Multigrid Analysis Research Articles

Multigrid solvers for isogeometric discretizations of the second biharmonic problem

We develop a multigrid solver for the second biharmonic problem in the context of Isogeometric Analysis (IgA), where we also allow a zero-order term. In a previous paper, the authors have developed an analysis for the first biharmonic problem based on Hackbusch’s framework. This analysis can only be extended to the second biharmonic problem if one assumes uniform grids. In this paper, we prove a multigrid convergence estimate using Bramble’s framework for multigrid analysis without regularity assumptions. We show that the bound for the convergence rate is independent of the scaling of the zero-order term and the spline degree. It only depends linearly on the number of levels, thus logarithmically on the grid size. Numerical experiments are provided which illustrate the convergence theory and the efficiency of the proposed multigrid approaches.

A Pattern Classification Distribution Method for Geostatistical Modeling Evaluation and Uncertainty Quantification

Geological models are essential components in various applications. To generate reliable realizations, the geostatistical method focuses on reproducing spatial structures from training images (TIs). Moreover, uncertainty plays an important role in Earth systems. It is beneficial for creating an ensemble of stochastic realizations with high diversity. In this work, we applied a pattern classification distribution (PCD) method to quantitatively evaluate geostatistical modeling. First, we proposed a correlation-driven template method to capture geological patterns. According to the spatial dependency of the TI, region growing and elbow-point detection were launched to create an adaptive template. Second, a combination of clustering and classification was suggested to characterize geological realizations. Aiming at simplifying parameter specification, the program employed hierarchical clustering and decision tree to categorize geological structures. Third, we designed a stacking framework to develop the multi-grid analysis. The contribution of each grid was calculated based on the morphological characteristics of TI. Our program was extensively examined by a channel model, a 2D nonstationary flume system, 2D subglacial bed topographic models in Antarctica, and 3D sandstone models. We activated various geostatistical programs to produce realizations. The experimental results indicated that PCD is capable of addressing multiple geological categories, continuous variables, and high-dimensional structures.

High‐Resolution Gridded Temperature and Salinity Fields From Argo Floats Based on a Spatiotemporal Four‐Dimensional Multigrid Analysis Method

AbstractThe shortage of in situ ocean observations greatly limits the development of gridded ocean subsurface environmental analysis products. Fully using the correlated information of the observations in time and space is an appropriate approach to this problem. Here we adopt a method of spatiotemporal four‐dimensional multigrid analysis. A weekly gridded Argo product is also constructed in the Northwest Pacific Ocean (0°–40°N, 120°–152°E) from 2005 to 2020, with a horizontal resolution of 0.5° × 0.5° and 65 layers in the upper 1,500 m. Then, the gridded Argo product is evaluated by using in situ Argo floats, satellite altimeter data, the Tropical Atmosphere Ocean/Triangle Trans‐Ocean Buoy Network (TAO/TRITON), high‐resolution shipboard observations and some global gridded Argo products. Compared with in situ Argo floats, the reconstructed fields retain the spatiotemporal information of Argo floats well. The maximum vertical root mean square errors of temperature and salinity between the gridded Argo product and in situ Argo floats are less than 0.47°C and 0.06 psu respectively. In addition, the product agrees well with independent observations of satellite altimeter data, TAO/TRITON. Compared with shipboard observations, the product can well reflect the structure of mesoscale eddies and detect some small‐scale water masses. Compared with global gridded Argo products, our product retains more mesoscale features and has spatiotemporal correlation.

