Linear Systems In Parallel Research Articles

Commentary: Pharmacokinetic Theory Must Consider Published Experimental Data.

Recently, we have proposed simple methodology to derive clearance and rate constant equations, independent of differential equations, based on Kirchhoff's Laws, a common methodology from physics used to describe rate-defining processes either in series or parallel. Our approach has been challenged in three recent publications, two published in this journal, but notably what is lacking is that none evaluate experimental pharmacokinetic data. As reviewed here, manuscripts from our laboratory have evaluated published experimental data, demonstrating that the Kirchhoff's Laws approach explains (1) why all of the experimental perfused liver clearance data appear to fit the equation that was previously believed to be the well-stirred model, (2) why linear pharmacokinetic systemic bioavailability determinations can be greater than 1, (3) why renal clearance can be a function of drug input processes, and (4) why statistically different bioavailability measures may be found for urinary excretion versus systemic concentration measurements. Our most recent paper demonstrates (5) how the universally accepted steady-state clearance approach used by the field for the past 50 years leads to unrealistic outcomes concerning the relationship between liver-to-blood Kpuu and hepatic availability FH , highlighting the potential for errors in pharmacokinetic evaluations based on differential equations. The Kirchhoff's Laws approach is applicable to all pharmacokinetic analyses of quality experimental data, those that were previously adequately explained with present pharmacokinetic theory, and those that were not. The publications that have attempted to rebut our position do not address unexplained experimental data, and we show here why their analyses are not valid. SIGNIFICANCE STATEMENT: The Kirchhoff's Laws approach to deriving clearance equations for linear systems in parallel or in series, independent of differential equations, successfully describes published pharmacokinetic data that has previously been unexplained. Three recent publications claim to refute our proposed methodology; these publications only make theoretical arguments, do not evaluate experimental data, and never demonstrate that the Kirchhoff methodology provides incorrect interpretations of experimental pharmacokinetic data, including statistically significant data not explained by present pharmacokinetic theory. We demonstrate why these analyses are invalid.

Domain decomposition based preconditioner combined local low-rank approximation with global corrections

To solve general sparse linear systems, this paper presents a domain decomposition based parallel preconditioner. Vertex-based partitioning is utilized to reorder the original coefficient matrix, resulting in a s×s block structure. Here, s is the number of subdomains used in the partition. Variables corresponding to the interface nodes are obtained by solving a linear system with coefficient matrix being the Schur complement S of the reordered matrix. Combining local low-rank correction approximation with a global low-rank correction technique to approximate the inverse of S, the method presented in this paper is different with previous Schur complement based preconditioners. The global low-rank correction terms are obtained by using the information comes from the local low-rank correction terms. In addition, variables corresponding to the interior nodes are computed by solving s small linear systems in parallel. Some numerical tests are presented to show the efficiency and robustness of the proposed preconditioner.

A parallelization strategy for hydro-mechanically coupled mechanized tunneling simulations

Abstract 3D computational simulations of the tunnel advancement in mechanized tunneling usually demand high computational resources. This paper addresses strategies to enable large scale simulations of the advancement process by means of iterative, parallel solvers, considering frictional contact between the tunnel boring machine and the soil, fully saturated soil with different permeabilities and a number of additional features inherently connected with tunnel simulations, which require special attention in regards to robustness. To improve the computational performance of mechanized tunnel simulations a solution strategy based on domain decomposition is proposed to solve the resulting discretized linear system in parallel using an appropriately accelerated iterative solver. The interaction between the TBM and the ground is modeled by means of robust contact algorithm based on penalty method. The domain decomposition scheme accounts for contact treatment in parallel, which becomes relevant when the TBM traverses through different domains in the mesh. Different variants of block preconditioners are investigated in regards to their performance to speed-up the convergence of the iterative solver for the block linear system arising from simulations of tunnel advancement in saturated soft soil, considering different permeabilities. A validation of the computational model and selected numerical examples to showcase the efficiency of the proposed method and to verify the applicability of this strategy to high performance tunneling simulations are presented.

Modified method of parallel matrix sweep

The topic of this paper refers to efficient parallel solvers of block-tridiagonal linear systems of equations. Such systems occur in numerous modeling problems and require usage of high-performance multicore computation systems. One of the widely used methods for solving block-tridiagonal linear systems in parallel is the original block-tridiagonal sweep method. We consider the algorithm based on the partitioning idea. Firstly, the initial matrix is split into parts and transformations are applied to each part independently to obtain equations of a reduced block-tridiagonal system. Secondly, the reduced system is solved sequentially using the classic Thomas algorithm. Finally, all the parts are solved in parallel using the solutions of a reduced system. We propose a modification of this method. It was justified that if known stability conditions for the matrix sweep method are satisfied, then the proposed modification is stable as well.

Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient Method

Pipelined Krylov subspace methods (also referred to as communication-hiding methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite system matrices the pipelined Conjugate Gradient method, p($l$l)-CG, outperforms its classic Conjugate Gradient counterpart on large scale distributed memory hardware by overlapping global communication with essential computations like the matrix-vector product, thus “hiding” global communication. A well-known drawback of the pipelining technique is the (possibly significant) loss of numerical stability. In this work a numerically stable variant of the pipelined Conjugate Gradient algorithm is presented that avoids the propagation of local rounding errors in the finite precision recurrence relations that construct the Krylov subspace basis. The multi-term recurrence relation for the basis vector is replaced by $\ell$l three-term recurrences, improving stability without increasing the overall computational cost of the algorithm. The proposed modification ensures that the pipelined Conjugate Gradient method is able to attain a highly accurate solution independently of the pipeline length. Numerical experiments demonstrate a combination of excellent parallel performance and improved maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm. This work thus resolves one of the major practical restrictions for the useability of pipelined Krylov subspace methods.

A robust hierarchical solver for ill-conditioned systems with applications to ice sheet modeling

A hierarchical solver is proposed for solving sparse ill-conditioned linear systems in parallel. The solver is based on a modification of the LoRaSp method, but employs a deferred-compression technique, which provably reduces the approximation error and significantly improves efficiency. Moreover, the deferred-compression technique introduces minimal overhead and does not affect parallelism. As a result, the new solver achieves linear computational complexity under mild assumptions and excellent parallel scalability. To demonstrate the performance of the new solver, we focus on applying it to solve sparse linear systems arising from ice sheet modeling. The strong anisotropic phenomena associated with the thin structure of ice sheets creates serious challenges for existing solvers. To address the anisotropy, we additionally developed a customized partitioning scheme for the solver, which captures the strong-coupling direction accurately. In general, the partitioning can be computed algebraically with existing software packages, and thus the new solver is generalizable for solving other sparse linear systems. Our results show that ice sheet problems of about 300 million degrees of freedom have been solved in just a few minutes using 1024 processors.

Parallel software platform INMOST: a framework for numerical modeling

The INMOST mathematical modeling toolkit helps a user to formulate and solve a problem of partial differential equations on general meshes in parallel. The current work covers: data structure description for efficient distributed unstructured mesh representation, interrelation of mesh elements with maximal flexibility of supported types of the mesh, treatment of ghost cells and distribution of mesh data for parallel execution, flexible templates for the implementation of numerical schemes, convenient framework for parallel linear systems assembly and solution. We also present aspects of the implementation and a simple example of application of INMOST to the solution of anisotropic diffusion problem. On this example we demonstrate the application of INMOST for all the stages of numerical modeling: construction of the distributed mesh, assignment of the problem data to the elements, problem discretization on local domain, solution of linear system in parallel. INMOST is a newly developed, flexible and efficient numerical analysis framework that provides scientists the infrastructure for designing highly scalable high performance applications for mathematical modeling.

Walking in the cloud

Despite its popularity, SimRank is computationally costly, in both time and space. In particular, its recursive nature poses a great challenge in using modern distributed computing power, and also prevents querying similarities individually. Existing solutions suffer greatly from these practical issues. In this paper, we break such dependency for maximum efficiency possible. Our method consists of offline and online phases. In offline phase, a length- n indexing vector is derived by solving a linear system in parallel. At online query time, the similarities are computed instantly from the index vector. Throughout, the Monte Carlo method is used to maximally reduce time and space. Our algorithm, called CloudWalker, is highly parallelizable, with only linear time and space. Remarkably, it responses to both single-pair and single-source queries in constant time. CloudWalker is orders of magnitude more efficient and scalable than existing solutions for large-scale problems. Implemented on Spark with 10 machines and tested on the web-scale clue-web graph with 1 billion nodes and 43 billion edges, it takes 110 hours for offline indexing, 64 seconds for a single-pair query, and 188 seconds for a single-source query. To the best of our knowledge, our work is the first to report results on clue-web, which is 10x larger than the largest graph ever reported for SimRank computation.

