Preconditioned Method Research Articles

Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization

Preconditioning and Regularization Enable Faster Reinforcement Learning Natural policy gradient (NPG) methods, in conjunction with entropy regularization to encourage exploration, are among the most popular policy optimization algorithms in contemporary reinforcement learning. Despite the empirical success, the theoretical underpinnings for NPG methods remain severely limited. In “Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization”, Cen, Cheng, Chen, Wei, and Chi develop nonasymptotic convergence guarantees for entropy-regularized NPG methods under softmax parameterization, focusing on tabular discounted Markov decision processes. Assuming access to exact policy evaluation, the authors demonstrate that the algorithm converges linearly at an astonishing rate that is independent of the dimension of the state-action space. Moreover, the algorithm is provably stable vis-à-vis inexactness of policy evaluation. Accommodating a wide range of learning rates, this convergence result highlights the role of preconditioning and regularization in enabling fast convergence.

A new accelerating technique for low speed flow: pseudo high speed method

This paper proposes a new accelerating technique for simulating low speed flows, termed as pSeudo High Speed method (SHS), which uses governing equations and numerical methods of compressible flows. SHS method has advantages of simple formula, easy manipulation, and only need to modify flux of Euler equations. It can directly employ the existing well-developed schemes of hyperbolic conservation laws. To verify the technique, several numerical experiments are performed, such as: flow past airfoils and flow past a cylinder. Analysis of SHS method and comparisons with some precondition methods are made numerically. All tests show that SHS method can greatly improve the efficiency of compressible method simulating low speed flow fields, which exhibits in accelerating the convergence rate and increasing the accuracy of the numerical results.

The method of distant preconditioning and results of its application in renal transplantation from the living family donor

Objective. Elaboration of method for the intraoperative ischemic-reperfusion trauma softening - distant ischemic preconditioning and investigation of results of its introduction into clinical practice.
 Materials and methods. The method of distant ischemic preconditioning elaborated have included four procedures the air inflation into the flap for the arterial pressure measuring by 40 mm Hg over the level of systolic arterial pressure, 5 min of duration in every one, with consequent 5-minute intervals for the air release from the flap. The procedure of a distant ischemic preconditioning was conducted after introduction into narcosis, but before the donor's nephrectomy in 30 patients (the main group). The procedure was not conducted in 30 such patients.
 Results. The procedure elaborated permits to enhance the glomerular filtration significantly in 6 and 12 mo, to reduce the rate of partial delay of the transplant functioning, its acute rejection and primary dysfunction.
 Conclusion. The procedure of a distant ischemic preconditioning elaborated improves the transplanted kidney function.

Numerical solutions to linear transfer problems of polarized radiation

Context. Numerical solutions to transfer problems of polarized radiation in solar and stellar atmospheres commonly rely on stationary iterative methods, which often perform poorly when applied to large problems. In recent times, stationary iterative methods have been replaced by state-of-the-art preconditioned Krylov iterative methods for many applications. However, a general description and a convergence analysis of Krylov methods in the polarized radiative transfer context are still lacking. Aims. We describe the practical application of preconditioned Krylov methods to linear transfer problems of polarized radiation, possibly in a matrix-free context. The main aim is to clarify the advantages and drawbacks of various Krylov accelerators with respect to stationary iterative methods and direct solution strategies. Methods. After a brief introduction to the concept of Krylov methods, we report the convergence rate and the run time of various Krylov-accelerated techniques combined with different formal solvers when applied to a 1D benchmark transfer problem of polarized radiation. In particular, we analyze the GMRES, BICGSTAB, and CGS Krylov methods, preconditioned with Jacobi, (S)SOR, or an incomplete LU factorization. Furthermore, specific numerical tests were performed to study the robustness of the various methods as the problem size grew. Results. Krylov methods accelerate the convergence, reduce the run time, and improve the robustness (with respect to the problem size) of standard stationary iterative methods. Jacobi-preconditioned Krylov methods outperform SOR-preconditioned stationary iterations in all respects. In particular, the Jacobi-GMRES method offers the best overall performance for the problem setting in use. Conclusions. Krylov methods can be more challenging to implement than stationary iterative methods. However, an algebraic formulation of the radiative transfer problem allows one to apply and study Krylov acceleration strategies with little effort. Furthermore, many available numerical libraries implement matrix-free Krylov routines, enabling an almost effortless transition to Krylov methods.