EAKF-Based Parameter Optimization Using a Hybrid Adaptive Method

Abstract To effectively reduce model bias and improve assimilation quality, we adopt a hybrid adaptive approach of ensemble adjustment Kalman filter (EAKF) and multigrid analysis (MGA), called EAKF-MGA, to implement parameter optimization as follows. For each assimilation cycle, observations are used to adjust the prior ensembles of both state variables and parameters using the EAKF without inflation. Then, the MGA is adaptively triggered to extract multiscale information from the observational residual to innovate the ensemble mean of the state once again. Results of biased twin experiments consisting of a barotropic spectral model and idealized observation systems show that the proposed EAKF-MGA is insensitive to state variance inflation and localization during the parameter optimization process, compared with the EAKF with adaptive inflation. We also find that computational efficiency is another important advantage of the EAKF-MGA for both state estimation and parameter estimation since extremely small ensemble size is allowed, while the EAKF with adaptive inflation does not work anymore. In essence, the EAKF-MGA is designed to estimate and correct systematic errors jointly with model’s state variables. Through alleviating biases, including the model bias caused by the biased parameter and the analysis bias resulting from the sampling noise given the limited ensemble size, it can be guaranteed that the analysis in the EAKF-MGA will be proceeded onward with the standard assumption of the unbiased model background field in modern data assimilation theory to be met.

Convergence Analysis of Inexact Two-Grid Methods: A Theoretical Framework

Multigrid is one of the most efficient methods for solving large-scale linear systems that arise from discretized partial differential equations. As a foundation for multigrid analysis, two-grid theory plays an important role in motivating and analyzing multigrid algorithms. For symmetric positive definite problems, the convergence theory of two-grid methods with exact solution of the Galerkin coarse-grid system is mature, and the convergence factor of exact two-grid methods can be characterized by an identity. Compared with the exact case, the convergence theory of inexact two-grid methods (i.e., the coarse-grid system is solved approximately) is of more practical significance, while it is still less developed in the literature (one reason is that the error propagation matrix of inexact coarse-grid correction is not a projection). In this paper, we develop a theoretical framework for the convergence analysis of inexact two-grid methods. More specifically, we present two-sided bounds for the energy norm of the error propagation matrix of inexact two-grid methods, from which one can readily obtain the identity for exact two-grid convergence. As an application, we establish a unified convergence theory for multigrid methods, which allows the coarsest-grid system to be solved approximately.

Local Fourier Analysis of Multigrid for Hybridized and Embedded Discontinuous Galerkin Methods

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 22 June 2020Accepted: 02 June 2021Published online: 19 August 2021Keywordspreconditioning, embedded and hybridized discontinuous Galerkin methods, geometric multigrid, local Fourier analysisAMS Subject Headings65F08, 65N30, 65N55Publication DataISSN (print): 1064-8275ISSN (online): 1095-7197Publisher: Society for Industrial and Applied MathematicsCODEN: sjoce3

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

SummaryIn this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher‐order finite‐element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse‐grid correction is assumed to be an ideal operator that annihilates the low‐frequency error components and leaves the high‐frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two‐grid convergence factor. While two‐grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is “between” two‐grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher‐order finite element discretizations, including for Q2‐Q1 (Taylor‐Hood) elements for the Stokes equations. Finally, we present two‐grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two‐grid and multigrid convergence factors.

Performance of preconditioned iterative linear solvers for cardiovascular simulations in rigid and deformable vessels.

Computing the solution of linear systems of equations is invariably the most time consuming task in the numerical solutions of PDEs in many fields of computational science. In this study, we focus on the numerical simulation of cardiovascular hemodynamics with rigid and deformable walls, discretized in space and time through the variational multiscale finite element method. We focus on three approaches: the problem agnostic generalized minimum residual (GMRES) and stabilized bi-conjugate gradient (BICGS) methods, and a recently proposed, problem specific, bi-partitioned (BIPN) method. We also perform a comparative analysis of several preconditioners, including diagonal, block-diagonal, incomplete factorization, multigrid, and resistance based methods. Solver performance and matrix characteristics (diagonal dominance, symmetry, sparsity, bandwidth and spectral properties) are first examined for an idealized cylindrical geometry with physiologic boundary conditions and then successively tested on several patient-specific anatomies representative of realistic cardiovascular simulation problems. Incomplete factorization preconditioners provide the best performance and results in terms of both strong and weak scalability. The BIPN method was found to outperform other methods in patient-specific models with rigid walls. In models with deformable walls, BIPN was outperformed by BICG with diagonal and Incomplete LU preconditioners.