Analysis of soil carbon transit times and age distributions using network theories

The long‐term soil carbon dynamics may be approximated by networks of linear compartments, permitting theoretical analysis of transit time (i.e., the total time spent by a molecule in the system) and age (the time elapsed since the molecule entered the system) distributions. We compute and compare these distributions for different network configurations, ranging from the simple individual compartment, to series and parallel linear compartments, feedback systems, and models assuming a continuous distribution of decay constants. We also derive the transit time and age distributions of some complex, widely used soil carbon models (the compartmental models CENTURY and Rothamsted, and the continuous‐quality Q‐Model), and discuss them in the context of long‐term carbon sequestration in soils. We show how complex models including feedback loops and slow compartments have distributions with heavier tails than simpler models. Power law tails emerge when using continuous‐quality models, indicating long retention times for an important fraction of soil carbon. The responsiveness of the soil system to changes in decay constants due to altered climatic conditions or plant species composition is found to be stronger when all compartments respond equally to the environmental change, and when the slower compartments are more sensitive than the faster ones or lose more carbon through microbial respiration.

Metric Based Multi-Timescale Control for Reducing Power in Embedded Systems

Digital control for embedded systems often requires low-power, hard real-time computation to satisfy high control-loop bandwidth, low latency, and low-power requirements. In particular, the emerging applications of Micro Electro-Mechanical Systems (MEMS) sensors, and their increasing integration, presents a challenging requirement to embed ultra-low power digital control architectures for these lithographically formed micro-structures. Controlling electromechanical structures of such a small scale, using naive digital controllers, can be prohibitively expensive (both in power and cost for portable or battery operated applications). In this paper, we describe the potential for control systems to be transformed into a set of co-operating parallel linear systems and demonstrate, for the first time, that this parallelization can reduce the total number of instructions executed, thereby reducing power, at the expense of controlled loss in control fidelity. Since the error tolerance of linear feedback control systems is mathematically well posed, this technique opens up a new, independent dimension for system optimization. We present a novel Computer-Aided Design (CAD) method to evaluate control fidelity, with varying timescales on the controller, and analyze the trade-off between performance and power dissipation. A CAD Metric for control fidelity is proposed and we demonstrate the potential for power savings using this decomposition on three different control problems.

Fast, exactk-space sample density compensation for trajectories composed of rotationally symmetric segments, and the SNR-optimized image reconstruction from non-Cartesian samples

A recently developed method for exact density compensation of non uniformly arranged samples relies on the analytically known cross-correlations of Fourier basis functions corresponding to the traced k-space trajectory. This method produces a linear system whose solution represents compensated samples that normalize the contribution of each independent element of information that can be expressed by the underlying trajectory. Unfortunately, linear system-based density compensation approaches quickly become computationally demanding with increasing number of samples (i.e., image resolution). Here, it is shown that when a trajectory is composed of rotationally symmetric interleaves, such as spiral and PROPELLER trajectories, this cross-correlations method leads to a highly simplified system of equations. Specifically, it is shown that the system matrix is circulant block-Toeplitz so that the linear system is easily block-diagonalized. The method is described and demonstrated for 32-way interleaved spiral trajectories designed for 256 image matrices; samples are compensated non iteratively in a few seconds by solving the small independent block-diagonalized linear systems in parallel. Because the method is exact and considers all the interactions between all acquired samples, up to a 10% reduction in reconstruction error concurrently with an up to 30% increase in signal to noise ratio are achieved compared to standard density compensation methods.

Parallel Newtonian Optimization without Hessian Approximation

The purpose of this paper is to introduce parallel algorithms based on the Newton method for solving non-linear unconstrained optimization problem in (MIMD) parallel computers by solving linear system in parallel using Gaussian Elimination method rather than finding inverse Hessian matrix to avoid the errors caused by evaluating the inverse matrix and also to increase computing power and reduce run time.

Studies of an infinite element method for acoustical radiation

Infinite element computations are very efficient for predicting the vibro-acoustic response and sensitivities of a vibrating structure for an exterior acoustic domain. In addition, domain decomposition methods are very powerful algorithms for solving large linear systems in parallel. In this paper, an infinite element method is proposed and analyzed for parallel computations purpose. An original formulation of this method with Lagrange multipliers defined on (semi-)infinite space is presented. The implementation aspects of this method in an industrial acoustic software (SYSNOISE) are discussed. New numerical results illustrate the efficiency of the proposed method for realistic acoustical radiation problems.

Versatile Two-Level Schwarz Preconditioners for Multiphase Flow

Numerically modeling groundwater flow on finely discretized two- and three-dimensional domains requires solution algorithms appropriate for distributed memory multiprocessor architectures. Multilevel and domain decomposition algorithms are appropriate for preconditioning or solving linear systems in parallel and have, therefore, been applied to linear models for saturated groundwater flow. These algorithms have also been incorporated into more complex nonlinear multiphase flow models in the context of a linearization procedure such as Newton's method. In this work, we study a class of parallel preconditioners based on two-level Schwarz domain decomposition applied in a nonlinear two-phase flow numerical model. The restriction and interpolation operators are based on an aggregation approach that has a straightforward implementation for a variety of applications arising in subsurface modeling: structured and unstructured discretizations, finite elements and finite differences, and multicomponent model equations. We present model formulations, results from numerical experiments, and a comparison of a standard one-level Schwarz method to three two-level aggregation-based methods.

