Transpose-free Quasi-minimal Residual Research Articles

Difüz Optik Tomografi’de Modellenmiş Meme Tümörü Benzeri Dokuların Teşhisi için Simülasyon Deneyleriyle Geri Çatım Algoritmalarının Uygulanması

In Diffuse Optical Tomography (DOT), data processing and reconstruction stages are crucial to obtain high-quality images. Thus, choosing suitable algorithms for the system is a critical choice. This study aims to determine an appropriate reconstruction algorithm for DOT imaging. There are several reconstruction algorithms used in DOT systems. Some algorithms have been improved for solving specific cases, and some still need to be improved. In this study, we used three algorithms for the reconstruction process: Singular Value Decomposition (SVD), Bi-Conjugated Gradient (Bi-CG), and Transpose Free Quasi Minimal Residual (TFQMR). In testing the algorithms, data of the simulation experiments have been used. The simulation experiments model the tumoral tissue within the breast. All three algorithms were produced correct images while the tumor close to the surface. In the case of the tumor that is not close to the breast surface, the tumor location on the images created by Bi-CG and SVD algorithms was not its actual location. However, the tumor location in the image created by the TFQMR algorithm was close to its actual location. Outcomes of the reconstruction algorithms were evaluated based on correctly defining the location of the tumors by using Mean Percentage Error (MPE), Mean Squared Error (MSE), and Mean Absolute Error (MAE) metrics. We have demonstrated the TFQMR algorithm is a more appropriate reconstruction technique for DOT systems. Thus, we have concluded that TFQMR can have the potential to be used in medical imaging systems.

Approaches to preclinical studies with heterogeneous breast phantom using reconstruction and three‐dimensional image processing algorithms for diffuse optical imaging

AbstractThe continuous‐wave back reflection diffuse optical tomography (rCWDOT) system is one of the new medical imaging modalities. This study examines the success of reconstruction and three‐dimensional (3D) image processing algorithms on data obtained from a heterogonous breast phantom by rCWDOT. Breast phantoms were prepared by putting a bit of spleen inside the tail. The spleen mimics a breast tumor since it has more blood than tail fat. rCWDOT was used to acquire data from the breast phantoms. The breast phantoms were reconstructed using the transpose‐free quasi‐minimal residual (TFQMR) reconstruction algorithm. Then, image processing algorithms were performed to improve the image quality. In image processing, 3D Gaussian filtering and bi‐cubic interpolation were used to enhance the appearance and remove noise from the images. After the image processing, the images were evaluated numerically using the peak signal‐to‐noise ratio (PSNR) method. It has been shown that the used reconstruction technique and image processing algorithms for a heterogeneous breast phantom provided 3D images that resemble actual ones. This study will help researchers use the most convenient reconstruction algorithm and image processing algorithms and perform preclinical experiments in this field.

Image reconstruction for diffuse optical tomography using bi‐conjugate gradient and transpose‐free quasi minimal residual algorithms and comparison of them

AbstractDiffuse optical tomography (DOT) is a new emerging modality in the diagnosis of soft tissue abnormalities. DOT image quality substantially depends on the reconstruction stage. In the literature, there are many reconstruction algorithms used in DOT systems. However, some algorithms were improved for solving specific cases but still need to be improved. The bi‐conjugate gradient (BiCG) enhanced is one of the conjugate gradient (CG)‐based reconstruction techniques for non‐Hermitian systems. The BiCG provides a solution to a non‐Hermitian system. However, it has erratic convergence in some cases. Therefore, DOT images reconstructed by BiCG can be at the wrong location and is inaccurate in some cases. In this study, we used continuous‐wave diffuse optical tomography (CW‐DOT) to acquire measurements from breast tissue phantoms with single or double inclusion at different depths and center‐to‐center separations and we have used the transpose free quasi minimal residual (TFQMR) reconstruction algorithm, improved as an alternative to BiCG for the first time in the CW‐DOT system. Moreover, we have experimentally proved that TFQMR is superior to BiCG in some specific cases for the first time in CW‐DOT. Therefore, we concluded that TFQMR has the potential to be able to be used in the reconstruction stage in CW‐DOT.

