Preconditioned Biconjugate Gradient Research Articles

From BEYONDPLANCK to COSMOGLOBE: Preliminary WMAP Q-band analysis

We present the first application of the COSMOGLOBE analysis framework by analyzing nine-year WMAP time-ordered observations that uses similar machinery to that of BEYONDPLANCK for the Planck Low Frequency Instrument (LFI). We analyzed only the Q-band (41 GHz) data and report on the low-level analysis process based on uncalibrated time-ordered data to calibrated maps. Most of the existing BEYONDPLANCK pipeline may be reused for WMAP analysis with minimal changes to the existing codebase. The main modification is the implementation of the same preconditioned biconjugate gradient mapmaker used by the WMAP team. Producing a single WMAP Q1-band sample requires 22 CPU-hrs, which is slightly more than the cost of a Planck 44 GHz sample of 17 CPU-hrs; this demonstrates that a full end-to-end Bayesian processing of the WMAP data is computationally feasible. In general, our recovered maps are very similar to the maps released by the WMAP team, although with two notable differences. In terms of temperature, we find a ∼2 μK quadrupole difference that most likely is caused by different gain modeling, while in polarization we find a distinct 2.5 μK signal that has been previously referred to as poorly measured modes by the WMAP team. In the COSMOGLOBE processing, this pattern arises from temperature-to-polarization leakage from the coupling between the CMB Solar dipole, transmission imbalance, and sidelobes. No traces of this pattern are found in either the frequency map or TOD residual map, suggesting that the current processing has succeeded in modeling these poorly measured modes within the assumed parametric model by using Planck information to break the sky-synchronous degeneracies inherent in the WMAP scanning strategy.

An efficient cascadic multigrid solver for 3-D magnetotelluric forward modelling problems using potentials

SUMMARY The fast and accurate 3-D magnetotelluric (MT) forward modelling is core engine of the interpretation and inversion of MT data. In this study, we develop an improved extrapolation cascadic multigrid method (EXCMG) to solve the large sparse complex linear system arising from the finite-element (FE) discretization on non-uniform orthogonal grids of the Maxwell’s equations using potentials. First, the vector Helmholtz equation and the scalar auxiliary equation are derived from the Maxwell’s equations using Coulomb-gauged potentials. The weighted residual method is adopted to discretize the weak formulation and assemble the FE equation. Secondly, carefully choosing the preconditioned complex stable bi-conjugate gradient method (BiCGStab) as multigrid smoother, we develop an improved EXCMG method on non-uniform grids to solve the resulting large sparse complex non-Hermitian linear systems. Finally, several examples including three standard testing models (COMMEMI3D-1, COMMEMI3D-2 and DTM1.0) and a topographic model are used to validate the accuracy and efficiency of the proposed multigrid solver. Numerical results show that the proposed EXCMG algorithm greatly improves the efficiency of 3-D MT forward modelling, is more efficient than some existing solvers, such as Pardiso, incomplete LU factorization preconditioned biconjugate gradients stabilized method (ILU-BiCGStab) and flexible generalized minimum residual method with auxiliary space Maxwell preconditioner (FGMRES-AMS), and capable to simulate large-scale problems with more than 100 million unknowns.

A three-dimensional finite difference model for ocean acoustic propagation and benchmarking for topographic effects.

A three-dimensional (3D) finite difference (FD) model with formal fourth-order accuracy has been developed for the ocean acoustic Helmholtz equation (HE), which can be used to address arbitrary bathymetry and provide more accurate benchmark solutions for other 3D underwater acoustic approximate models. The derivatives in the acoustic HE are numerically discretized based on regular grids, and the perfectly matched layer is introduced to absorb unphysical reflections from the boundaries where Sommerfeld radiation conditions are deployed. The system of linear equations is solved using a parallel matrix-free geometric multigrid preconditioned biconjugate gradient stabilized iteration method, and the code (named COACH) is run on the Tianhe-2 supercomputer in China. Four 3D topographic benchmark acoustic cases-a wedge waveguide, Gaussian canyon, conical seamount, and corrugated seabed-are simulated to test the present FD model, and the maximum number of grid points reaches 33.15 × 109 in the wedge waveguide case, running in parallel with 988 central processing unit cores. Furthermore, the accuracy and generality of the present model have been verified by solution comparisons with other available 3D acoustic propagation models, and the two-dimensional and 3D transmission loss contours are presented to facilitate the distinguishing among the acoustic field features of these cases.

