Large Systems Of Equations Research Articles

A model for pressure driven analysis-design of water distribution networks

Pressure Driven Analysis (PDA) method provides the ability to simulate Water Distribution Networks (WDNs) more realistically. Most of the PDA-based models use either the Newton–Raphson method or the gradient method for simulation. The computational cost of these models is high, so that in the solution process of them a large system of equations is solved in each iteration. This paper presents a new model for PDA-based design and analysis of WDNs. The proposed model developed employing the graph portioning approach is named GPDA model. This model provides the PDA-based design and analysis of WDNs without any need of iterative computations for governing equations. Two example networks are designed and analyzed using the proposed model and the results are discussed. The obtained results confirm the ability and computational efficiency of the GPDA model for the design and analysis of WDNs.

Express Diagnostic Method of the Stress State of Weakly Magnetized Ferromagnetic Rods

A method for express diagnostics of weakly magnetized ferromagnetic rods is proposed. It is based on measuring the stray magnetic field of the rod. We analyze the inverse magnetostatic problem and formulate the conditions when its regularization is possible. In this case, it is possible to connect the value of the average transverse component of magnetization with the results of the measured stray magnetic field. The distributions of transverse magnetization values were experimentally studied using rods made of SV08G2S [transliterated from Russian] and ER-308LSi steels as an example. The studied samples were subjected to impact plastic deformation, tensile deformation, and torsion deformation. It is shown that this method allows to register areas with stress concentrations without additional magnetization of the test sample. A feature of the method is the use of once calculated system functions without solving large systems of equations and using unstable spectral methods. The method is applicable for cylindrical objects of small radius far from the edge of the test sample, when the magnetization in the cross section of the sample can be considered as constant.

CONSTRUCTION OF FRACTAL SURFACES

The paper approaches the construction of fractal surfaces of interpolation and approximation on the basis of a fractal perturbation of any mapping defined on a rectangle. Conditions for the differentiability of these elements are also provided. The fractal surfaces obtained may be used for the approximation of real-world data. The method proposed does not require any restriction on the type of data. Furthermore, the present approach does not imply the solution of large systems of equations. The paper considers both the continuous and the discontinuous case.

Explicit Iterative Methods of Second Order and Approximate Inverse Preconditioners for Solving Complex Computational Problems

Explicit Exact and Approximate Inverse Preconditioners for solving complex linear systems are introduced. A class of general iterative methods of second order is presented and the selection of iterative parameters is discussed. The second order iterative methods behave quite similar to first order methods and the development of efficient preconditioners for solving the original linear system is a decisive factor for making the second order iterative methods superior to the first order iterative methods. Adaptive preconditioned Conjugate Gradient methods using explicit approximate preconditioners for solving efficiently large sparse systems of algebraic equations are also presented. The generalized Approximate Inverse Matrix techniques can be efficiently used in conjunction with explicit iterative schemes leading to effective composite semi-direct solution methods for solving large linear systems of algebraic equations.

Error Estimators for Proper Generalized Decomposition in Time-Dependent Electromagnetic Field Problems

Due to fine discretization in space and time, the simulation of transient electromagnetic phenomena results in a large system of equations. To cope with this computational effort, model order reduction techniques can be employed. To assess the accuracy of the solution of the reduced model, an error estimation is crucial. A commonly used approach consists of the evaluation of the deviation between the reduced and the full model. This yields a loss of the a-priori property of the Proper Generalized Decomposition. To overcome this problem two a-priori criteria are presented in this paper.

Cloaking of a circular cylindrical elastic inclusion from antiplane elastic waves and resonance effects

Cloaking of a circular cylindrical elastic inclusion embedded in a homogeneous linear isotropic elastic medium from antiplane elastic waves is studied. The transformation or change-of-variables method is used to determine the material properties of the cloak and the homogenization theory of composites is used to construct a multilayered cloak consisting of many bi-material cells. The large system of algebraic equations associated with this problem is solved by using the concept of multiple scattering with wave expansion coefficient matrices. Numerical results for cloaking of an elastic inclusion and a rigid inclusion are compared with the case of a cavity. It is found that while the cloaking patterns for the three cases are similar, the major difference is that standing waves are generated in the elastic inclusion and the multilayered cloak cannot prevent the motion inside the elastic inclusion, even though the cloak seems nearly perfect. Waves can penetrate into and cause vibrations inside the elastic inclusion, where the amplitude of standing waves depend on the material properties of the inclusion but are very much reduced when compared to the case when there is no cloak. For a prescribed mass density, the displacements inside the elastic cylinder decrease as the shear modulus increases. Moreover, the cloaking of the elastic inclusion over a range of wavenumbers is also investigated. There is significant low frequency scattering even if the cloak consists of a large number of layers. When the wavenumber increases, the multilayered cloak is not effective if the cloak consists of an insufficient number of layers. Resonance effects that occur in cloaking of elastic inclusions are also discussed.