Boundary Element Algorithm for Modeling and Simulation of Dual-Phase Lag Bioheat Transfer and Biomechanics of Anisotropic Soft Tissues

The main aim of this paper is to propose a new boundary element algorithm for describing thermomechanical interactions in anisotropic soft tissues. The governing equations are studied based on the dual-phase lag bioheat transfer and Biot’s theory. Due to the advantages of convolution quadrature boundary element method (CQBEM), such as low CPU usage, low memory usage and suitability for treatment of soft tissues that have complex shapes, it is a versatile and powerful method for modeling of bioheat distribution in anisotropic soft tissues and the related deformation. The resulting linear systems for bioheat and mechanical equations are solved by Transpose-free quasi-minimal residual (TFQMR) solver with a dual-threshold incomplete LU factorization technique (ILUT) preconditioner that reduces the iterations number and total CPU time. Numerical results demonstrate the validity, efficiency and accuracy of the proposed algorithm and technique.

A parallel VOF IB pressure-correction method for simulation of multiphase flows

This paper develops an improved pressure correction method, coupled with volume-of-fluid (VOF) and immersed boundary (IB) methods, to accelerate multiphase flow computations. A momentum interpolation method (MIM) is used to avoid the issue of a checkerboard pressure field on a non-staggered grid system. A transpose-free quasi-minimal residual (TFQMR) method is applied to reduce the CPU time when solving the Poisson equation for the pressure correction procedure. A suitable initial guess for the TFQMR method is investigated to enhance the stability of the matrix solver. The IB method is applied to treat the solid-wall boundary conditions while OpenMP is used to reduce CPU time. Two physical problems – dam breakage and a water wave impacting on a tall structure – are simulated and compared with known data. The improved numerical scheme has CPU speeds at least 8 times faster than the initial numerical scheme for simulating two-phase flows. With regard to the parallel computational efficiency, using six cores with the OpenMP allows the improved scheme to have 77% parallel efficiency. These modifications greatly enhance overall computational accuracy and efficiency.

Jacobian-Free Newton Krylov based iterative strategies in frequency-domain analyses

This study examines the theory and performance of Jacobian-Free Newton Krylov (JFNK) methods for the efficient, iterative solution of steady state dynamics of linear vibrating systems in the frequency domain. Currently, most commercial FEM algorithms employ use direct factorization of large system matrices to achieve steady state solution. Such approaches are usually quite demanding on memory and CPU requirements. Some implementations exploit iterative solutions, but still use large, assembled system matrices, requiring significant memory. The methods investigated in this work avoid the formation of a matrix completely, minimizing memory requirements and enabling much larger problems to be performed on desktop computers. Here, the Conjugate-Gradient (CG) and Transpose Free Quasi Minimal Residual (TFQMR) algorithms are studied as possibilities. These methods are of particular interest for adaptation to finite element software which uses explicit transient dynamics, because such software's optimal architecture prevents the formation and solution of a stiffness (or a tangent stiffness) matrix. In order to demonstrate the advantages of these algorithms, we choose examples that use the JFNK-CG and JFNK-TFQMR technique to show computational advantages over regular matrix based solutions, with significant decrease in memory requirements.

A transpose-free quasi-minimal residual variant of the CORS method for solving non-Hermitian linear systems

Recently, the conjugate A-orthogonal residual squared (CORS) method is often competitive and superior to other Krylov subspace methods for solving complex non-Hermitian linear systems from many realistic problems. However, like the conjugate gradient squared (CGS) method, the CORS method often exhibits irregular convergence behavior with wild oscillations in the residual norm. To try and remedy this problem, we develop a transpose-free quasi-minimal residual (TFQMR) variant of the CORS method, which leads to smooth convergence curves with a convergence rate similar to the CORS method. We also give convergence results for this new method. Finally, numerical examples are reported to illustrate the effectiveness of the proposed method.