Local Fourier Analysis of Balancing Domain Decomposition By Constraints Algorithms

Local Fourier analysis is a commonly used tool for the analysis of multigrid and other multilevel algorithms, providing both insight into observed convergence rates and predictive analysis of the performance of many algorithms. In this paper, for the first time, we adapt local Fourier analysis to examine variants of two- and three-level balancing domain decomposition by constraints (BDDC) algorithms, to better understand the eigenvalue distributions and condition number bounds on these preconditioned operators. This adaptation is based on a modified choice of basis for the space of Fourier harmonics that greatly simplifies the application of local Fourier analysis in this setting. The local Fourier analysis is validated by considering the two-dimensional Laplacian and predicting the condition numbers of the preconditioned operators with different sizes of subdomains. Several variants are analyzed, showing that the two- and three-level performance of the “lumped” variant can be greatly improved when used in multiplicative combination with a weighted diagonal scaling preconditioner, with weight optimized through the use of local Fourier analysis.

Improving EnKF-Based Initialization for ENSO Prediction Using a Hybrid Adaptive Method

Abstract Probabilistic forecasts, which are usually initialized by an ensemble Kalman filter (EnKF), are known to be better than deterministic (or one member) forecasts for the El Niño–Southern Oscillation (ENSO) phenomenon. Because of sampling errors caused by a finite ensemble and the errors related to model biases associated with the physical parameterizations, dynamic core, model resolution, and so on, a state-of-the-art inflation method is commonly used in the standard EnKF to increase the prior variance so as to avoid filter divergence. However, the optimal inflation factor is almost prohibitive in reality because of vast computational cost. An adaptive EnKF and multigrid analysis hybrid approach without inflation is presented to compensate for the abovementioned limitations of the standard EnKF. In this study, the adaptive approach is applied to an intermediate coupled model for ENSO prediction. Gridded observations of daily-mean sea surface temperature (SST) anomalies from the Advanced Very High Resolution Radiometer (AVHRR) during January 1982–December 2012 are assimilated into the model to initialize a 2-yr ENSO hindcast. Results show that compared to the standard EnKF that uses multiplicative variance inflation, the adaptive method can reduce analysis errors by 63% for both the daily SST anomaly and the Niño-1+2 SST anomaly. The prediction skill of Niño-1+2 SST anomaly is consistently enhanced, especially for phase forecast. For SST anomaly forecasting, the advantage of the adaptive method mainly occurs in the eastern equatorial Pacific and the northern boundary of the intermediate coupled model.

Reconstruction of Typhoon Structure Using 3-Dimensional Doppler Radar Radial Velocity Data with the Multigrid Analysis: A Case Study in an Idealized Simulation Context

Extracting multiple-scale observational information is critical for accurately reconstructing the structure of mesoscale circulation systems such as typhoon. The Space and Time Mesoscale Analysis System (STMAS) with multigrid data assimilation developed in Earth System Research Laboratory (ESRL) in National Oceanic and Atmospheric Administration (NOAA) has addressed this issue. Previous studies have shown the capability of STMAS to retrieve multiscale information in 2-dimensional Doppler radar radial velocity observations. This study explores the application of 3-dimensional (3D) Doppler radar radial velocities with STMAS for reconstructing a 3D typhoon structure. As for the first step, here, we use an idealized simulation framework. A two-scale simulated “typhoon” field is constructed and referred to as “truth,” from which randomly distributed conventional wind data and 3D Doppler radar radial wind data are generated. These data are used to reconstruct the synthetic 3D “typhoon” structure by the STMAS and the traditional 3D variational (3D-Var) analysis. The degree by which the “truth” 3D typhoon structure is recovered is an assessment of the impact of the data type or analysis scheme being evaluated. We also examine the effects of weak constraint and strong constraint on STMAS analyses. Results show that while the STMAS is superior to the traditional 3D-Var for reconstructing the 3D typhoon structure, the strong constraint STMAS can produce better analyses on both horizontal and vertical velocities.