A novel nonlinear system identification technique for modeling cochlear transduction using otoacoustic emissions data

A novel mathematical technique is presented for modeling the linear and nonlinear properties of the cochlear transduction process. This new nonlinear system identification procedure (Bendat, 1990) has several advantages over classical techniques used for analyzing nonlinear systems. With this technique, nonlinear systems may be modeled as a linear system in parallel with any number of finite-memory nonlinear systems. Higher-order nonlinear terms can be more easily computed with this technique than with others such as volterra kernel analysis. Furthermore, a broadband signal such as a transient signal can be used to characterize the cochlea over a wide frequency band. This makes the application of this procedure ideal for modeling cochlear transduction process with transient evoked otoacoustic emissions data. Otoacoustic emissions were recorded in response to finite impulse response (FIR) pulses from 11 subjects with normal hearing. System identification procedures were implemented using third- and fifth-order polynomial models. Based on the input and output records, system properties were estimated. Coherence measures tested the goodness-of-fit of the chosen model. The identified systems were used to predict the systems’ response to two-tone stimuli. The fifth-order model predictions were in better accordance with previously reported data on acoustic distortion products.

Fast parallel algorithms for matrix inversion and linear systems solution

An approach to preconditioning linear systems is presented, which is well suitable for parallel implementation. Such approach leads to efficient parallel linear systems solvers, as well as to new schemes for matrix inversion, in some special cases.

Spectral analysis of non-linear systems involving square-law operations

This paper contains a number of useful theoretical formulas to analyze the frequency domain properties of Gaussian input data passing through non-linear square-law systems. Special bispectral density functions are defined and applied that are functions of a single variable. From measurements of input data and output data only, results are obtained to identify the separate frequency response functions for two models of linear systems in parallel with non-linear square-law systems. Non-linear coherence functions are defined from these models which determine the proportion of the output spectrum due to the non-linear operations. Together with ordinary coherence functions, a measured output spectrum for these models can be decomposed into three components representing the linear operations, the non-linear operations, and the remaining uncorrelated noise effects. This material indicates also how to analyze other types of non-linear models by employing similar techniques.

Necessary and Sufficient Conditions for the Controllability and Observability of General Composite Systems

Necessary and sufficient conditions for the controllability and observability of general series parallel or feedback composite linear multivariable systems of the class investigated by Davison and wang(1) are obtained by exploiting certain properties of the generalised resultant(7), (8). These conditions are expressed directly in terms of the parameters of the state equations of the sub-systems and in terms of the interconnecting parameters of the composite systems in the manner of Davison and Wang(1), but the present conditions require no computations of eigenvalue spectra.

Optimum filtering with a class of nonlinear systems

This paper considers the problem of nonlinear filtering and interpolation of random input in the presence of noise. A special class of nonlinear systems is used which is composed of a linear system in parallel with a zero-memory nonlinearity (ZNL) preceded by a variable delay. The optimization problem is simplified if the inputs are restricted to be of the separable class. The optimum system of this class is shown to be implemented in terms of the optimum linear filter and the optimum ZNL filter. Furthermore, a suboptimal scheme can also be constructed by adding a ZNL to the optimum linear system which improves its performance. The dependence of the error improvement on the delay is also considered, and the best choice of the delay is illustrated by an example.

A mathematical model for nonlinear hydrologic systems

A new type of mathematical model for hydrologic systems is introduced. This decomposition model represents nonlinear time-lag systems, in decomposed form, by a series combination of parallel linear time-lag systems and parallel nonlinear no-time-lag systems. Thus the assumption that hydrologic systems are linear is no longer postulated. The theory of the decomposition model and its use in the prediction of the system output, given the input, are discussed. Computational algorithms for model construction and for its use in output prediction are given. A number of decomposition models have been constructed for a small watershed. These computational experiments established the applicability of the proposed method and its practicality in terms of ease of application and small computational cost. It is also shown how specific parameters of a given system can be determined. These parameters include length of the record that is needed to obtain a model of reasonable quality, length of the maximum time lag in a given system and its degree of nonlinearity, and details of the model structure. In particular, for a 2.33-acre watershed it was found that a record of 3000 hours suffices for model derivation, that the system maximum time lag was approximately 5 hours, and that three linear time-lag systems and up to fourth-order nonlinear, no-time-lag systems had to be included in the model.