Biconjugate Gradient Research Articles

Comparison the iterative solvers for large sparse matrix in 3D electromagnetic forward modelling

In 3D electromagnetic (EM) forward modeling, an analytical solution is generally not available. Numerical solution is commonly applied to solve the forward modeling problems, mostly based on iterative solvers. The efficiency of EM forward modeling is critical for the development of practical inversion for EM data. The Krylov subspace solvers are widely used to solve frequency-domain EM forward modeling problems. However, these solvers converge remarkably more slowly as the operating period increases. This can be improved by the use of preconditioner and divergence correction. Multigrid (MG) solver is efficient for solving EM forward modelling problems without the use of preconditioner and divergence correction. In this paper, a MG solver is compared with Bi-Conjugate Gradients Stabilized (BCG) solvers with different preconditioners. They are compared, in terms of iteration number and computing time, indicating the MG solver is much more efficient.

MATLAB package for discrete dipole approximation by graphics processing unit: Fast Fourier Transform and Biconjugate Gradient

A MATLAB package describing discrete dipole approximation (MPDDA) is introduced to calculate the optical properties of arbitrary shaped plasmonic nanoparticles (NPs). The polarizability function, induced dipoles, and dipole interaction matrix are discussed. To calculate the dipole moments, Fast Fourier Transform (FFT) and Biconjugate Gradient (BCG) were used to reduce the computational time and memory. To further accelerate the computational procedure, we used MATLAB functions and toolboxes supported by the graphics processing unit (GPU) and execute them on GPU by supplying a “gpuArray” argument. Using the GPU, the computation cost significantly decreased when compared to the CPU. We also simulated the optical properties of plasmonic NPs, such as extinction, absorption, and scattering efficiencies, electric field enhancement around monomeric and dimeric structures using this package. The accuracy and capabilities of the code have been confirmed by comparing with Mie theory (exact solution) and ADDA code (a DDA package developed in C).

Numerical simulation of a prostate tumor growth model by the RBF-FD scheme and a semi-implicit time discretization

The aim of this work consists of finding a suitable numerical method for the solution of the mathematical model describing the prostate tumor growth, formulated as a system of time-dependent partial differential equations (PDEs), which plays a key role in the field of mathematical oncology. In the literature on the subject, there are a few numerical methods for solving the proposed mathematical model. Localized prostate cancer growth is known as a moving interface problem, which must be solved in a suitable stable way. The mathematical model considered in this paper is a system of time-dependent nonlinear PDEs that describes the interaction between cancer cells, nutrients, and prostate-specific antigen (PSA). Here, we first derive a non-dimensional form of the studied mathematical model using the well-known non-dimensionalization technique, which makes it easier to implement different numerical techniques. Afterward, the analysis of the numerical method describing the two-dimensional prostate tumor growth problem, based on radial basis function-generated finite difference (RBF-FD) scheme, in combination with a first-order time discretization has been done. The numerical technique we use, does not need the use of any adaptivity techniques to capture the features in the interface. The discretization leads to solving a linear system of algebraic equations solved via the biconjugate gradient stabilized (BiCGSTAB) algorithm with zero-fill incomplete lower–upper (ILU) preconditioner. Comparing the results obtained in this investigation with those reported in the recent literature, the proposed approach confirms the ability of the developed numerical scheme. Besides, the effect of choosing constant parameters in the mathematical model is verified by many simulations on rectangular and circular domains.

Numerical subspace algorithms for solving the tensor equations involving Einstein product

AbstractIn this article, we propose some subspace methods such as the conjugate residual, generalized conjugate residual, biconjugate gradient, conjugate gradient squared and biconjugate gradient stabilized methods based on the tensor forms for solving the tensor equation involving the Einstein product. These proposed algorithms keep the tensor structure. The convergence analysis shows that the proposed methods converge to the solution of the tensor equation for any initial value. Some numerical results confirm the feasibility and applicability of the proposed algorithms in practice.