The Convergence Estimation of the Parallel Algorithm of the Linear Cauchy Problem Solution for Large Systems of First-Order Ordinary Differential Equations Using the Solution as Expansion over Orthogonal Polynomials

This paper is devoted to the algorithm of the linear Cauchy problem solution for large systems of first-order ordinary differential equations using parallel calculations. The proof of the convergence of the iteration process using the solution as expansion over orthogonal polynomials for the interval [0,1] is presented. The features of this algorithm are its simplicity, the opportunity to get a solution by parallel calculations, and also the possibility to get a solution for nonlinear problems by changing the operator using the solution from the iteration process.

Structure-preserving vs. standard particle-in-cell methods: The case of an electron hybrid model

We applied two numerical methods both belonging to the class of finite element particle-in-cell methods to a four-dimensional (one dimension in real space and three dimensions in velocity space) hybrid plasma model for electrons in a stationary, neutralizing background of ions. Here, the term hybrid means that (energetic) electrons with velocities close to the phase velocities of the model's characteristic waves are treated kinetically, whereas electrons that are much slower than the phase velocity are treated with fluid equations. The two developed numerical schemes are based on standard finite elements on the one hand and on structure-preserving geometric finite elements on the other hand. We tested and compared the schemes in the linear and in the nonlinear stage. We show that the structure-preserving algorithm leads to better results in both stages. This can be related to the fact that the spatial discretization results in a large system of ordinary differential equations that exhibits a noncanonical Hamiltonian structure. To such systems special time integration schemes with good conservation properties can be applied.

Generalized Multiscale Finite Element method for multicontinua unsaturated flow problems in fractured porous media

In this paper, we present a multiscale method for simulations of the multicontinua unsaturated flow problems in heterogeneous fractured porous media. The mathematical model is described by the system of Richards equations for each continuum that is coupled by the specific transfer term. To illustrate the idea of our approach, we consider a dual continua background model with discrete fractures networks that is generalized as a multicontinua model for unsaturated fluid flow in the complex heterogeneous porous media. We present fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) approach. In this model, we construct an unstructured fine grid that takes into account complex fracture geometries for two and three dimensional formulations. Due to construction of the unstructured grid, the fine grid approximation leads to the very large system of equations. For reduction of the discrete system size, we develop a multiscale method for coarse grid approximation of the coupled problem using Generalized Multiscale Finite Element Method (GMsFEM). In this method, we construct coupled multiscale basis functions that are used to construct highly accurate coarse grid approximation. The multiscale method allowed us to capture detailed interactions between multiple continua. The adaptive approach is investigated, where we consider two approaches for multiscale basis functions construction: (1) based on the spectral characteristics of the local problems and (2) using simplified multiscale basis functions. We investigate accuracy of the proposed method for the several test problems in two and three dimensional formulations. We present a comparison of the relative error for different number of basis functions and for adaptive approach. Numerical results illustrate that the presented method provides accurate solution of the unsaturated multicontinua problem on the coarse grid with huge reduction of the discrete system size.

Propagation of chaos for mean field rough differential equations

We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type dXt=V(Xt,L(Xt))dt+F(Xt,L(Xt))dWt, where W is a random rough path and L(Xt) is the law of Xt. We prove propagation of chaos, and provide also an explicit optimal convergence rate. The analysis is based upon the tools we developed in our companion paper (Electron. J. Probab. 25 (2020) 21) for solving mean field rough differential equations and in particular upon a corresponding version of the Itô-Lyons continuity theorem. The rate of convergence is obtained by a coupling argument developed first by Sznitman for particle systems with Brownian inputs.

Numerical solution of two-energy-group neutron noise diffusion problems with fine spatial meshes

The paper presents the development of a strategy for the fine-mesh full-core computation of neutron noise in nuclear reactors. Reactor neutron noise is related to fluctuations of the neutron flux induced by stationary perturbations of the properties of the system. Its monitoring and analysis can provide useful insights in the reactor operations. The model used in the work relies on the neutron diffusion approximation and requires the solution of both the criticality (eigenvalue) and neutron noise equations. A high-resolution spatial discretization of the equations is important for an accurate evaluation of the neutron noise because of the strong gradients that may arise from the perturbations. Considering the size of a nuclear reactor, the application of a fine mesh generates large systems of equations which can be challenging to solve. Then, numerical methods that can provide efficient solutions for these kinds of problems using a reasonable computational effort, are investigated. In particular the power method accelerated with the Chebychev or JFNK-based techniques for the eigenvalue problem, and GMRES with the Symmetric Gauss-Seidel, ILU, SPAI preconditioners for the solution of linear systems, are evaluated with the computation of neutron noise in the case of localized perturbations in 1-D and 2-D simplified reactor cores and in a 3D realistic reactor core.

