct solution method in combination with the Woodbury matrix identity, to compute the inverse of a low‐rank corrected stiffness matrix relatively cheaply. The second is a variation of the traditional incomplete LU preconditioned conjugate gradient method, wherein the preconditioner is the complete factorisation of a previous analysis step's stiffness matrix. For both the approaches, optimal points at which the factorisation is recomputed are determined such that the total analysis time is minimised. Comparison and validation against a traditional parallel direct sparse solver, with regard to a two‐dimensional (2D) and three‐dimensional (3D) benchmark study, illustrates the improved performance of the Woodbury‐based direct solver over its counterparts, especially for large 3D problems.

LMI-Based Robust Stabilization of a Class of Input-Constrained Uncertain Nonlinear Systems with Application to a Helicopter Model

This paper is concerned with the robust stabilization of a class of continuous-time nonlinear systems, with an application to the pitch dynamics of a simple helicopter model, via an affine state-feedback control law using the linear matrix inequality (LMI) approach. The nonlinear dynamics is subject to norm-bounded parametric uncertainties and disturbances. In addition, the problem of actuator nonlinearity is addressed by considering the saturation effect of the control law. We demonstrate first that the synthesis problem of the saturated controller is expressed in terms of bilinear matrix inequalities (BMIs). Thanks to the Schur complement lemma and the matrix inversion lemma, we convert these BMIs into LMIs allowing the simultaneous computation of the two gains of the affine controller. Furthermore, we address in this work the estimation problem of the domain of attraction using the invariant set concept. This is solved by computing the largest attractive invariant ellipsoid. Compared with previous works, the research procedure of such ellipsoidal set is achieved in a single step with a reduced number of LMI constraints and then with less conservative conditions. A portfolio of numerical results is presented. The effectiveness and robustness of the proposed saturated controller in the stabilization of the adopted helicopter pitch model toward parametric uncertainties and disturbances are illustrated through simulation results.

An Efficient Parallel Computing Method for the Steady-State Analysis of Electric Machines Using the Woodbury Formula

This article describes a parallel computing method for the steady-state analysis of electric machines. In this method, due to the low rank of the coupling sub-matrix, the Woodbury formula is exploited to develop a highly efficient parallel algorithm to solve the formulated linearized block matrix. The algorithm and the analysis of the computational complexity are worked out in detail. The performance of the algorithm is demonstrated by an application example.

Efficient calculation of interior scattering from cavities with small modifications

The analysis of the interior scattering from open cavities with small modifications is an important task in designing a stealthy jet engine. Previous research has shown the magnetic field integral equation with the Kirchhoff approximation can be used to calculate the cavity interior scattering. However, it must repeat the expensive method of moments (MoM) solution even when the cavity is modified only slightly. In this Letter, the efficient method based on the partitioned-inverse formula and the Sherman–Morrison–Woodbury formula is employed to address this problem. It can avoid the repeated MoM direct solution. We only need to solve the lower-upper (LU) decomposition of the impedance matrix of the original cavity, and can efficiently derive the solution of the modified cavities via matrix identities without loss of accuracy. Numerical results are given to demonstrate the performance of the proposed approach.

On the Convexity of Bit Depth Allocation for Linear MMSE Estimation in Wireless Sensor Networks

Energy efficiency is crucial for a wireless sensor network (WSN) since its nodes are generally powered by energy sources of limited capacity, such as batteries. The bit depth used to quantize the sensor signal samples heavily influences energy consumption, as it strongly impacts the amount of information to be transmitted between the sensor nodes. Bit depth allocation problems seek to assign a certain bit depth to each sensor signal such that energy consumption is minimized while respecting a performance constraint. For multi-channel signal estimation tasks these problems are generally non-convex, and they are often solved through simplifying assumptions or through convex relaxation. However, for linear minimum mean squared error (MMSE) estimation, we show how the matrix inversion lemma allows to transform the MMSE constraint into a convex constraint, which can then be interpreted as a constraint on the excess MMSE due to quantization. As a result, as long as the cost function representing energy consumption is convex, this class of bit depth allocation problems is convex, i.e., if the bit depth variable is relaxed to a real-valued variable. This guarantees global optimality up to discretization of the obtained solution.

Block Iterative STMV Algorithm and Its Application in Multi-Targets Detection

STMV beamforming algorithm needs inversion operation of matrix, and its engineering application is limited due to its huge computational cost. This paper proposed block iterative STMV algorithm based on one-phase regressive filter, matrix inversion lemma and inversion of block matrix. The computational cost is reduced approximately as 1/4 M times as original algorithm when array number is M. The simulation results show that this algorithm maintains high azimuth resolution and good performance of detecting multi-targets. Within 1 - 2 dB directional index and higher azimuth discrimination of block iterative STMV algorithm are achieved than STMV algorithm for sea trial data processing. And its good robustness lays the foundation of its engineering application.