An anisotropic multilevel preconditioner for solving the Helmholtz equation with unequal directional sampling intervals

An efficient finite-difference method for solving the isotropic Helmholtz equation relies on a discretization scheme and an appropriate solver. Accordingly, we have adopted an average-derivative optimal scheme that has two advantages: (1) it can be applied to unequal directional sampling intervals and (2) it requires less than four grid points of sampling per wavelength. Direct methods are not of interest for industry-sized problems due to the high memory requirements; Krylov subspace methods such as the biconjugate gradient stabilized method and the flexible generalized minimal residual method that combine a multigrid-based preconditioner are better alternatives. However, standard geometric multigrid algorithms fail to converge when there exist unequal directional sampling intervals; this is called anisotropic grids in terms of the multigrid. We first review our previous research on 2D anisotropic grids: the semicoarsening strategy, line-wise relaxation operator, and matrix-dependent interpolation were used to modify the standard V-cycle multigrid algorithms, resulting in convergence. Although directly extending to the 3D case by substituting line relaxation for plane relaxation deteriorates the convergence rate considerably, we then find that a multilevel generalized minimal residual preconditioner-combined semicoarsening strategy is more suitable for anisotropic grids and the convergence rate is faster in the 2D and 3D cases. The results of the numerical experiments indicate that the standard geometric multigrid does not work for anisotropic grids, whereas our method demonstrates a faster convergence rate than the previous method.

A spectral element method to compute approximations of the anisotropic diffusion operator with bidimensional tensor coefficient

We derived a continuous Galerkin spectral element method to compute approximations of the anisotropic diffusion operator with the tensorial coefficient of order two. After writing the operator in the weak form, we calculated spatial integration using the Legendre base. Then, we applied three different methods for the solution of the associated linear system: The Lower-Upper factorization, the biconjugate gradient method and the biconjugate gradient stabilized method with the Richardson preconditioner. To validate the algorithm, we present a convergence study for the Poisson equation when varying the functions of the tensor coefficient. Results show how the algorithm can be used to construct solvers for more complex problems like the anisotropic diffusion-reaction equation.

BICGSTAB-FFT Method of Moments with NURBS for Analysis of Planar Generic Layouts Embedded in Large Multilayer Structures

BICGSTAB-FFT method of moment (MM) scheme is proposed to analyze several levels of planar generic layouts embedded in large multilayer structures when the layout geometries are modeled by NURBS surfaces. In this scheme, efficient computation of normalized error defined in iterative bi-conjugate gradient stabilized (BICGSTAB) method for large multilayer structure analysis problems is implemented. The efficient computation is based on pulse expansion with dense equi-spaced mesh of generalized rooftop basis functions (BFs) defined on NURBS surfaces and equivalent periodic problem (EPP) in order to apply fast Fourier transforms (FFT). Moreover, efficient computation of Green’s functions for multilayer structure is implemented for near and far field regions. Experimental and numerical validations of whole printed reflect array antennas of electrical size between 8 and 16 times the vacuum wavelengths are shown. In these validations, CPU time consumptions of the proposed method are obtained with results between few minutes and half an hour using a conventional laptop.

Damage detection of cylindrical shells based on Sander's theory and model updating using incomplete modal data considering random noises

In this paper, a new system of damage detection equations for damage identification of cylindrical shells is developed based on classic finite element method, incomplete noisy modal data and model updating. Classic finite element technique in conjunction with Sander's thin shell theory is deployed in order to construct the finite element model of cylindrical shells and compute vibration data. To overcome the difficulty of incomplete data, “system equivalent reduction expansion process” (SEREP) method is applied to find a proper complete approximation of incomplete modal data. The proposed system is usually ill-posed due to noisy data and ill-conditioned system derived from the model updating process. To tackle this issue, a constraint as a regularization parameter is employed in model updating problem to differentiate potentially damaged elements and undamaged elements. This constraint is imposed on the model updating process by defining a Reduction Index (RI) and forcing severities of undamaged elements to tend to zero. Furthermore, the presented system is non-square and large depending on the number of structural unknowns. For these reasons, Biconjugate gradient (BCG) method along with an appropriate preconditioner are employed to solve the system of equations. Not only does the preconditioner stabilize the solution against numerical errors and inaccuracies caused by noisy data, but also it alters the system to a square one. Numerical studies are performed to demonstrate the efficiency of the presented method. Moreover, a statistical analysis is carried out to evaluate the effect of different sequences of random noises on the damage detection results. Findings show that the proposed method is capable of detecting the severities and sites of damages for cylindrical shells in the presence of noisy data.