A comparison of one-step and two-step W-methods and peer methods with approximate matrix factorization

One- and Two-step W-methods and peer methods are considered combined with an Approximate Matrix Factorization (AMF) for the numerical solution of large systems of stiff Ordinary Differential Equations (ODEs) coming from the spatial discretization of parabolic Partial Differential Equations (PDEs) of convection–diffusion–reaction type. Although one-step AMF W-methods do not require starting values and have larger stability regions, they may suffer from order reduction for such PDE problems when time-dependent Dirichlet boundary conditions are imposed. Here by numerical tests it is illustrated that two-step AMF W-methods and AMF peer methods avoid this order reduction (in stiff sense, i.e. for a fixed spatial grid) due to their high stage order. On the other hand, while AMF W-methods are linearly implicit, AMF peer methods require the solution of nonlinear equations in order to compute an advancing solution. In this case, the predictor considered for the internal stages influences both stability and accuracy. However, with only one Newton iteration, for a given number of stages, the three classes of methods have a similar computational cost.

Reduced Model to Compute AC Losses of Twisted Multifilamentary Superconductors

Classical numerical techniques remain very computationally expensive to model the magnetization of twisted multifilamentary superconductors in three dimensions because they require solving a large system of equations. In this paper, a reduced model is proposed to approximate the ac losses of these types of wires. It is established through the projection of the physical problem in a helical Frenet basis and the use of periodic boundary conditions on a slice of the wire. The reduced model on the slice leads to systems of more reasonable size leading to smaller and faster simulations than when the whole wire is modeled. Simulation results show that the proposed method leads to computed losses similar to those obtained on a whole wire. Several simulation cases are proposed to prove the validity of the proposed model, and a comparison to some ac loss measurements is proposed to show the effectiveness of the proposed approach. The implementation relies on the H-formulation and is solved with the open-source solver, Getdp.

Improving locality of explicit one-step methods on GPUs by tiling across stages and time steps

The performance of explicit parallel methods solving large systems of ordinary differential equations (ODEs) on GPUs is often memory bound. Therefore, locality optimizations, such as kernel fusion, are desirable. This paper exploits a special property of a large class of right-hand-side (RHS) functions to enable the fusion of computations of blocks of components of dependent stages of the method. This allows the derivation of tilings of the stages not only within one time step, but also spanning several successive time steps. Our approach is based on a representation of the ODE method by a data flow graph and allows efficient GPU code with fused kernels to be generated automatically for user-defined tilings. In particular, we investigate two generalized tiling strategies, trapezoidal and hexagonal tiling, and two different partitionings, which are evaluated experimentally for several different high- and low-order Runge–Kutta (RK) methods.

Computational modeling of therapy on pancreatic cancer in its early stages

More than eighty percent of pancreatic cancer involves ductal adenocarcinoma with an abundant desmoplastic extracellular matrix surrounding the solid tumor entity. This aberrant tumor microenvironment facilitates a strong resistance of pancreatic cancer to medication. Although various therapeutic strategies have been reported to be effective in mice with pancreatic cancer, they still need to be tested quantitatively in wider animal-based experiments before being applied as therapies. To aid the design of experiments, we develop a cell-based mathematical model to describe cancer progression under therapy with a specific application to pancreatic cancer. The displacement of cells is simulated by solving a large system of stochastic differential equations with the Euler–Maruyama method. We consider treatment with the PEGylated drug PEGPH20 that breaks down hyaluronan in desmoplastic stroma followed by administration of the chemotherapy drug gemcitabine to inhibit the proliferation of cancer cells. Modeling the effects of PEGPH20 + gemcitabine concentrations is based on Green’s fundamental solutions of the reaction–diffusion equation. Moreover, Monte Carlo simulations are performed to quantitatively investigate uncertainties in the input parameters as well as predictions for the likelihood of success of cancer therapy. Our simplified model is able to simulate cancer progression and evaluate treatments to inhibit the progression of cancer.

Hierarchical Template Matching for 3D Myocardial Tracking and Cardiac Strain Estimation

Myocardial tracking and strain estimation can non-invasively assess cardiac functioning using subject-specific MRI. As the left-ventricle does not have a uniform shape and functioning from base to apex, the development of 3D MRI has provided opportunities for simultaneous 3D tracking, and 3D strain estimation. We have extended a Local Weighted Mean (LWM) transformation function for 3D, and incorporated in a Hierarchical Template Matching model to solve 3D myocardial tracking and strain estimation problem. The LWM does not need to solve a large system of equations, provides smooth displacement of myocardial points, and adapt local geometric differences in images. Hence, 3D myocardial tracking can be performed with 1.49 mm median error, and without large error outliers. The maximum error of tracking is up to 24% reduced compared to benchmark methods. Moreover, the estimated strain can be insightful to improve 3D imaging protocols, and the computer code of LWM could also be useful for geo-spatial and manufacturing image analysis researchers.

