Fast Multigrid Algorithm Research Articles

Fast Multigrid Algorithm for Non-Linear Simulation of Intact and Damaged Ship Motions in Waves

This paper proposes a fast multigrid algorithm to simulate the non-linear motion of ships in both intact and damaged conditions. The simulations of ship motions in waves are known to require much time to calculate due to the strong non-linear interactions between ship and waves. To improve the calculation efficiency while retaining the accuracy, a prediction-correction strategy was designed to accelerate the simulation through three sets of locally refined meshes. The flow field was first estimated in a coarse mesh and then mapped to a locally refine mesh for further higher-fidelity corrections. A partitioned radial basis function (PRBF) method is proposed to interpolate and reconstruct the flow field for the refined mesh. A new two-phase flow solver was developed with a fast multigrid algorithm based on the Reynolds-averaged Navier–Stokes equations (RANSE). The new solver was applied to study the non-linear behavior of a damaged ship in beam waves and the effect of damaged compartments on ship rolling motion. Validation against the solution with the original method of single set meshes and experimental data indicates that the proposed algorithm yields satisfactory results while saving 30–40% of the computational time.

A fast multi grid algorithm for 2D diffeomorphic image registration model

In Han and Wang (2020), a 2D diffeomorphic image registration model is proposed to eliminate mesh folding. To solve the 2D diffeomorphic model, a diffeomorphic fractional-order image registration algorithm (DFIRA for short) is proposed in Han and Wang (2020). DFIRA achieves a satisfactory image registration result but it costs too much CPU time. To accelerate DFIRA, we propose a fast multi grid algorithm for 2D diffeomorphic image registration model in this paper. This algorithm achieves a satisfactory image registration result by using only one-tenth CPU time of DFIRA. At the same time, no mesh folding occurs in proposed algorithm. Furthermore, convergence analysis of the proposed algorithm is also presented. Moreover, numerical tests are also performed to show that the proposed algorithm is competitive compared with some other algorithms.

An Effective Diffeomorphic Model and Its Fast Multigrid Algorithm for Registration of Lung CT Images

Abstract Image registration is the process of aligning sets of similar, but different, intensity image functions to track changes between the images. In medical image problems involving lung images, variational registration models are a very powerful tool which can aid in effective treatment of various lung conditions and diseases. However, a common drawback of many variational models, such as the diffusion model and even optic flow models, is the lack of control of folding in the deformations leading to physically inaccurate transformations. For this reason, such models are generally not suitable for real life lung imaging problems where folding cannot occur. There are two approaches offering reliable solutions (though not necessarily accurate). The first approach is a parametric model such as the affine registration model, still widely used in many applications, but is unable to track local changes or yield accurate results. The second approach is to impose an extra constraint on the transformation of registration at the cost of increased non-linearity. An alternative to the second approach, achieving diffeomorphic transforms without adding any constraints, is an inverse consistent model such as by Christensen and Johnson (2001) from computing explicitly both the forward and inverse transforms. However, one must deal with the strong non-linearity in the formulation. In this paper we first propose a simplified inverse consistent model to avoid the inclusion of strong non-linearities and then a fast non-linear multigrid (NMG) technique to overcome the extra computational work required by the inverse consistent model. Experiments, performed on real medical CT images, show that our proposed inverse consistent model is robust to both parameter choice and non-folding in the transformations when compared with diffusion type models.

3DRISM Multigrid Algorithm for Fast Solvation Free Energy Calculations.

In this paper we present a fast and accurate method for modeling solvation properties of organic molecules in water with a main focus on predicting solvation (hydration) free energies of small organic compounds. The method is based on a combination of (i) a molecular theory, three-dimensional reference interaction sites model (3DRISM); (ii) a fast multigrid algorithm for solving the high-dimensional 3DRISM integral equations; and (iii) a recently introduced universal correction (UC) for the 3DRISM solvation free energies by properly scaled molecular partial volume (3DRISM-UC, Palmer et al., J. Phys.: Condens. Matter2010, 22, 492101). A fast multigrid algorithm is the core of the method because it helps to reduce the high computational costs associated with solving the 3DRISM equations. To facilitate future applications of the method, we performed benchmarking of the algorithm on a set of several model solutes in order to find optimal grid parameters and to test the performance and accuracy of the algorithm. We have shown that the proposed new multigrid algorithm is on average 24 times faster than the simple Picard method and at least 3.5 times faster than the MDIIS method which is currently actively used by the 3DRISM community (e.g., the MDIIS method has been recently implemented in a new 3DRISM implicit solvent routine in the recent release of the AmberTools 1.4 molecular modeling package (Luchko et al. J. Chem. Theory Comput. 2010, 6, 607-624). Then we have benchmarked the multigrid algorithm with chosen optimal parameters on a set of 99 organic compounds. We show that average computational time required for one 3DRISM calculation is 3.5 min per a small organic molecule (10-20 atoms) on a standard personal computer. We also benchmarked predicted solvation free energy values for all of the compounds in the set against the corresponding experimental data. We show that by using the proposed multigrid algorithm and the 3DRISM-UC model, it is possible to obtain good correlation between calculated and experimental results for solvation free energies of aqueous solutions of small organic compounds (correlation coefficient 0.97, root-mean-square deviation <1 kcal/mol).