K[formula omitted] — Open-source library for the shifted Krylov subspace method of the form [formula omitted

We develop Kω, an open-source linear algebra library for the shifted Krylov subspace methods. The methods solve a set of shifted linear equations (zkI−H)x(k)=b(k=0,1,2,…) for a given matrix H and a vector b, simultaneously. The leading order of the operational cost is the same as that for a single equation. The shift invariance of the Krylov subspace is the mathematical foundation of the shifted Krylov subspace methods. Applications in materials science are presented to demonstrate the advantages of the algorithm over the standard Krylov subspace methods such as the Lanczos method. We introduce benchmark calculations of (i) an excited (optical) spectrum and (ii) intermediate eigenvalues by the contour integral on the complex plane. In combination with the quantum lattice solver HΦ, Kω can realize parallel computation of excitation spectra and intermediate eigenvalues for various quantum lattice models. Program summaryProgram Title: Kω [kéi-óumig@]CPC Library link to program files:http://dx.doi.org/10.17632/mt928nz5r3.1Developer’s repository link:https://github.com/issp-center-dev/KomegaLicensing provisions: GNU Lesser General Public License Version 3.Programming language: Fortran 90External routines/libraries: BLAS library, LAPACK library (Used in the sample program), MPI library (Optional).Nature of problem: Efficient algorithms, called shifted Krylov subspace algorithms, designed to solve the shifted linear equations.Solution method: Shifted conjugate gradient method, Shifted conjugate orthogonal conjugate gradient method, shifted bi-conjugate gradient method.Additional comments: The present paper is accompanied by a frozen copy of Kω release 2.0.0 that is made publicly available on GitHub (repository https://github.com/issp-center-dev/Komega, commit hash fd5455328b102ec4fa13432496e41c404a0f5a9d).

Simulation of 3-D Electromagnetic Scattering and Inverse Scattering by Arbitrary Anisotropic Dielectric Objects Embedded in Layered Arbitrary Anisotropic Media

This communication studies 3-D electromagnetic (EM) scattering and inverse scattering of arbitrary anisotropic dielectric objects embedded in multilayered media. Different from previous work, the scatterer is not only arbitrary anisotropic, but also the arbitrary anisotropy of the layered background media is formed by rotation of the principal axes of uniaxial media. In addition, the scatterers are allowed to be placed across layer boundaries. The forward model is formulated by the electric field integral equation (EFIE), solved by the biconjugate gradient stabilized (BCGS) method, and accelerated by the fast Fourier transform (FFT). The time complexity is analyzed for the BCGS-FFT applied to layered arbitrary anisotropic media. In the inversion model, the variational Born iterative method (VBIM) is employed to reconstruct the arbitrary anisotropic profiles of both the permittivity and conductivity. Numerical simulations are performed to verify the feasibility and antinoise ability of the forward and inverse solvers in the circumstance of layered arbitrary anisotropic background media.