An Adaptive Compensatory Approach of the Fixed Localization in the EnKF

Abstract Although the fixed covariance localization in the ensemble Kalman filter (EnKF) can significantly increase the reliability of background error covariance, it has been demonstrated that extreme impact radii can cause the EnKF to lose some useful information. Tuning an optimal impact radius, on the other hand, is always difficult for a general circulation model. The EnKF multiscale analysis (MSA) approach was presented to make up for the above-mentioned drawback of the fixed localization. As a follow-up, this study presents an adaptive compensatory approach to further improve the performance of the EnKF-MSA. The new method adaptively triggers a multigrid analysis (MGA) to extract multiscale information from the observational residual after the EnKF without inflation is completed at each analysis step. Within a biased twin experiment framework consisting of a barotropic spectral model and an idealized observing system, the performance of the adaptive method is examined. Results show that the MGA reduces the computational cost of the MSA by 93%. On the assimilation quality, the adaptive method has an incremental improvement over the EnKF-MSA. That is, the adaptive EnKF-MGA reduces to the EnKF without inflation, which is better than the EnKF-MSA, for moderate impact radii. The proposed scheme works for a broader range of impact radii than the standard EnKF (i.e., the EnKF with inflation). For extreme impact radii, the adaptive EnKF-MGA can produce smaller assimilation errors than the standard EnKF and shorten the spinup period by 53%. In addition, the computational cost of the MGA is negligible relative to that of the standard EnKF.

Multigrid analysis of spatially resolved hepatitis C virus protein simulations

Viruses are a major challenge to human health and prosperity. This holds true for various viruses which are either threatening Europe (like Dengue and Yellow fever) or which are currently causing big health problems like the hepatitis C virus (HCV). HCV causes chronic liver diseases like cirrhosis and cancer and is the main reason for liver transplantations. Exploring biophysical properties of virus-encoded components and viral life cycle is an exciting new area of current virological research. In this context, spatial resolution is an aspect that has not yet been received much attention despite strong biological evidence suggesting that intracellular spatial dependence is a crucial factor in the viral replication process. We are developing first spatio-temporal resolved models which mimic the behavior of the important components of virus replication within single liver cells. HCV replication is strongly associated to the intracellular Endoplasmatic Reticulum (ER) network. Here, we present the computational basis for the estimation of the diffusion constant of a central component of HCV genome (viral RNA) replication, namely the NS5a protein, on the surface of realistic reconstructed ER geometries. The basic surface partial differential equation (sPDE) evaluations are performed with UG4 using fast massively parallel multigrid solvers. The numerics of the simulations are studied in detail. Integrated concentrations within special subdomains correspond to experimental FRAP time series. In particular, we analyze the refinement stability in time and space for these integrated concentrations based on diffusion sPDEs upon large unstructured surface grids using heuristic values for the NS5a diffusion constant. This builds up a solid basis for future research not included in this presentation. e.g. the presented refinement stability analysis of the single sPDEs allows for parameter estimations for the NS5a diffusion constant. Our advanced Finite Volume/multigrid techniques also could be applied for studying life cycles of other viruses.

マグネトロンスパッタ法におけるイオン衝撃の評価

Momentum of ion bombardment in sputtering deposition process, which strongly depends on internal stress of thin film, has been evaluated regarding to a new parameter Pi we previously proposed for fine-tuning of mechanical, optical, electrical and magnetic properties. In this study, we demonstrate to estimate the Pi defined as (i/a) p, where i the ion flux, a the atom flux and p the ion’s momentum, by means of Langmuir probe and multi-grid analyzer in a function of negative bias voltage applied on substrates Vsub. As a result, it was found that the Pi was roughly proportional to Vsub except on lower voltage than 30 V. This indicates that the Pi should be measured plasma-diagnostically under incident ion energy as low as plasma potential.

Multigrid Algorithms for $hp$-Discontinuous Galerkin Discretizations of Elliptic Problems

We present W-cycle $h$-, $p$-, and $hp$-multigrid algorithms for the solution of the linear system of equations arising from a wide class of $hp$-version discontinuous Galerkin discretizations of elliptic problems. Starting from a classical framework in geometric multigrid analysis, we define a smoothing and an approximation property, which are used to prove uniform convergence of the W-cycle scheme with respect to the discretization parameters and the number of levels, provided the number of smoothing steps is chosen of order $p^{2+\mu}$, where $p$ is the polynomial approximation degree and $\mu=0,1$. A discussion on the effects of employing inherited or noninherited sublevel solvers is also presented. Numerical experiments confirm the theoretical results.