Hybrid multi-projection method using sparse approximate inverses on GPU clusters

The state-of-the-art supercomputing infrastructures are equipped with accelerators, such as graphics processing units (GPUs), that operate as coprocessors for each workstation of the distributed memory system. The multi-projection type methods are a class of algebraic domain decomposition methods based on semi-aggregation techniques. The multi-projection type methods have improved convergence behavior, as the number of subdomains increases, due to the corresponding augmentation of the semi-aggregated local linear systems with more coarse components, while the number of fine components is reduced. Moreover, limited amount of communications among the workstations is required by the proposed method. The utilization of the available GPUs allows an increase in the number of subdomains along with finer-grained parallelism, leading to improved performance. A load-balancing algorithm that ensures the concurrency of the computations on multicore processors and GPUs is proposed. Flexible parallel preconditioned Krylov subspace iterative methods enhanced with multi-projection type methods have been designed appropriately in order to have improved performance, compared to CPU-only or GPU-only executions, by exploiting the available CPUs and GPUs of the distributed memory system concurrently. The unsymmetric local linear systems are solved by the preconditioned Bi-Conjugate Gradient STABilized (BiCGSTAB) method enhanced with the modified generic factored approximate sparse inverse preconditioner, whereas the preconditioned conjugate gradient (CG) method along with the symmetric factored approximate sparse inverse preconditioner is used for the symmetric positive definite local coefficient matrices. Numerical results regarding the convergence behavior, the performance, and the scalability of the proposed method for several problems are given.

Digital Estimation and Compensation of Analog Errors in Frequency-Interleaved ADCs

A novel digital compensation scheme for measuring, estimating and correcting linear weakly time-varying analog errors in frequency-interleaved analog-to-digital converters (FI-ADCs) is presented. This method features three important improvements over existing approaches: First, the Wigner–Ville distribution (WVD) is used to better estimate the nonstationary analog channel frequency response (ACFR) spectrum. Secondly, the estimated ACFR spectrum is approximated with a rational polynomial model using the [Formula: see text]-norm metric. The corresponding [Formula: see text]-norm minimization problem is solved using a primal-relaxed dual global optimization (PRD-GOP) method. Thirdly, the digital compensation circuitry is designed utilizing a preconditioned biconjugate gradient stabilized (BICGSTAB) algorithm that yields a computationally efficient solution. Numerical experimentations have been conducted and the outcomes validate the feasibility and superior performance of this proposed method.

A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time–space fractional diffusion equation

A block lower triangular Toeplitz system arising from time-space fractional diffusion equation is discussed. For efficient solutions of such the linear system, the preconditioned biconjugate gradient stabilized method and flexible general minimal residual method are exploited. The main contribution of this paper has two aspects: (i) A block bi-diagonal Toeplitz preconditioner is developed for the block lower triangular Toeplitz system, whose storage is of $\mathcal{O}(N)$ with $N$ being the spatial grid number; (ii) A new skew-circulant preconditioner is designed to fast calculate the inverse of the block bi-diagonal Toeplitz preconditioner multiplying a vector. Numerical experiments are given to demonstrate the efficiency of our preconditioners.

High-Frequency Electrothermal Characterization of TSV-Based Power Delivery Network

In this paper, high-frequency electrothermal characteristics of the power delivery network (PDN) are investigated for through-silicon-via (TSV)-based 3-D ICs by utilizing a self-developed electrothermal co-simulation solver. The solver circularly solves the full-wave electromagnetic equation and the steady heat conduction equation using the finite-element method. The preconditioned biconjugate gradient method combined with the element-by-element approximate factorization method is employed to speed up the simulation and save memory cost. Based on the solver, the impacts of the excitation condition and dielectric loss tangent are analyzed for a one-chip power grid, while the influence of TSV location is studied for a two-chip PDN structure. In the modeling, both the conductor and dielectric losses are taken into account, and the temperature dependence of material parameters is treated appropriately. As results, PDN impedance and self-heating-induced temperature rise are emphatically analyzed in a wide frequency range, and the electric field and temperature distributions of several resonance modes are presented. The results would be beneficial for the design and thermal management of 3-D PDNs.