Iterative Preconditioned Methods in Krylov Spaces: Trends of the 21st Century

Iterative Preconditioned Methods in Krylov Spaces: Trends of the 21st Century

Modulus-based inexact non-alternating preconditioned splitting method for linear complementarity problems

For the large sparse linear complementarity problem, by reformulating it as implicit fixed-point equations, we establish a modulus-based iteration method with the aid of an inexact non-alternating preconditioned matrix splitting iteration method used as an inner iteration to solve the module equations rapidly in each iteration step. The convergence properties of this method are carefully demonstrated under certain conditions. Numerical results, including the numbers of iteration steps and the computing times, validate that our method is superior to the other three iteration methods.

BDDC algorithms for advection-diffusion problems with HDG discretizations

In this paper, a preconditioned GMRES method is developed and analyzed for solving the linear system from advection-diffusion equations with the hybridizable discontinuous Galerkin (HDG) discretization. The preconditioner is the balancing domain decomposition methods (BDDC), one of the most popular nonoverlapping domain decomposition methods. For large viscosity, if the subdomain size is small enough, the number of iterations is independent of the number of subdomains and depends only slightly on the subdomain problem size. The convergence deteriorates when the viscosity decreases. These results are similar to those with the standard finite element discretizations. Numerical results of two examples in two dimensions are provided to confirm the theory.

A preconditioned two-sweep shift splitting method for non-Hermitian positive definite linear systems

In this paper, to find the numerical solution of the linear systems with the non-Hermitian positive definite matrix, a preconditioned two-sweep shift splitting (PTSS) iteration method is introduced and its convergent conditions are discussed. When the PTSS method is used as a preconditioner, the eigenvalue distribution of the corresponding preconditioned matrix is given. Numerical experiments are reported to the efficiency of the PTSS method and the PTSS preconditioner.

Tusas: A fully implicit parallel approach for coupled phase-field equations

We develop a fully-coupled, fully-implicit approach for phase-field modeling of solidification in metals and alloys. Predictive simulation of solidification in pure metals and metal alloys remains a significant challenge in the field of materials science, as microstructure formation during the solidification process plays a critical role in the properties and performance of the solid material. Our simulation approach consists of a finite element spatial discretization of the fully-coupled nonlinear system of partial differential equations at the microscale, which is treated implicitly in time with a preconditioned Jacobian-free Newton-Krylov method. The approach is algorithmically scalable as well as efficient due to an effective preconditioning strategy based on algebraic multigrid and block factorization. We implement this approach in the open-source Tusas framework, which is a general, flexible tool developed in C++ for solving coupled systems of nonlinear partial differential equations. The performance of our approach is analyzed in terms of algorithmic scalability and efficiency, while the computational performance of Tusas is presented in terms of parallel scalability and efficiency on emerging heterogeneous architectures. We demonstrate that modern algorithms, discretizations, and computational science, and heterogeneous hardware provide a robust route for predictive phase-field simulation of microstructure evolution during additive manufacturing.

Fast Parallel Solver for the Space-time IgA-DG Discretization of the Diffusion Equation

We consider the space-time discretization of the diffusion equation, using an isogeometric analysis (IgA) approximation in space and a discontinuous Galerkin (DG) approximation in time. Drawing inspiration from a former spectral analysis, we propose for the resulting space-time linear system a multigrid preconditioned GMRES method, which combines a preconditioned GMRES with a standard multigrid acting only in space. The performance of the proposed solver is illustrated through numerical experiments, which show its competitiveness in terms of iteration count, run-time and parallel scaling.