Radial basis function-generated finite difference scheme for simulating the brain cancer growth model under radiotherapy in various types of computational domains

Background and ObjectivesWe extend the original mathematical model, i.e., Swanson’s reaction-diffusion equation to the surfaces with no boundary, and we find a new numerical method based on a meshless approach for solving numerically Swanson’s reaction-diffusion model in the square and on the sphere. MethodsTo solve numerically the Swanson’s reaction-diffusion model and its extension version, a collocation meshless technique, namely radial basis function-generated finite difference (RBF-FD) scheme is employed for approximating the spatial variables in the square domain and on the sphere, respectively. Also, to approximate the time variable of the studied models, a first-order semi-implicit backward Euler scheme is used. The resulting fully discrete scheme is a linear system of algebraic equations per time step that is solved via the biconjugate gradient stabilized (BiCGSTAB) iterative algorithm with a zero-fill incomplete lower-upper (ILU) preconditioner. ResultsThe numerical simulations show the growth of untreated and treated brain tumors with radiotherapy using estimated and clinical data (given from magnetic resonance imaging (MRI) scans of patients). Moreover, the results reported here can be used for improving the treatment strategies of the invasive brain tumor. ConclusionsUsing the developed numerical scheme in this paper, we can simulate the behavior of the invasive form of brain tumor response to radiotherapy. Also, we can see the effects of radiation response on the brain tumor cell concentration of individual patients. The proposed meshless technique, which is applied for solving numerically the studied model, does not depend on any background mesh or triangulation for approximation in comparison with mesh-dependent methods. Moreover, we apply this technique to the sphere via any set of distributed points easily.

Development of Two-Dimensional Non-Hydrostatic Wave Model Based on Central-Upwind Scheme

In this study, a two-dimensional depth-integrated non-hydrostatic wave model is developed. The model solves the governing equations with hydrostatic and non-hydrostatic pressure separately. The velocities under hydrostatic pressure conditions are firstly obtained and then modified using the biconjugate gradient stabilized method. The hydrostatic front approximation (HFA) method is used to deal with the wave breaking issue, and after the wave breaks, the non-hydrostatic model is transformed into the hydrostatic shallow water model, where the non-hydrostatic pressure and vertical velocity are set to zero. Several analytical solutions and laboratory experiments are used to verify the accuracy and robustness of the developed model. In general, the numerical simulations are in good agreement with the theoretical or experimental results, which indicates that the model is able to simulate large-scale wave motions in practical engineering applications.

MAGNETIC FIELD EFFECT ON THE HEAT TRANSFER IN A NANOFLUID FILLED LID DRIVEN CAVITY WITH JOULE HEATING

In this paper, the effects of magnetic field, Joule heating and volumetric heat generation on the heat transfer and fluid flow in a Cu-Water nanofluid filled lid driven cavity using enhanced streamfunction–velocity method are investigated. The cavity is heated by a uniform volumetric heat density and side walls have constant temperature. The top wall moves with constant velocity in +x direction, while no-slip boundary conditions are imposed on the other walls of the cavity. An inclined fixed magnetic field is applied to the left side wall of the cavity. The dimensionless governing equations are solved numerically for the stream function and temperature using finite difference method for various Richardson(Ri), Reynolds(Re), Hartmann (Ha), Eckert(Ec)numbers, magnetic field angle(α) and solid volume fraction of the nanofluid() in MATLAB software. To discretize the streamfunction-velocity formulation, a five point constant coefficient second-order compact finite difference approximation which avoids difficulties inherent in the conventional streamfunction–vorticity and primitive variable formulations is used. The stream function equation is solved using fast Poisson's equation solver on a rectangular grid (POICALC function in MATLAB) and the temperature equation is solved using Jacobi bi-conjugate gradient stabilized (BiCGSTAB) method. The heat transfer within the cavity is characterized by Nusselt number (Nu1). The results show that Nu1 is significantly increased by increasing Ri and  and increasing the Reynolds number enhances convective cooling. The heat transfer within the cavity is decreased by increasing Hartmann number which improves conduction heat transfer and reduces Nu1. Joule heating has a negative effect on the convection within the cavity and convection is decreased by increasing the value of Ec. It can be investigated that Nu1 is decreased by increasing Ec due to the strong distortion effect of Joule heating on convection current of heat transfer.