Analytical Gradients for the MSINDO-sCIS and MSINDO-UCIS Method: Theory, Implementation, Benchmarks, and Examples.

Analytical expressions for the sCIS (scaled configuration interaction singles) and UCIS (unrestricted CIS) energy gradients are presented for the semiempirical method MSINDO. The theoretical background of the derivation of the analytical gradients is presented, and the implementation into the MSINDO program package is described. The computational efficiency of the underlying Z-vector method is greatly enhanced by making use of the transpose-free quasiminimal residual (TFQMR) algorithm. Benchmark timing tests are compared to the widely used TD-B3LYP approach. For a statistical evaluation of the accuracy of MSINDO-sCIS, geometry optimizations are performed for a small set of organic molecules in selected excited states. The obtained results are compared to CASPT2 and TD-B3LYP/TZVP. In order to demonstrate the applicability of the present approach to periodic systems within the cyclic cluster model, we present first calculations of the excited state structure of ethyne adsorbed on the NaCl (100) surface.

A Study of preconditioned Krylov subspace methods with reordering for linear systems from a biphasic v–p finite element formulation

A study was conducted on combinations of preconditioned iterative methods with matrix reordering to solve the linear systems arising from a biphasic velocity–pressure (v–p) finite element formulation used to simulate soft hydrated tissues in the human musculoskeletal system. Krylov subspace methods were tested due to the symmetric indefiniteness of our systems, specifically the generalized minimal residual (GMRES), transpose-free quasi-minimal residual (TFQMR), and biconjugate gradient stabilized (BiCGSTAB) methods. Standard graph reordering techniques were used with incomplete LU (ILU) preconditioning. Performance of the methods was compared on the basis of convergence rate, computing time, and memory requirements. Our results indicate that performance is affected more significantly by the choice of reordering scheme than by the choice of Krylov method. Overall, BiCGSTAB with one-way dissection (OWD) reordering performed best for a test problem representative of a physiological tissue layer. The preferred methods were then used to simulate the contact of the humeral head and glenoid tissue layers in glenohumeral joint of the shoulder, using a penetration-based method to approximate contact. The distribution of pressure and stress fields within the tissues shows significant through-thickness effects and demonstrates the importance of simulating soft hydrated tissues with a biphasic model.

Iterative solution techniques for boundary element method (BEM) equations

Boundary-element discretization of the Kirchhoff-Helmholtz integral equation gives rise to a linear system of equations. This system may be solved directly or iteratively. Application of direct solvers is quite common but turns out to be inefficient for large scale problems with 10<th>000 unknowns and more. These systems can be solved on behalf of iterative methods. This paper is dedicated to testing performance of four iterative solvers being the Restarted Bi-Conjugate Gradient Stabilized algorithm (RBiCGStab), the Conjugate Gradient method applied to the normal equations (CGNR), the Generalized Minimal Residual (GMRes), and the Transpose Free Quasi Minimal Residual (TFQMR). For that, it is distinguished between internal and external problems. Performance of iterative solvers is investigated with respect to different parameters. It turns out that sophisticated preconditioning is required for the hyper- singular external Burton and Miller formulation of nonsmooth meshes.

By how much can residual minimization accelerate the convergence of orthogonal residual methods

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.

Two-dimensional fluid simulation of plasma reactors for the immobilization of krypton

Direct current glow discharge plasma sources are being developed as reactors for immobilization of radioactive krypton gas recovered from the off-gas stream of a nuclear fuel reprocessing plant. In order to design these reactors, two-dimensional simulations have been performed using a fluid model coupled with Poisson's equation. Nonlinear governing equations were expressed by a fully implicit scheme and the equations were solved using the Newton–Krylov method. We investigated algorithm performance of two methods from the Krylov projection techniques, namely the Bi-CGSTAB method and the transpose-free quasi-minimal residual (TFQMR) method. The results showed that the Bi-CGSTAB method converged considerably faster than the TFQMR method. Direct current glow discharge plasmas in three types of krypton immobilization apparatus were simulated to study the differences in electrode geometry. Simulation results agreed qualitatively with measurements of electron density.