Quarter-Sweep Preconditioned Relaxation Method, Algorithm and Efficiency Analysis for Fractional Mathematical Equation

Research into the recent developments for solving fractional mathematical equations requires accurate and efficient numerical methods. Although many numerical methods based on Caputo’s fractional derivative have been proposed to solve fractional mathematical equations, the efficiency of obtaining solutions using these methods when dealing with a large matrix requires further study. The matrix size influences the accuracy of the solution. Therefore, this paper proposes a quarter-sweep finite difference scheme with a preconditioned relaxation-based approximation to efficiently solve a large matrix, which is based on the establishment of a linear system for a fractional mathematical equation. The paper presents the formulation of the quarter-sweep finite difference scheme that is used to approximate the selected fractional mathematical equation. Then, the derivation of a preconditioned relaxation method based on a quarter-sweep scheme is discussed. The design of a C++ algorithm of the proposed quarter-sweep preconditioned relaxation method is shown and, finally, efficiency analysis comparing the proposed method with several tested methods is presented. The contributions of this paper are the presentation of a new preconditioned matrix to restructure the developed linear system, and the derivation of an efficient preconditioned relaxation iterative method for solving a fractional mathematical equation. By simulating the solutions of time-fractional diffusion problems with the proposed numerical method, the study found that computing solutions using the quarter-sweep preconditioned relaxation method is more efficient than using the tested methods. The proposed numerical method is able to solve the selected problems with fewer iterations and a faster execution time than the tested existing methods. The efficiency of the methods was evaluated using different matrix sizes. Thus, the combination of a quarter-sweep finite difference method, Caputo’s time-fractional derivative, and the preconditioned successive over-relaxation method showed good potential for solving different types of fractional mathematical equations, and provides a future direction for this field of research.

Assessment of a symmetry-preserving JFNK method for atmospheric convection

Numerical simulations of nonhydrostatic atmospheric flow, based on linearly decoupled semi-implicit or fully-implicit techniques, usually solve linear systems by a pre-conditioned Krylov method without preserving the skew-symmetry of convective operators. We propose to perform atmospheric simulations in such a fully-implicit manner that the difference operators preserve both the skew-symmetry and the tightly nonlinear coupling of the differential operators. We demonstrate that a symmetry-preserving Jacobian-free Newton-Krylov (JFNK) method mimics a balance between convective transport and turbulence dissipation. We present a wavelet method as an effective symmetry preserving discretization technique. The symmetry-preserving JFNK method for solving equations of nonhydrostatic atmospheric flows has been examined using two benchmark simulations of penetrative convection – a) dry thermals rising in a neutrally stratified and stably stratified environment, and b) urban heat island circulations for effects of the surface heat flux H0 varying in the range of 25≤H0≤930 Wm−2. The results show that an eddy viscosity model provides the necessary dissipation of the subgrid-scale modes, while the symmetry-preserving JFNK method provides the conservation of mass and energy at a satisfactory level. Comparisons of the results from a laboratory experiment of heat island circulation and a field measurement of potential temperature also suggest the modelling accuracy of the present symmetry-preserving JFNK framework.

A Preconditioned Variant of the Refined Arnoldi Method for Computing PageRank Eigenvectors