Minimization of entropy generation due to MHD double diffusive mixed convection in a lid driven trapezoidal cavity with various aspect ratios

Entropy generation minimization has significant importance in fluid flow, heat and mass transfer in an enclosure to get the maximum efficiency of a system and to reduce the loss of energy. In the present study, the analysis of mixed convection fluid flow, heat and mass transfer with heat line and mass line concept and entropy generation due to the effects of fluid flow, heat flow, mass flow and magnetic field in a trapezoidal enclosure with linearly heated and diffusive left wall, uniformly heated and diffusive lower wall, cold and nondiffusive right wall, adiabatic and zero diffusion gradient top wall have been reported. Parametric studies for the wide range of Prandtl number (Pr = 0.7 for air cooling system and Pr = 1000 for the engines filled with olive or engine oils), Rayleigh number (Ra = 103–105), aspect ratio (A = 0.5–1.5) and inclination angle of the cavity (ϕ = 45°–90°) have been performed, which help to construct the perfect shape of cavity in many engineering and physical applications so that the entropy is minimum to get the maximum efficiency of any system. The finite-difference approximation has been used to find out the numerical solutions. Biconjugate Gradient Stabilized (BiCGStab) method is used to solve the discretized nonhomogeneous system of linear equations.

Incomplete block‐matrix factorization of M‐matrices using two‐step iterative method for matrix inversion and preconditioning

Using the general method of Owe Axelsson given in 1986 for incomplete factorization of M‐matrices in block‐matrix form, we give a recursive approach to construct incomplete block‐matrix factorization of M‐matrices by proposing a two‐step iterative method for the approximation of the inverse of diagonal pivoting block matrices at each stage of the recursion. For various predescribed tolerances in the accuracy of the approximation of the inverses, the obtained incomplete block‐matrix factorizations are used to precondition the iterative methods as one‐step stationary iterative (OSSI) method and biconjugate gradient stabilized method (BI‐CGSTAB). Certain applications are conducted on M‐matrices occurring from the discretization of two Dirichlet boundary value problems of Laplace's equation on a rectangle using finite difference method. Numerical results justify that the given incomplete block‐matrix factorization of M‐matrices using the two‐step iterative method to approximate the inverse of diagonal pivoting block matrices at each stage of the recursion give robust preconditioners. The obtained results are presented through tables and figures.

A novel linear solver for simulating highly heterogeneous black oil reservoirs

A great deal of computational power and time is necessary for the simulation of highly heterogeneous fractured reservoirs with complicated geometry. Simulating such largescale complex reservoirs with both minimum time and maximum accuracy has consistently remained a topic of concern to reservoir simulation engineers worldwide. The currently used linear solvers in well-known commercial simulators that utilize classical methods such as Nested Factorization (NF) and Incomplete LU Factorization (ILU) as their preconditioners, all suffer from severe convergence problems in models with high heterogeneity and/or large number of non-neighbor connections. To overcome such problems, a novel Adaptive Algebraic Multi-grid algorithm (AAMG) is proposed to be used in a constrained pressure residual (CPR) preconditioner, abbreviated as CPR-AAMG, in order to solve the governing discretized three-dimensional partial differential equations. The implemented CPR preconditioner which uses AAMG as the preconditioner for the elliptic pressure equation, and the Blocked ILU for the hyperbolic saturation equations, is a robust way for damping both smooth pressure field and high frequency saturation field errors in giant reservoir models. The findings of the current study illustrate that the developed Biconjugate Gradient Stabilized (BiCG-Stab) solver preconditioned by CPR-AAMG is capable of achieving acceptable results of high efficiency and robustness in the SPE10 project which is renowned as an enormous challenge for linear solvers due to the highly heterogeneous porosity and permeability values. The novelty of the proposed CPR-AAMG preconditioner is its ability to solve three-dimensional black oil models of giant, complex and heterogeneous fractured reservoirs without any convergence problems with reduction in time and required computational power.

