Inverse Of System Matrix Research Articles

Weighted sum of harmonic signals for direct imaging in magnetic particle imaging

Objective. Magnetic particle imaging (MPI) is a novel radiation-free medical imaging modality that can directly image superparamagnetic iron oxide tracers (SPIOs) with high sensitivity, temporal resolution, and good spatial resolution. The MPI reconstruction task can be formulated mathematically as a Fredholm integral problem, but the concrete inversion is not easily possible because of the particle dynamics or scanner issues. Measurement based system matrix inversion takes these factors into account, but prior measurement and calibration are time consuming. Approach. We proposed a direct imaging method based on the weighted sum of harmonic signals. The harmonic signals with spatial information are obtained by the short-time Fourier transform, and odd harmonic components are selected for recombination and then mapped to the sampling trajectory to image the concentration distribution of SPIOs. In addition, we adopt a normalized-weighted sum of harmonics to improve the resolution of the native image. Main results. The effectiveness of the proposed method is verified by simulation imaging experiments and our in-house scanner-based experiments. Quantitative evaluation results show that compared with traditional methods, the structural similarity improved by 48%, mean square error decreased by 88%, and signal-to-artifact ratio increased by 2.5 times. Significance. The proposed method can rapidly image the concentration distribution of nanoparticles without any prior calibration measurements and reduce the blur of MPI images without deconvolution, which has the potential to be implemented as a multi-patch imaging method in MPI.

Fast Direct Surface Integral Equation Solution for Electromagnetic Scattering Analysis With Skeletonization Factorization

A fast direct surface integral equation (SIE) solver based on a novel skeletonization factorization scheme is proposed for electromagnetic scattering from electrically large and complex conducting objects. A novel skeletonization strategy is utilized to accelerate the skeletonization with the Huygens’ principle and proxy surface. Different from the auxiliary Rao–Wilton–Glisson (RWG) bases used in conventional methods, the constant basis functions are employed to discretize Huygens’ surface in the novel skeletonization strategy. By doing these, the number of basis functions on the surface can be greatly reduced. Moreover, the skeleton basis functions in selected neighboring groups are used to account the neighboring interactions. With these ideas, the dimensions of proxy matrix constructed for selecting skeleton can be greatly reduced. Next, a recursive skeletonization factorization (RSF) is proposed to further enhance the computational efficiency. The inverse of system matrix can be expressed as the multiplication factorization form with RSF rather than conventional recursively additive low-rank update. The computational time would be significantly saved with the application of RSF. A series of the numerical results are presented to show both the accuracy and effectiveness of the proposed method.

Sensitivity Analysis of Topological Subgraphs within Water Distribution Systems

Sensitivity analysis of the actual hydraulic state of water distribution systems is a valuable tool with number of applications in hydraulic systems analysis. Sensitivity matrices include the information of the response of the hydraulic state variables (flows, pressures) to changes in model parameters (e.g. demands, roughness, control parameters) for a specific hydraulic state of the system. For calculation, there exists in addition to finite difference approximations also exact solutions that include the inversion of the system matrix (the Schur Complement of the Jacobian of the hydraulic network equations). In combination with hydraulic network simulations, the factorization (for example Cholesky matrix decomposition) of the matrix that has already been done by the hydraulic solver in the computation process can be used for the efficient calculation of the inverse matrix. However, for large realworld networks the sensitivity calculation is a time and memory consuming process because the inverse of the system matrix of a connected network has no zero elements. In this paper a new method is presented that allows for the exact calculation of sensitivities of a particular subgraph of interest, the topological minor or supergraph within a water distribution system network graph. Supernodes are the most important nodes in terms of connectivity redundancy within the network graph. Superlinks replace all pipes (links) in series between two supernodes. It will be shown that the sensitivities that are calculated for the subgraph deliver exactly the same results as the inversion of the entire system matrix reduced to supernodes. This paper focuses on the derivation of the equations for the reduced system matrix inversion of the topological subgraph. In addition, the paper includes the proof of equivalence of the matrix inverses for the topological minor subgraph. This inverse represents the fundamental sensitivities of nodal heads and pipe flows with respect to nodal demands in demand driven analysis. The results presented can be extended to other sensitivities, since the matrix inverse in question is included in all other derived parameter sensitivities.

