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This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

This work examines the behaviour of flow and heat transmission in the presence of hybrid nanofluid in thermal radiation, heat generation, and magnetohydrodynamics. The hybrid state in this model is represented by two different fluids, TiO (titanium dioxide) and Ag (silver). The enclosure is wavy and slanted, with curving walls on the left and right. The finite difference approximation method was utilized to resolve the fundamental equations after they were non-dimensionalized, which are further reduced to a fourth-order bi-harmonic equation and are numerically solved based on the biconjugate gradient-stabilized approach method. The simulations are performed with various Rayleigh numbers, Hartmann numbers, an inclination angle of the enclosure, radiation parameters, heat generation parameters, inclination angle of the magnetic field, and volume fraction of hybrid nanoparticles. The streamlines, isotherms, and average Nusselt number contours are used to depict the thermo-fluid patterns. The findings show that the average Nusselt number relies on and increases as rises. The investigation’s findings demonstrated that the transfer of heat on the heated bottom wall significantly increases with the Rayleigh number (Ra = 10 and 10 ). At a cavity inclination of 45°, interesting multi-vortex structures are observed. The results of this study may enhance the effectiveness of solar collectors, heat exchangers, and other similar systems that depend on convective heat transfer in nature.

The numerical simulation of the tumor-induced angiogenesis process is an useful tool for the prediction of this mechanism and drug targeting using anti-angiogenesis strategy. In the current paper, we study numerically on the continuous mathematical model of tumor-induced angiogenesis in two-dimensional spaces. The studied model is a system of nonlinear time-dependent partial differential equations, which describes the interactions between endothelial cell, tumor angiogenesis factor and fibronectin. We first derive the global weak form of the model and discretize the time variable via a semi-implicit backward Euler method. To approximate the spatial variables of the studied model, we use a meshless technique, namely element-free Galerkin. Also, the shape functions of moving least square and moving Kriging approximations are used in this method. The main difference between two meshless methods proposed here is that the shape functions of moving least squares approximation do not satisfy Kroncker’s delta property, while moving Kriging technique satisfies this property. Also, both techniques do not require the generation of a mesh for approximation, but a background mesh is needed to compute the numerical integrations, which are appeared in the derived global weak form. The full-discrete scheme obtained here gives the linear system of algebraic equations that is solved via an iterative method, namely biconjugate gradient stabilized with zero-fill incomplete lower upper (ILU) preconditioner. Some numerical simulations are provided to illustrate the ability of the presented numerical methods, which show the endothelial cell migration in response to the tumor angiogenesis factors during angiogenesis process as well.

The main aim of this paper is to present new and simple numerical methods for solving the time-dependent transport equation on the sphere in spherical coordinates. We use two techniques, namely generalized moving least squares and moving kriging least squares to find the new formulations for approximating the advection operator in spherical coordinates such that any singularities have been omitted. Another feature of the methods considered here is that they do not depend on the background mesh or triangulation for approximation, which they could be applied on the transport equation in spherical coordinates easily with different distribution points. Furthermore, due to the eigenvalue stability of the dicretized advection operator via two proposed approximations, an implicit-explicit linear multistep method has been applied to discretize the time variable. The fully discrete scheme obtained here yields a linear system of algebraic equations at each time step, which is solved via the biconjugate gradient stabilized algorithm with zero-fill incomplete lower-upper (ILU) preconditioner. Three well-known test problems, namely “solid body rotation”, “vortex roll-up” and “deformational flow” are solved to demonstrate our developments. Also for the first test problem, we apply a simple positivity-preserving filter at the end of each time step, which keeps the transported variable positive.

This study analyzes and evaluates the performance of various solvers and preconditioners for reservoir simulations of CO2 injection and long-term storage using the model 11B of SPE CSP (Society of Petroleum Engineers, 11th Comparative Solution Project) and the MATLAB Reservoir Simulation Toolbox (MRST). The SPE CSP 11 model serves as a benchmark for testing numerical methods for solving partial differential equations (PDEs) in reservoir simulations. The research focuses on the Biconjugate Gradient Stabilized (BiCGSTAB) and Loose Generalized Minimum Residual (LGMRES) solver methods, as well as multiple preconditioning techniques designed to improve convergence rates and reduce computational effort for CO2 storage. Extensive simulations were performed to compare the performance of different solver-preconditioner combinations under varying reservoir conditions, leveraging MRST’s flexible simulation capabilities. Key performance metrics, including iteration counts and computational time, were analyzed for the project. The results reveal trade-offs between computational efficiency and solution accuracy, providing valuable insights into the effectiveness of each approach. This study offers practical guidance for reservoir engineers and researchers seeking to analyze and optimize simulation workflows within MRST by identifying the strengths and limitations of specific solver-preconditioner combinations for complex linear systems.

We capitalize upon the known relationship between pairs of orthogonal and minimal residual methods (or, biorthogonal and quasi-minimal residual methods) in order to estimate how much smaller the residuals or quasi-residuals of the minimizing methods can be compared to those of the corresponding Galerkin or Petrov–Galerkin method. Examples of such pairs are the conjugate gradient (CG) and the conjugate residual (CR) methods, the full orthogonalization method (FOM) and the generalized minimal residual (GMRES) method, the CGNE and BiCG versions of applying CG to the normal equations, as well as the biconjugate gradient (BiCG) and the quasi-minimal residual (QMR) methods. Also the pairs consisting of the (bi)conjugate gradient squared (CGS) and the transpose-free QMR (TFQMR) methods can be added to this list if the residuals at half-steps are included, and further examples can be created easily.The analysis is more generally applicable to the minimal residual (MR) and quasi-minimal residual (QMR) smoothing processes, which are known to provide the transition from the results of the first method of such a pair to those of the second one. By an interpretation of these smoothing processes in coordinate space we deepen the understanding of some of the underlying relationships and introduce a unifying framework for minimal residual and quasi-minimal residual smoothing. This framework includes the general notion of QMR-type methods.

This paper synthesizes formally orthogonal polynomials, Gaussian quadrature in the complex plane and the bi-conjugate gradient method together with an application. Classical Gaussian quadrature approximates an integral over (a region of) the real line. We present an extension of Gaussian quadrature over an arc in the complex plane, which we call complex Gaussian quadrature. Since there has not been any particular interest in the numerical evaluation of integrals over the long history of complex function theory, complex Gaussian quadrature is in need of motivation. Gaussian quadrature in the complex plane yields approximations of certain sums connected with the bi-conjugate gradient method. The scattering amplitude cTA−1b is an example where A is a discretization of a differential–integral operator corresponding to the scattering problem and b and c are given vectors. The usual method to estimate this is to use cTx(k). A result of Warnick is that this is identically equal to the complex Gaussian quadrature estimate of 1/λ. Complex Gaussian quadrature thereby replaces this particular inner product in the estimate of the scattering amplitude.