Quadratic finite element and preconditioning methods for options pricing in the SVCJ model

We consider option pricing problems in the stochastic volatility jump diffusion model with correlated and contemporaneous jumps in both the return and the variance processes (SVCJ). The option value function solves a partial integro-differential equation (PIDE). We discretize this PIDE in space by the quadratic FE method and integrate the resulting ordinary differential equation in time by an implicit-explicit Euler based extrapolation scheme. The coefficient matrix of the resulting linear systems is block penta-diagonal with penta-diagonal blocks. The preconditioned biconjugate gradient stabilized (PBiCGSTAB) method is used to solve the linear systems. According to the structure of the coefficient matrix, several preconditioners are implemented and compared. The performance of preconditioning techniques for solving block-tridiagonal systems resulting from the linear FE discretization of the PIDE is also investigated. The combination of the quadratic FE for spatial discretization, the extrapolation scheme for time discretization, and the PBiCGSTAB method with an appropriate preconditioner is found to be very efficient for solving the option pricing problems in the SVCJ model. Compared to the standard second order linear finite element method combined with the popular successive over-relaxation (SOR) linear system solver, the proposed method reduces computational time by about twenty times at the accuracy level of 1 cent and more than fifty times at the accuracy level of 0.1 cent for the barrier option example tested in the paper.

Parallel multigrid algorithms based on generic approximate sparse inverses: an SMP approach

New parallel computational techniques are introduced for the parallelization of Generic Approximate Sparse Inverse multigrid methods, based on Portable Operating System Interface for UniX (POSIX) threads, for multicore systems. Parallelization of the Generic Approximate Sparse Inverse Matrix (GenAspI) algorithm is achieved based on a new computational approach, namely "strip," which utilizes the data independence of the rows assigned in each available processor. Additionally, new parallel computational techniques are proposed for the parallelization of a modified multigrid V-Cycle method, based on POSIX Threads, for multicore systems. The modified V-Cycle utilized a Parallel PGenAspI Preconditioned Bi-Conjugate Gradient STABilized (BiCGSTAB) as a coarse solver to ensure better parallel performance of the multigrid method. For parallelization purposes, a replication of the multigrid method function is executed on each processor with different index bands and with proper synchronization points to ensure less thread-creation overhead and to maximize parallel performance. Theoretical estimates on speedups and efficiency are also presented. Finally, numerical results for the performance of the PGenAspI algorithm and the PGenAspI---MGV method for solving classical two-dimensional boundary value problems on multicore computer systems are presented. The implementation issues of the proposed method are also discussed using POSIX threads on multicore systems.

Modified nodal cubic spline collocation for three-dimensional variable coefficient second order partial differential equations

We formulate a fourth order modified nodal cubic spline collocation scheme for variable coefficient second order partial differential equations in the unit cube subject to nonzero Dirichlet boundary conditions. The approximate solution satisfies a perturbed partial differential equation at the interior nodes of a uniform $N\times N\times N$ partition of the cube and the partial differential equation at the boundary nodes. In the special case of Poisson's equation, the resulting linear system is solved by a matrix decomposition algorithm with fast Fourier transforms at a cost $O(N^3\log N)$ . For the general variable coefficient diffusion-dominated case, the system is solved using the preconditioned biconjugate gradient stabilized method.

Calculation of normal mode spectra in laterally heterogeneous earth models using an iterative direct solution method

Normal mode observations play an important role in studying broad-scale lateral variations in the Earth. Such studies require the calculation of accurate synthetic spectra in realistic earth models, and this remains a computationally challenging problem. Here, we describe a new implementation of the direct solution method for calculating normal mode spectra in laterally heterogeneous earth models. In this iterative direct solution method, the mode-coupling equations are solved in the frequency-domain using the preconditioned biconjugate gradient algorithm, and the time-domain solution is recovered using a numerical inverse Fourier transform. A number of example calculations are presented to demonstrate the accuracy and efficiency of the method for performing large ‘full coupling’ calculations as compared to methods based on matrix diagonalization and the traditional direct solution method.

A Parallel Preconditioned Bi-Conjugate Gradient Stabilized Solver for the Poisson Problem

We present a parallel Preconditioned Bi-Conjugate Gradient Stabilized(BICGstab) solver for the Poisson problem. Given a real, nosymmetric and positive definite coefficient matrix , the parallized Preconditioned BICGstab -solver is able to find a solution for that system by exploiting the massive compute power of todays GPUs.Comparing sequential CPU implementations and that algorithm.we achieve a speed up from 8 to 10 depending on the dimension of the coefficient matrix. Additionally the concept of preconditioners to decrease the time to find a solution is evaluated using the AINV method.