The PageRank model computes the stationary distribution of a Markov random walk on the linking structure of a network, and it uses the values within to represent the importance or centrality of each node. This model is first proposed by Google for ranking web pages, then it is widely applied as a centrality measure for networks arising in various fields such as in chemistry, bioinformatics, neuroscience and social networks. For example, it can measure the node centralities of the gene-gene annotation network to evaluate the relevance of each gene with a certain disease. The networks in some fields including bioinformatics are undirected, thus the corresponding adjacency matrices are symmetry. Mathematically, the PageRank model can be stated as finding the unit positive eigenvector corresponding to the largest eigenvalue of a transition matrix built upon the linking structure. With rapid development of science and technology, the networks in real applications become larger and larger, thus the PageRank model always desires numerical algorithms with reduced algorithmic or memory complexity. In this paper, we propose a novel preconditioning approach for solving the PageRank model. This approach transforms the original PageRank eigen-problem into a new one that is more amenable to solve. We then present a preconditioned version of the refined Arnoldi method for solving this model. We demonstrate theoretically that the preconditioned Arnoldi method has higher execution efficiency and parallelism than the refined Arnoldi method. In plenty of numerical experiments, this preconditioned method exhibits noticeably faster convergence speed over its standard counterpart, especially for difficult cases with large damping factors. Besides, this superiority maintains when this technique is applied to other variants of the refined Arnoldi method. Overall, the proposed technique can give the PageRank model a faster solving process, and this will possibly improve the efficiency of researches, engineering projects and services where this model is applied.

Preconditioned iterative method for nonsymmetric saddle point linear systems

In this paper, a new preconditioned iterative method is presented to solve a class of nonsymmetric nonsingular or singular saddle point problems. The implementation of the proposed preconditioned Krylov subspace method avoids solving inverse of Schur complement and only needs to solve one linear sub-system at each step, which implies that it may save considerable costs. Theoretical convergence analysis, including the bounds of eigenvalues and eigenvectors, the degree of the minimal polynomial of the preconditioned matrix, are discussed in details. Moreover, a novel algebraic estimation technique for finding a practical iteration parameter is presented, which is very effective and practical even for large scale problems. At last, some numerical examples are carried, showing that the theoretical results are valid and convincing.

Activity of ROS-induced processes in the combined preconditioning with amtizol before and after cerebral ischemia in rats

Introduction: The dose-dependent effect of reactive oxygen species (ROS) in tissues in preconditioning (PreC) and oxidative stress, as well as NO-synthase participation in mitochondrial ROS production determined the study aim – to assess the impact of the neuroprotective method of combined preconditioning (CPreC) on free radical reactions (FRRs) in brain in normoxia and in cerebral ischemia, including in NO-synthase blockade.
 Materials and methods: The intensity of FRR by iron-induced chemiluminescence (CL), the content of lipid peroxidation products and antioxidant enzyme activity were investigated 1 hr (early period) and 48 hrs (delayed period) after CPreC (amtizol and hypobaric hypoxia) in Wistar rat brain. Some animal groups were operated (common carotid artery bilateral ligation) 1 hr and 48 hrs after CPreC, as well as with preliminary introduction of L-NAME and aminoguanidine.
 Results and discussion: In normoxia, CPreC led to increase the CLmaximum level (Fmax) in the delayed PreC period. The amount of thiobarbituric acid reactive products (TBA-RP), activity of superoxide dismutase (SOD) and catalase in mitochondrial fraction of rat brain did not change in comparison with the intact control in both PreC periods. In cerebral ischemia, oxidative stress was observed. The CPreC use before ischemia caused a decrease in CL parameters and TBA-RP in brain, the maintenance of SOD and high catalase activity. NO-synthase inhibitors partially abolished the antioxidant effect of CPreC in ischemia.
 Conclusion: CPreC had no influence on FRRs in brain tissue in normoxia, but prevented their excessive activation after ischemia, especially in the delayed period. NO-synthase was involved in the CPreC neuroprotection.

The Effect of Lower Body Anaerobic Pre-Loading on Upper Body Ergometer Time Trial Performance.