Regularity and multigrid analysis for Laplace-type axisymmetric equations

Consider axisymmetric equations associated with Laplace-type operators. We establish full regularity estimates in high-order Kondrat′vetype spaces for possible singular solutions due to the non-smoothness of the domain and to the singular coefficients in the operator. Then, we show suitable graded meshes can be used in high-order finite element methods to achieve the optimal convergence rate, even when the solution is singular. Using these results, we further propose multigrid V-cycle algorithms solving the system from linear finite element discretizations on optimal graded meshes. We prove the multigrid algorithm is a contraction, with the contraction number independent of the mesh level. Numerical tests are provided to verify the theorem.

Theoretical bounds for algebraic multigrid performance: review and analysis

SUMMARYAlgebraic multigrid methods continue to grow in robustness as effective solvers for the large and sparse linear systems of equations that arise in many applications. Unlike geometric multigrid approaches, however, the theoretical analysis of algebraic multigrid is less predictive of true performance. Multigrid convergence factors naturally depend on the properties of the relaxation, interpolation, and coarse‐grid correction routines used, yet without the tools of Fourier analysis, optimal and practical bounds for algebraic multigrid are not easily quantified. In this paper, we survey bounds from existing literature, with particular focus on the predictive capabilities of the theory, and provide new results relating existing bounds. We highlight the impact of these theoretical observations through several model problems and discuss the role of theoretical bounds on practical performance. Copyright © 2014 John Wiley & Sons, Ltd.

Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids

In this paper we generalize the analysis of classical multigrid and two-level overlapping Schwarz methods for 2nd order elliptic boundary value problems to problems with large discontinuities in the coefficients that are not resolved by the coarse grids or the subdomain partition. The theoretical results provide a recipe for designing hierarchies of standard piecewise linear coarse spaces such that the multigrid convergence rate and the condition number of the Schwarz preconditioned system do not depend on the coefficient variation or on any mesh parameters. An assumption we have to make is that the coarse grids are sufficiently fine in the vicinity of cross points or where regions with large diffusion coefficients are separated by a narrow region where the coefficient is small. We do not need to align them with possible discontinuities in the coefficients. The proofs make use of novel stable splittings based on weighted quasi-interpolants and weighted Poincare-type inequalities. Numerical experiments are...

Multigrid analysis for the time dependent Stokes problem

Certain implicit time stepping procedures for the incompressible Stokes or Navier-Stokes equations lead to a singular-perturbed Stokes type problem at each time step. The paper presents a convergence analysis of a geometric multigrid solver for the system of linear algebraic equations resulting from the disretization of the problem using a finite element method. Several smoothing iterative methods are considered: a smoother based on distributive iterations, the Braess-Sarazin and inexact Uzawa smoother. Convergence analysis is based on smoothing and approximation properties in special norms. A robust (independent of time step and mesh parameter) estimate is proved for the two-grid and multigrid W-cycle convergence factors.

Algebraic analysis of aggregation-based multigrid

A convergence analysis of two-grid methods based on coarsening by (unsmoothed) aggregation is presented. For diagonally dominant symmetric (M-)matrices, it is shown that the analysis can be conducted locally; that is, the convergence factor can be bounded above by computing separately for each aggregate a parameter, which in some sense measures its quality. The procedure is purely algebraic and can be used to control a posteriori the quality of automatic coarsening algorithms. Assuming the aggregation pattern is sufficiently regular, it is further shown that the resulting bound is asymptotically sharp for a large class of elliptic boundary value problems, including problems with variable and discontinuous coefficients. In particular, the analysis of typical examples shows that the convergence rate is insensitive to discontinuities under some reasonable assumptions on the aggregation scheme. Copyright © 2010 John Wiley & Sons, Ltd.