A mimetic iterative scheme for solving biharmonic equations

A new mimetic iterative scheme for solving general biharmonic equations under Robin’s conditions is presented. This approach combines recently developed mimetic techniques for partial differential equations (PDEs) with an efficient iterative scheme based on a global conjugate gradient and a local preconditioned biconjugate gradient methods. The elegant matrix formulation of mimetic methods allows us to present an original convergence analysis of the proposed scheme. Numerical experiments provide strong evidences of the advantages of the novel approach.

Spectral Chebyshev Collocation for the Poisson and Biharmonic Equations

This paper is concerned with the spectral Chebyshev collocation solution of the Dirichlet problems for the Poisson and biharmonic equations in a square. The collocation schemes are solved at a cost of $2N^3+O(N^2\log N)$ operations using an appropriate set of basis functions, a matrix diagonalization algorithm, and fast Fourier transforms. For the biharmonic problem, the resulting Schur complement system is solved by a preconditioned biconjugate gradient method. An application of the Poisson spectral preconditioner is discussed for the solution of a variable coefficient spectral problem. Numerical results confirm the efficiency of the proposed algorithms and the spectral and polynomial accuracy of the collocation schemes for smooth and singular solutions, respectively.

PRECONDITIONED BI-CONJUGATE GRADIENT METHOD FOR RADIATIVE TRANSFER IN SPHERICAL MEDIA

A robust numerical method called the Preconditioned Bi-Conjugate Gradient (Pre-BiCG)method is proposed for the solution of radiative transfer equation in spherical geometry.A variant of this method called Stabilized Preconditioned Bi-Conjugate Gradient (Pre-BiCG-STAB) is also presented. These are iterative methods based on the construction of a set of bi-orthogonal vectors. The application of Pre-BiCG method in some benchmark tests show that the method is quite versatile, and can handle hard problems that may arise in astrophysical radiative transfer theory.

Preconditioning the FETI Method for Accelerating the Solution of Large EM Scattering Problems

This letter addresses the convergence rate of the Finite Element Tearing and Interconnecting (FETI) algorithm when applied to the solution of large electromagnetic scattering problems formulated via the finite element method (FEM). The FETI-H formulation is applied to derive a preconditioner for the scalar Helmholtz operator with inhomogeneous media. Examples are presented which show increased speedup over the nonpreconditioned FETI method as well as dramatic speedup as compared to a traditional diagonally preconditioned BiConjugate Gradient Method (BiCGM) solver

Performance of discrete dipole approximation for prediction of amplitude and phase of electromagnetic scattering by particles

Electromagnetic scattering provides useful signatures for nonintrusive particle characterization. Scattered wave which carries characteristic information about particles is identified completely by its intensity, polarization state and phase. Recent developments in measurement techniques have enabled measurement of phase of the scattered wave which is a source of additional information about particles. In the present study, accuracy of discrete dipole approximation (DDA) in predicting amplitude and phase of scattered wave is investigated via publicly available DDSCAT code by Draine and Flatau, which is a well-established tool for DDA and has found wide range of applications in the literature due to its flexibility. DDSCAT routine is modified to enable accurate computation of phase of complex amplitude scattering matrix (ASM) elements as well as their magnitude. DDA method was implemented by using lattice dispersion relation for dipole polarizabilities, generalized prime factor algorithm for fast-Fourier transformation and pre-conditioned bi-conjugate gradient method with stabilization for the solution of the complex linear system of equations. Accuracy of ASM elements predicted by DDA is assessed on single sphere problems with various size parameters and refractive indices by validation against Mie theory solutions. Excellent agreement between predictions and exact solutions proves the reliability of the modified DDSCAT code for prediction of amplitude and phase of scattered electromagnetic wave. Applicability conditions and requirements of the present DDA application to ensure accurate prediction of complete set of scattering parameters are mapped for single spheres, on an extensive domain of size parameters and refractive indices. A correlation is presented to estimate the magnitude and phase errors associated with given size parameter, refractive index and cubic lattice subdivision. Assessment of computational time requirements for different optical constants shows that implementation of DDA with the present specifications is unfeasible for size parameters larger than 4 when Re( m)>2 and Im( m)<0.1 at the same time, due to slow convergence rate.