Discrete–time Output Feedback Sliding Mode Control of a Large Pressurized Heavy Water Reactor

In this paper a novel method is presented to design a sliding mode spatial control for a large Pressurized HeavyWater Reactor (PHWR) using a new formulation of Multirate Output Feedback (MROF). In the new formulation of MROF, the outputs of the system should be some of the states of the system or system should be in special form. The non–linear model of PHWR including xenon and iodine dynamics is characterized by 70 state variables and 14 inputs and outputs each. Linear nodal model is obtained by linearizing the non–linear dynamic equations of the reactor about the full power operating point. The 14 outputs of the PHWR are the power levels in 14 zones, also these are the 14 states of the system. The PHWR model is perfectly suitable for the application of this new formulation in which the states of the system can be computed using reduced system matrix inversion. The PHWR is an ill–conditioned system and the computation of the states of the system using existing MROF require the larger matrix inversions which sometimes may not possible. The proposed approach avoids this difficulty and produces the similar result as it is produced by the existing technique. The proposed control method does not require state information of the system for feedback purposes and hence may be easier to implement. From simulation of the non–linear model of the reactor in representative transients, the proposed control scheme is found to be superior to other methods.

Advanced analysis of coupled buffeting response of bridges: a complex modal decomposition approach

This paper presents a framework based on a complex modal decomposition technique for predicting coupled buffeting response of bridges in both time and frequency domains. The coupled equations of motion in structural modal coordinates with frequency dependent aeroelastic self-excited terms are approximated by frequency independent state-space equations, without augmented aerodynamic states, which retain the complex modal properties of the original system. These equations are then decomposed into a set of uncoupled equations of motion for buffeting response analysis. The frequency dependent unsteady buffeting characteristics and their spanwise correlation are considered in both frequency and time domain analyses instead of invoking the customary quasi-steady assumption. This framework significantly enhances computational efficiency in the frequency domain analysis by avoiding system matrix inversion at each discretized frequency when evaluating the transfer function matrix. Furthermore, it also offers simulation of buffeting response in the time domain that includes frequency dependence of buffeting and self-excited forces. A detailed discussion concerning the complex modal frequencies, damping ratios, mode shapes, and the significance of structural modes on the multimode coupled buffeting response is provided. This helps to glean additional insight and to improve our understanding of the underlying physics of wind–structure interactions. Examples of long span suspension bridges are provided to illustrate the proposed scheme and to demonstrate its effectiveness.

A robust, parallelizable, O(m), a posteriori recursive least squares algorithm for efficient adaptive filtering

This article presents a new recursive least squares (RLS) adaptive algorithm. The proposed computational scheme uses a finite window by means of a lemma for the system matrix inversion that is, for the first time, stated and proven here. The new algorithm has excellent tracking capabilities. Moreover, its particular structure allows for stabilization by means of a quite simple method. Its stabilized version performs very well not only for a white noise input but also for nonstationary inputs as well. It is shown to follow music, speech, environmental noise, etc., with particularly good tracking properties. The new algorithm can be parallelized via a simple technique. Its parallel form is very fast when implemented with four processors.

A practical algorithm for the efficient computation of eigenvector sensitivities

Derivatives of eigenvalues and eigenvectors have become increasingly important in the development of modern numerical methods for areas such as structural design optimization, dynamic system identification and dynamic control, and the development of effective and efficient methods for the calculation of such derivatives has remained to be an active research area for several decades. In this paper, a practical algorithm has been developed for efficiently computing eigenvector derivatives of generalized symmetric eigenvalue problems. For eigenvector derivative of a separate mode, the computation only requires the knowledge of eigenvalue and eigenvector of the mode itself and an inverse of system matrix accounts for most computation cost involved. In the case of two close modes, the modal information of both modes is required and the eigenvector derivatives can be accurately determined simultaneously at minor additional computational cost. Further, the proposed method has been extended to the case of practical structural design where structural modifications are made locally and the eigenderivatives of the modes concerned before are still of interest. By combining the proposed algorithm together with the proposed inverse iteration technique and singular value decomposition theory, eigenproperties and their derivatives can be very efficiently computed. Numerical results from a practical finite element model have demonstrated the practicality of the proposed method. The proposed method can be easily incorporated into commercial finite element packages to improve the computational efficiency of eigenderivatives needed for practical applications.