Fast Iterative Algorithms for Total Variation Based Multiplicative Noise Removal Model

This paper presents fast iterative algorithms for solution of PDEs arisen from minimization of multiplicative noise removal model [14]. This model may be regarded as an improved version of the Total Variation (TV) de-noising models. For the TV and the multiplicative noise removal models, their associated Euler-Lagrange equations are highly nonlinear Partial Differential Equations (PDEs). For this model a very slow explicit time marching method has been reported. The main contribution we present in this paper is the implementation of the fixed point, semi-implicit and additive operator splitting schemes which do not yield good results. Consequently a fast and efficient multi-grid method with AOS as smoother is developed. Numerical experiments are presented to show the good performance of the fast multi-grid algorithm.

A Fourth-Order Variational Image Registration Model and Its Fast Multigrid Algorithm

Several partial differential equations (PDEs) based variational methods can be used for deformable image registration, mainly differing in how regularization for deformation fields is imposed [J. Modersitzki, Numerical Methods for Image Restoration, Oxford University Press, Oxford, 2004]. On one hand for smooth problems, models of elastic-, diffusion-, and fluid-image registration are known to generate globally smooth and satisfactory deformation fields. On the other hand for nonsmooth problems, models based on the total variation (TV) regularization are better for preserving discontinuities of the deformation fields. It is a challenge to design a deformation model suitable for both smooth and nonsmooth deformation problems. One promising model that is based on a curvature type regularizer and appears to deliver excellent results for both problems is proposed and studied in this paper. A related work due to B. Fischer and J. Modersitzki [J. Math. Imaging Vision, 18 (2003), pp. 81–85] and then refined by S. Henn and K. Witsch [Multiscale Model. Simul., 4 (2005), pp. 584–609] used an approximation of the mean curvature and obtained improved results over previous models. However, this paper investigates the full curvature model and finds that the new model is more robust than approximated curvature models and leads to further improvement. Associated with the new model is the apparent difficulty in developing a fast algorithm as the system of two coupled PDEs is highly nonlinear and of fourth order so standard application of multigrid methods does not work. In this paper, we first propose several fixed-point type smoothers. Then we use both local Fourier analysis and experiments to select the most effective smoother which turns out to be a primal-dual based method. Finally we use the recommended smoother to propose a nonlinear multigrid algorithm for the new model. Numerical tests using both synthetic and realistic images not only confirm that the proposed curvature model is more robust in registration quality for a wide range of applications than previous work [J. Modersitzki, Numerical Methods for Image Restoration, Oxford University Press, Oxford, 2004] and [S. Henn and K. Witsch, Multiscale Model. Simul., 4 (2005), pp. 584–609], but also that the proposed algorithm is fast and accurate in delivering visually pleasing registration results.

On High-Order Denoising Models and Fast Algorithms for Vector-Valued Images

Variational techniques for gray-scale image denoising have been deeply investigated for many years; however, little research has been done for the vector-valued denoising case and the very few existent works are all based on total-variation regularization. It is known that total-variation models for denoising gray-scaled images suffer from staircasing effect and there is no reason to suggest this effect is not transported into the vector-valued models. High-order models, on the contrary, do not present staircasing. In this paper, we introduce three high-order and curvature-based denoising models for vector-valued images. Their properties are analyzed and a fast multigrid algorithm for the numerical solution is provided. AMS subject classifications: 68U10, 65F10, 65K10.