Phase-type approximation of stochastic Petri nets for analysis of manufacturing systems

NonMarkovian stochastic Petri nets (SPN) have received special attention due to their functionality in reflecting nonexponential dynamic behavior encountered in modeling and analysis of real systems. In this paper, a novel analysis approach, based on phase-type approximation, is proposed to provide transient and steady-state probabilities and determine performance measures of these nonMarkovian SPN. The approach can accommodate a wide variety of nonexponential distributions and provide a stronger mechanism than other methods proposed to date for analyzing system performance. The proposed procedure primarily consists of three steps. First, all generally distributed transitions are fitted with phase-type transitions. Next, the nonMarkovian SPN with the approximated phase-type transitions is converted into a Markov chain. Last, transient-state probabilities are obtained by employing the uniformization method and steady-state probabilities are determined by utilizing the preconditioned biconjugate gradient method. Pertinent performance measures can be computed by using these probabilities. The proposed methodology is validated through a real example with respect to its accuracy and speed.

Development of a Local Grid-Refinement Technique for Accurate Representation of Cavity-Completed Wells in Reservoir Simulators

Summary In this paper the development of a three-dimensional, single-, and two-phase fluid flow model to study flow performances of cavity-completed wells is presented. A static local grid refinement (LGR) is embedded into the model in order to adequately capture the flow behavior around cavity completions using a reasonable number of grid cells. The model adopts an approach that is significantly different from the current practice in which cavity-completed wells are incorporated into computations by assigning large wellbore radii and/or large negative skin values to a conventional wellbore model. The model utilizes the residual equation rather than a wellbore equation to determine the flow into cavity-completed wells. Cavity-completed wells are described as a collection of fine rectangular grid blocks. This representation is referred to as the cavity approach. The constructed fine grid lines are terminated at the coarse/fine grid block interfaces by implementing the LGR technique developed. The technique subdivides selected coarse grid blocks into fine rectangular grid blocks in all three dimensions that are known as window blocks. The flow equations generated for window and coarse grid cells are then solved simultaneously by using the preconditioned biconjugate gradient stabilized iterative method. First, a radial-cylindrical wellbore is approximated using the cavity representation. The model results from single phase, slightly compressible fluid flow simulation with the cavity approach are, then, compared against the analytical solutions. In these comparisons, very good matches are obtained for both flow rate and sandface pressure specifications at the wellbore. The model is further extended to single phase, compressible, and two-phase fluid flow conditions in both conventional and coalbed reservoirs. The model results are, in this case, compared with the existing numerical models that utilize Peaceman's wellbore model. Again, close agreement in flow response is obtained for a series of comparisons. Finally, the cavity approach is tested by modeling flow performances of irregularly shaped, cavity-completed wells in conventional and coalbed reservoirs. During the creation of cavity completions in coal seams, it is hypothesized that fracture networks in certain directions are generated around cavities. The effects of fractures are usually incorporated into a simulation study by modifying the permeability values around the wellbore. The cavity approach presented in this paper provides not only a better approximation to the shape of the cavity but also a better representation of the altered permeability values around the cavity since fine grid cells are already in place around cavity completions.

Efficient Pseudospectral Flow Simulations in Moderately Complex Geometries

A computationally efficient pseudospectral method is developed for incompressible flow simulations in two-dimensional geometries involving periodicity in one direction and significant surface deformations. A pseudoconformal mapping is used to map the flow domain into a rectangle, thereby establishing an orthogonal curvilinear coordinate system within which the governing equations are formulated. The time integration of the spectrally discretized, two-dimensional momentum equations is performed by a second-order mixed explicit/implicit time integration scheme. The satisfaction of the continuity equation is obtained through the solution of a Poisson equation for the pressure and the use of the influence matrix technique. A highly efficient iterative solver has been developed for the solution of a generalized Stokes problem at each time step based on a spectrally preconditioned biconjugate gradient algorithm, which exhibits almost linear scalability, requiring an orderNlog2Nnumber of operations, whereNis the number of unknowns. Numerical results are presented for two-dimensional steady, oscillatory, and peristaltic flows within an undulating channel, which agree well with previous results that have appeared in the literature.