Least Square Parallel Extreme Learning Machine for Modeling NOx Emission of a 300MW Circulating Fluidized Bed Boiler

It is very important to establish an accurate combustion characteristics model of a boiler to reduce NOx emission. In this paper, a novel least square parallel extreme learning machine (LSPELM) is firstly proposed, all of whose weights and thresholds are determined by using least square method twice. Then, LSPELM is applied to 11 classical regression problems to test the validity. The experimental results show that, compared with other methods, LSPELM with a few hidden neurons can achieve good generalization and stability. Next, using Moore-Penrose generalized inverse theory and Woodbury formula, an online learning way of LSPELM (OLSPELM) based on sample increment is also proposed. If the samples of the current time are the same as those of the last time, the weights and thresholds of OLSPELM remain unchanged and are not updated. Only when the input samples of two times are different, can the weights and thresholds of OLSPELM be updated adaptively. Finally, LSPELM and OLSPELM are employed to successfully establish offline and online models of NOx emission concentration for a 300WM circulating fluidized bed boiler. The simulation results also show that LSPELM and OLSPELM have better nonlinear generalization ability and stability performance than some other state-of-the-art models. So, the proposed LSPELM and OLSPELM have good application value.

Robust Hierarchical-Optimization RLS Against Sparse Outliers

This letter fortifies the recently introduced hierarchical-optimization recursive least squares (HO-RLS) against outliers which contaminate infrequently linear-regression models. Outliers are modeled as nuisance variables and are estimated together with the linear filter/system variables via a sparsity-inducing (non-)convexly regularized least-squares task. The proposed outlier-robust HO-RLS builds on steepest-descent directions with a constant step size (learning rate), needs no matrix inversion (lemma), accommodates colored nominal noise of known correlation matrix, exhibits small computational footprint, and offers theoretical guarantees, in a probabilistic sense, for the convergence of the system estimates to the solutions of a hierarchical-optimization problem: Minimize a convex loss, which models a-priori knowledge about the unknown system, over the minimizers of the classical ensemble LS loss. Extensive numerical tests on synthetically generated data in both stationary and non-stationary scenarios showcase notable improvements of the proposed scheme over state-of-the-art techniques.

An adjoint-free four-dimensional variational data assimilation method via a modified Cholesky decomposition and an iterative Woodbury matrix formula

In this paper, we propose an efficient and practical implementation of a four-dimensional variational ensemble Kalman filter (4D-EnKF) via a modified Cholesky decomposition. The main scope of our method is to avoid the intrinsic needed of adjoint models in the four-dimensional context. As it is well known, in practice, adjoint models can be labor-intensive to develop and computationally expensive to run. We avoid the use of adjoint models by taking snapshots of an ensemble of model realizations at observation times. Then, we employ a modified Cholesky decomposition on those ensembles to build control spaces, which in turn are employed to estimate analysis increments and to mitigate the impact of sampling noise. We discuss a matrix-free implementation of our 4D-EnKF formulation via the Woodbury matrix identity. Experimental tests are performed by using the Lorenz 96 model and an atmospheric general circulation model. The numerical results reveal that the accuracy of our proposed filter implementation outperforms those of traditional 4D-EnKF formulations in terms of L-2 error norms and root-mean-square error values.

Improved Woodbury approximation approach for inelasticity-separated solid model analysis

Solid elements are widely implemented in conventional finite element method (FEM), but they also consume more computing resources relative to the bar, beam and shell elements because of the high dimensional tangent stiffness matrix recalculation and factorization. In this work, an inelasticity-separated solid element model and an efficient numerical solution procedure are proposed for the material nonlinear analysis of three-dimensional (3D) entity structures. This solid element model was developed within the framework of the inelasticity-separated finite element method (IS-FEM) presented in prior studies [1], the computational efficiency of IS-FEM is better than that of conventional FEM for the local material nonlinearity problem when using the Woodbury formula. The tangent stiffness matrix of a structure is expressed by the sum of the invariant global stiffness and another changing Schur complement modification with a low-rank. However, a nonlinearity problem with solid elements may have more inelastic degrees of freedom (IDOFs), thus, the efficiency of IS-FEM is low due to the high-rank Schur complement modification. This study improved the solution procedure of the Woodbury approximation Method (WAM), which applies the combined approximation (CA) approach within the framework of the Woodbury formula. The new derived solution scheme only contains limited back-substitutions of the global initial stiffness matrix and product of the sparse matrix and vector during a certain iteration step, which completely avoids the Schur complement matrix factorization and the constant matrix precalculation and storage process. Time complexity analysis indicates that the efficiency of the improved methodology is greatly enhanced relative to both the conventional FEM and the IS-FEM with the exact Woodbury formula, and the ratio of the efficiency improvement increases with an increase in the degrees of freedom (DOFs). Finally, the numerical examples demonstrate the validity and efficiency of the presented solid element model and the improved solution procedure.