A Fast Multigrid Algorithm for Energy Minimization under Planar Density Constraints

The two-dimensional layout optimization problem reinforced by the efficient space utilization demand has a wide spectrum of practical applications. Formulating the problem as a nonlinear minimization problem under planar equality and/or inequality density constraints, we present a linear time multigrid algorithm for solving a correction to this problem. The method is demonstrated in various graph drawing (visualization) instances.

Dynamic models for substrate coupling in mixed-mode systems

In modern monolithic integrated circuits, substrate coupling is a major concern in mixed-mode systems design. Verification of such systems implies the availability of accurate and simulation-efficient substrate coupling models. Traditionally, for frequencies up to a few gigahertz, pure resistive models have been considered sufficient. However, with increasing frequencies of operation, dynamic models become mandatory. The authors motivate the use of dynamic resistive-capacitive (RC) models of substrate coupling as a natural extension to the standard purely resistive models. They propose an extraction methodology that starts with information about the process parameters and circuit&apos;s contact layout, and leads to a contact-to-contact RC element model. The underlying algorithm is based on a finite difference discretisation of the substrate, leading to a large tridimensional mesh which is reduced by means of a fast multigrid algorithm. Unlike standard model order reduction algorithms which can produce models of similar accuracy to state-space descriptions, the proposed method leads to a realisable RC model that can trivially be incorporated into circuit simulation tools. As a first approximation, such a model is shown to correspond to a single time-constant system. Furthermore, it is shown that this time constant can be computed from knowledge of the conductivity and permittivity of a single dominant layer. It is verified that this formulation can accurately model substrate coupling effects for frequencies up to several tens of gigahertz.

Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution

We propose a novel method for image reconstruction from nonuniform samples with no constraints on their locations. We adopt a variational approach where the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms: (1) the sum of squared errors at the specified points and (2) a quadratic functional that penalizes the lack of smoothness. We search for a solution that is a uniform spline and show how it can be determined by solving a large, sparse system of linear equations. We interpret the solution of our approach as an approximation of the analytical solution that involves radial basis functions and demonstrate the computational advantages of our approach. Using the two-scale relation for B-splines, we derive an algebraic relation that links together the linear systems of equations specifying reconstructions at different levels of resolution. We use this relation to develop a fast multigrid algorithm. We demonstrate the effectiveness of our approach on some image reconstruction examples.

Fast multigrid algorithm for quasisingular classes of diffusion problems

In this paper it is shown that the fast multigrid algorithm (FMA) first proposed by the author in [A. D. Javadian, Fast multigrid algorithm for quasisingular classes of elliptic problems. Russ. J. Numer. Anal. Math. Modelling (2001) 16, No. 5, 385–408.] is applicable to the solution of a sufficiently broad family of quasisingulsar classes of diffusion problems. The optimal accuracy in the sense of the Kolmogorov N-width is attained using O (| ln ε | 3N) operations, where ε is the minimum value of the coefficient in the problem.

Fast multigrid algorithm for quasisingular classes of diffusion problems

In this paper it is shown that the fast multigrid algorithm (FMA) first proposed by the author in [A. D. Javadian, Fast multigrid algorithm for quasisingular classes of elliptic problems. Russ. J. Numer. Anal. Math. Modelling (2001) 16, No. 5, 385–408.] is applicable to the solution of a sufficiently broad family of quasisingulsar classes of diffusion problems. The optimal accuracy in the sense of the Kolmogorov N-width is attained using O (| ln ε | 3N) operations, where ε is the minimum value of the coefficient in the problem.

Fast multigrid algorithm for quasisingular classes of elliptic problems

For certain quasisingular classes of elliptic problems (QCEP) we have obtained condensed finite element grids to get an optimal approximate solution in the sense of the Kolmogorov N-width. But the computational cost of any method of solving the system of finite element equations is high.

A Fast Multigrid Algorithm for Isotropic Transport Problems I: Pure Scattering

The authors present a multigrid method for solving the one-dimensional (1-D) slab-geometry $S_N $ equations with isotropic scattering and no absorption. This scheme is highly compatible with massively parallel computer architectures and represents a first step toward similar multigrid methods for the $S_N $ equations in curvilinear and multidimensional geometries. Extensive theoretical analyses are given for our scheme which indicate that it is extremely efficient. In fact, the method is so efficient that it very nearly represents an exact solution technique. Results from calculations are presented which validate the theoretical results. The case with absorption is treated in a sequel to this paper [Manteuffel et al., A fast multigrid algorithm for transport problems II: With absorption, SIAM J. Sci. Comput., submitted].