A Preconditioned Krylov Subspace Iterative Methods for Inverse Source Problem by Virtue of a Regularizing LM-DRBEM

In this paper, we investigate an effectual methodology to solve the problem of identifying the source term standing on a combined discontinuous dual reciprocity boundary element method with a regularized Levenberg–Marquardt algorithm. The mathematical model is described by optimizing the variation between the measured potential in some points within the problem domain and the estimated one by the discontinuous dual reciprocity boundary element approximation. To provide a qualitative evaluation, the derived dense and non symmetric matrix system of the forward problem is solved by virtue of the LU factorization and using some preconditioned Krylov subspace iterative algorithms, expressly the biconjugate gradient stabilized, the generalized minimum residual and the squared conjugate gradient method. Due to noise interference, a regularized Levenberg–Marquardt algorithm is adopted to retrieve the unknown source term. Extensive experiments evince greater effectiveness and stability criteria of the proposed approach.

Efficient numerical methods for the nonlinear two-sided space-fractional diffusion equation with variable coefficients

In this paper, a second order accurate time and space difference method is proposed to solve the nonlinear two-sided space-fractional diffusion equation with variable coefficients. We combine the alternating direction implicit method and Bi-conjugate gradient stabilized iterative solver to develop an efficient numerical method for a two dimensional, nonlinear, two-sided space-fractional diffusion equation with variable coefficients. Furthermore, the proposed method is proved to be uniquely solvable, stable and convergent. Finally, some numerical results verify the accuracy and efficiency of our computational approach.

Multiparametric Electromagnetic Inversion of 3-D Biaxial Anisotropic Objects Embedded in Layered Uniaxial Media Using VBIM Enhanced by Structural Consistency Constraint

This article presents the 3-D inversion of biaxial anisotropic objects embedded in layered uniaxial media by the variational Born iterative method (VBIM). Although previously only the isotropic complex permittivity profile was reconstructed, here we retrieve all the six diagonal anisotropic permittivity and conductivity parameters simultaneously. In the forward model, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is used to solve the volume electric field integral equation (EFIE). In the inversion model, VBIM is employed to minimize the cost function. In order to improve the inversion results, we propose a novel structural consistency constraint (SCC) applied to VBIM. The SCC is based on the Monte Carlo method and reduces the computational domain in the Born iteration. The major new contribution of this work is to combine the deterministic inversion algorithm VBIM with the stochastic Monte Carlo method to reconstruct all the diagonal anisotropic parameters of the scatterers. Several typical numerical models are simulated, and the results validate the efficiency, antinoise ability as well as the accuracy of VBIM-SCC for reconstructing 3-D biaxial anisotropic objects embedded in multilayered uniaxial media.

Generalized moving least squares approximation for the solution of local and non-local models of cancer cell invasion of tissue under the effect of adhesion in one- and two-dimensional spaces

The main aim of this study is to solve numerically the mathematical models showing cancer cell invasion of tissue with/without considering the effect of cell-cell and cell-matrix adhesion. The mathematical models studied here are the systems of time-dependent reaction-diffusion-taxis equations in one- and two-dimensional spaces, which are formulated in the local and non-local forms. There are some difficulties in finding their solutions via numerical methods. The main difficulty is to compute the non-local term appearing in one of the studied models, which causes more CPU time during simulations. The current paper aims to overcome this problem, where a new meshless method, namely generalized moving least squares (GMLS) approximation in space and a semi-implicit backward differential formula of first-order (SBDF1) in time have been applied. Based on GMLS theory, the non-local term is approximated without any difficulties. Moreover, a simple method based on the GMLS technique is presented to implement the boundary conditions. The obtained discrete scheme for both mathematical models is a linear system of algebraic equations per time step. The biconjugate gradient stabilized (BiCGSTAB) algorithm with zero–fill incomplete lower–upper (ILU) preconditioner is used to solve the obtained linear system at each time step. At the end of this paper, some simulation results are reported to show the behavior of cancer cell invasion in the local model, and the non-local model due to reduction of cell-cell adhesion and increasing cell-matrix adhesion in one- and two-dimensional spaces, where two different types of distribution points have been considered in the square domain. The computational algorithms of the GMLS approximation and the developed numerical method for solving the non-local (local) model are included in the Appendix.