Pre-competitive conditioning has become a substantial part of successful performance. In addition to temperature changes, a metabolic conditioning can have a significant effect on the outcome, although the right dosage of such a method remains unclear. The main goal of the investigation was to measure how a lower body high-intensity anaerobic cycling pre-load exercise (HIE) of 25 s affects cardiorespiratory and metabolic responses in subsequent upper body performance. Thirteen well-trained college-level male cross-country skiers (18.1 ± 2.9 years; 70.8 ± 7.6 kg; 180.6 ± 4.7 cm; 15.5 ± 3.5% body fat) participated in the study. The athletes performed a 1000-m maximal double-poling upper body ergometer time trial performance test (TT) twice. One TT was preceded by a conventional low intensity warm-up (TTlow) while additional HIE cycling was performed 9 min before the other TT (TThigh). Maximal double-poling performance after the TTlow (225.1 ± 17.6 s) was similar (p > 0.05) to the TThigh (226.1 ± 15.7 s). Net blood lactate (La) increase (delta from end of TT minus start) from the start to the end of the TTlow was 10.5 ± 2.2 mmol L−1 and 6.5 ± 3.4 mmol L−1 in TThigh (p < 0.05). La net changes during recovery were similar for both protocols, remaining 13.5% higher in TThigh group even 6 min after the maximal test. VCO2 was lower (p < 0.05) during the last 400-m split in TThigh, however during the other splits no differences were found (p < 0.05). Respiratory exchange ratio (RER) was significantly lower in TThigh in the third, fourth and the fifth 200 m split. Participants individual pacing strategies showed high relation (p < 0.05) between slower start and faster performance. In conclusion, anaerobic metabolic pre-conditioning leg exercise significantly reduced net-La increase, but all-out upper body performance was similar in both conditions. The pre-conditioning method may have some potential but needs to be combined with a pacing strategy different from the usual warm-up procedure.

Preconditioned deconvolution method for high-resolution ghost imaging

Ghost imaging (GI) can nonlocally image objects by exploiting the fluctuation characteristics of light fields, where the spatial resolution is determined by the normalized second-order correlation function g ( 2 ) . However, the spatial shift-invariant property of g ( 2 ) is distorted when the number of samples is limited, which hinders the deconvolution methods from improving the spatial resolution of GI. In this paper, based on prior imaging systems, we propose a preconditioned deconvolution method to improve the imaging resolution of GI by refining the mutual coherence of a sampling matrix in GI. Our theoretical analysis shows that the preconditioned deconvolution method actually extends the deconvolution technique to GI and regresses into the classical deconvolution technique for the conventional imaging system. The imaging resolution of GI after preconditioning is restricted to the detection noise. Both simulation and experimental results show that the spatial resolution of the reconstructed image is obviously enhanced by using the preconditioned deconvolution method. In the experiment, 1.4-fold resolution enhancement over Rayleigh criterion is achieved via the preconditioned deconvolution. Our results extend the deconvolution technique that is only applicable to spatial shift-invariant imaging systems to all linear imaging systems, and will promote their applications in biological imaging and remote sensing for high-resolution imaging demands.

Impact of Three Methods of Ischemic Preconditioning on Ischemia-Reperfusion Injury in a Pig Model of Liver Transplantation

Background Ischemic preconditioning (IPC), either direct (DIPC) or remote (RIPC), is a procedure aimed at reducing the harmful effects of ischemia-reperfusion (I/R) injury. Objectives To assess the local and systemic effects of DIPC, RIPC, and both combined, in the pig liver transplant model. Materials and methods Twenty-four pigs underwent orthotopic liver transplantation and were divided into 4 groups: control, direct donor preconditioning, indirect preconditioning at the recipient, and direct donor with indirect recipient preconditioning. The recorded parameters were: donor and recipient weight, graft-to-recipient weight ratio (GRWR), surgery time, warm and cold ischemia time, and intraoperative hemodynamic values. Blood samples were collected before native liver removal (BL) and at 0 h, 1 h, 3 h, 6 h, 12 h, 18 h, and 24 h post-reperfusion for the biochemical tests: aspartate aminotransferase (AST), alanine aminotransferase (ALT), alkaline phosphatase (ALP), gamma-glutamyl transferase (GGT), creatinine, BUN (blood urea nitrogen), lactate, total and direct bilirubin. Histopathological examination of liver, gut, kidney, and lung fragments were performed, as well as molecular analyses for expression of the apoptosis-related BAX (pro-apoptotic) and Bcl-XL (anti-apoptotic) genes, eNOS (endothelial nitric oxide synthase) gene, and IL-6 gene related to inflammatory ischemia-reperfusion injury, using real-time polymerase chain reaction (RT-PCR). Results There were no differences between the groups regarding biochemical and histopathological parameters. We found a reduced ratio between the expression of the BAX gene and Bcl-XL in the livers of animals with IPC versus the control group. Conclusions DIPC, RIPC or a combination of both, produce beneficial effects at the molecular level without biochemical or histological changes.