Monte Carlo Method for Solving ODE Systems

The Monte Carlo method is applied to solve Cauchy problems for a system of linear and nonlinear ordinary differential equations. The Monte Carlo method is relevant for the solution of large systems of equations and in the case of small smoothness of initial functions. In this case, the system is reduced to an equivalent system of integral equations of the Volterra type. For linear systems, this transformation allows removing constraints connected with a convergence of a majorizing process. Examples of estimates of solution functionals are provided, and a behavior of their variances are discussed. In the general case, a solution interval is divided into finite subintervals, on which the nonlinear function is approximated by a polynomial. The obtained integral equation is solved by using branched Markov chains with absorption. Algorithm parallelization problems arising in this case are discussed in this paper. A one-dimensional cubic equation is considered as an example. A choice of transition densities of branching is discussed. A method of generations is described in detail. Numerical results are compared with a solution obtained by the Runge–Kutta method.

Error indicators and adaptive refinement of the discrete thin plate spline smoother

The discrete thin plate spline is a data fitting and smoothing technique for large datasets. Current research only uses uniform grids for this discrete smoother, which may require a fine grid to achieve a certain accuracy. This leads to a large system of equations and high computational costs. Adaptive refinement adapts the precision of the solution to reduce computational costs by refining only in sensitive regions. The error indicator is an essential part of the adaptive refinement as it identifies whether certain regions should be refined. Error indicators are well researched in the finite element method, but they might not work for the discrete smoother as data may be perturbed by noise and not uniformly distributed. Two error indicators are presented: one computes errors by solving an auxiliary problem and the other uses the bounds of the finite element error. Their performances are evaluated and compared with 2D model problems.
 
 References H. Chui and A. Rangarajan. A new point matching algorithm for non-rigid registration. Comput. Vis. Image Und., 89 (23): 114141, 2003. doi:10.1016/S1077-3142(03)00009-2. W. F. Mitchell. A comparison of adaptive refinement techniques for elliptic problems. ACM T. Math. Software, 15 (4): 326347, 1989. doi:10.1145/76909.76912. S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. SIAM J. Numer. Anal., 41(1):208234, 2003. doi:10.1137/S0036142901383296. G. Sewell. Analysis of a finite element method. Springer-Verlag, 1985. doi:10.1007/978-1-4684-6331-6. R. Sprengel, K. Rohr, and H. S. Stiehl. Thin-plate spline approximation for image registration. In P. IEEE EMBS, volume 3, pages 11901191. IEEE, 1996. doi:10.1109/IEMBS.1996.652767. L. Stals. Efficient solution techniques for a finite element thin plate spline formulation. J. Sci. Comput., 63(2):374409, 2015. doi:10.1007/s10915-014-9898-x. G. Wahba. Spline models for observational data, volume 59 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1990. doi:10.1137/1.9781611970128.

Efficient modeling of distributions of polymer properties using probability generating functions and parallel computing

High-fidelity models of polymer processes should include the prediction of distributions of polymer properties, including multivariate distributions. Deterministic models with this capability usually involve a large system of equations, which compromises the model performance in terms of CPU time. The probability generating function (pgf) technique is a powerful method for modeling distributions of polymer properties, including multivariate distributions. It can be applied to systems described by complex kinetic mechanism and requires no a priori assumptions about the distribution shape. The structure of this modeling method makes it particularly suitable for parallel computing. This work describes the application of the pgf technique for modeling uni- and bi-variate distributions of polymer properties with parallelization of the model code. It is shown that accurate results can be achieved in very short running times, which makes the technique suitable for models to be employed in optimization and online control tasks.

LARGE SYSTEM OF SEEMINGLY UNRELATED REGRESSIONS: A PENALIZED QUASI-MAXIMUM LIKELIHOOD ESTIMATION PERSPECTIVE

In this article, using a shrinkage estimator, we propose a penalized quasi-maximum likelihood estimator (PQMLE) to estimate a large system of equations in seemingly unrelated regression models, where the number of equations is large relative to the sample size. We develop the asymptotic properties of the PQMLE for both the error covariance matrix and model coefficients. In particular, we derive the asymptotic distribution of the coefficient estimator and the convergence rate of the estimated covariance matrix in terms of the Frobenius norm. The model selection consistency of the covariance matrix estimator is also established. Simulation results show that when the number of equations is large relative to the sample size and the error covariance matrix is sparse, the PQMLE outperforms other contemporary estimators.