Analysis of scattering by finite periodic dielectric structures using TFQMR-FFT

In this letter, the scattering of finite periodic dielectric structures is analyzed using the volume-integral moment method enhanced by the transpose-free quasiminimal residual (TFQMR) algorithm and fast Fourier transforms (FFTs). To efficiently use TFQMR–FFT, the convergent rate and accuracy of the TFQMR–FFT related to mesh division are numerically investigated for this special structure. An approach to determine reflection coefficients from the bistatic radar cross section (bi-RCS) is presented. Various numerical results are given to demonstrate the scattering characteristics of finite periodic dielectric structures. © 2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 28: 221–224, 2001.

Circulants, symmetry and time-independent wavepacket scattering

Time-independent implementations of wavepacket scattering theory usually require iterative solution of a linear system Ax= b, in which the primary representation is often a spatial grid. We show that for Fourier basis functions, while A is complex-symmetric on a spatial grid, it will generally not be so in the corresponding spectral representation, since the persymmetry properties of the grid-based potential matrix govern the spectral symmetry properties of A. For an illustrative test-case, correct scattering information is yielded on the grid using the symmetric quasi-minimal residual (QMR) method and spectrally via unsymmetric transpose-free QMR.

Simple and efficient computation of electromagnetic fields in arbitrarily shaped inhomogeneous dielectric bodies using transpose-free QMR and FFT

A simple and efficient numerical method is presented for computing electromagnetic fields in three-dimensional (3-D) inhomogeneous-dielectric bodies. The method employs a two-stage discretization to convert an integro-differential equation into an implicit system of linear algebraic equations. This discrete system is then solved using a transpose-free quasi-minimal residual (TFQMR) algorithm, which avoids the calculation of the multiplication between the transpose of the system matrix and a vector. The simple multiplication between the system matrix and a vector required in the TFQMR algorithm is calculated efficiently using only six fast Fourier transforms (FFTs). Numerical results for strongly inhomogeneous and lossy spheres show that the method has a stable convergence behavior and excellent numerical performance.

Méthodes itératives pour les fluides incompressibles

ABSTRACT This paper presents an overview of different iterative methods Conjugate gradient (CG), Biconjugate gradient (BICG), conjugate gradient squared (CGS), Biconjugate gradient stabilized (BICGSTAB), transpose-free quasi minimal residual (TFQMR), full orthogonal method (FOM) et Generalized minimal residual (GMRES) for solving finite element systems of Navier-Stokes incompressible flows. To accelerate convergence of those methods, preconditionner based on incomplete Gauss factorisation (ILU) is used. The accent is put on the necessity to renumber the unknowns to guaranty the convergence of the iterative methods. The importance of a variable stop criterion of iterative methods for each Newton-Raphson step is underlined.

Comparison of variants of the bi‐conjugate gradient method for compressible Navier‐Stokes solver with second‐moment closure

AbstractVariants of the bi‐conjugate gradient (Bi‐CG) method are used to resolve the problem of slow convergence in CFD when it is applied to complex flow field simulation using higher‐order turbulence models. In this study the Navier‐Stokes and Reynolds stress transport equations are discretized with an implicit, total variation diminishing (TVD), finite volume formulation. The preconditioning technique of incomplete lower‐upper (ILU) factorization is incorporated into the conjugate gradient square (CGS), bi‐conjugate gradient stable (Bi‐CGSTAB) and transpose‐free quasi‐minimal residual (TFQMR) algorithms to accelerate convergence of the overall itertive methods. Computations have been carried out for separated flow fields over transonic bumps, supersonic bases and supersonic compression corners. By comparisons of the convergence rate with each other and with the conventional approximate factorization (AF) method it is shown that the Bi‐CGSTAB method gives the most efficient convergence rate among these methods and can speed up the CPU time by a factor of 2·4–6·5 as compared with the AF method. Moreover, the AF method may yield somewhat different results from variants of the Bi‐CG method owing to the factorization error which introduces a higher level of convergence criterion.