Effects of freeze-thaw and chemical preconditioning on the consolidation properties and microstructure of landfill sludge

At present, a large amount of landfill sludge(LS) has been accumulated all over the world. For environmental and engineering purposes, there is an urgent need for deep dewatering and volume reduction of LS. The deep dewatering of LS mainly uses the method of chemical preconditioning and mechanical dewatering, which is easy to cause environmental pollution and is not conducive to the subsequent resource treatment of LS. To find a more environmentally friendly and efficient method for deep dewatering of LS, an in-situ treatment method combining freeze-thaw and vacuum preloading was proposed. In this paper, based on the existing research, through compression consolidation test and MIP, SEM micro test, the consolidation properties and microstructure of LS after freeze-thaw and chemical preconditioning were studied, and the vacuum consolidation principle of different preconditioning was explored. The results show that: Both FeCl3 and freeze-thaw preconditioning can increase the permeability coefficient and consolidation coefficient by one to two orders of magnitude; After freeze-thaw preconditioning, the void ratio of sludge decreases and the permeability coefficient increases; Under low consolidation pressure, the mechanical properties of the two kinds of pretreated sludge changed significantly; The original sludge is mainly composed of small pores. After FeCl3 conditioning, the large pores and mesopores increased significantly, while the small pores decreased. After freeze-thaw, the large pores and mesopores increase greatly, while the small pores decrease greatly; The original sludge is in the form of a dispersive flocculent structure with many impurities. After freeze-thaw, the intercluster pores increase, showing a honeycomb structure. After FeCl3 conditioning, the sludge structure is more compact and uniform. The change of microstructure and consolidation characteristics of sludge after conditioning reflects the difference of two different preconditioning mechanisms.

Numerical investigation of conjugate heat transfer in a microchannel with a hydrophobic surface utilizing nanofluids under a magnetic field

Conjugate heat transfer in a microchannel with a slip boundary condition imposed on the channel's walls by a uniform magnetic field is studied. The working fluid consists of a Water/Ag mixture nanofluid. A preconditioned lattice Boltzmann method (LBM), formed by combining the incompressible LBM with the regular LBM, is applied to the velocity field and temperature field, respectively. The microchannel's upper wall is thermally isolated when a constant heat flux is imposed on the basin of the microchannel. The simulations are carried out under a variety of different conditions, e.g., various Reynold numbers, Re = 50 and 150, nanoparticle concentrations (φ = 0, 3%), slip coefficients (0 ≤ B ≤ 0.03), and Hartmann numbers (0 ≤ Ha ≤ 30). Surface hydrophobicity results in a reduction of surface friction of up to 46% at B = 0.03 and Ha = 30. The surface friction reductions at Ha = 0, 10, and 20 are 15%, 27%, and 38%, respectively. These results indicate that as the surface slip increases, the drag resisting the fluid dynamics decreases. Moreover, adding the nanoparticles to the base flow can improve the heat transfer by 50%. Besides, using the magnetic field increase the shear stress and, consequently, the drag force dramatically (up 340%). On the other hand, the magnetic field enhances the heat transfer by improving the fluid velocity near the wall, while its effect on the Nu number improvement is not more than 20%. As a result, the magnetic power should be controlled to achieve the best heat transfer performance with the lowest pumping energy consumption.