GMRES Research Articles

Preconditioned SAV-leapfrog finite difference methods for spatial fractional Cahn–Hilliard equations

Combining the scale auxiliary variable (SAV) approach with leapfrog finite difference methods, an unconditional energy-stable, non-couple and linearly implicit numerical scheme is presented for solving spatial fractional Cahn–Hilliard equations. The fully-discrete scheme gives an indefinite and ill-conditioned system of linear algebraic equations, whose coefficient matrix is a 2 × 2 block with block Toeplitz matrix. The preconditioned minimum residual method with a positive preconditioner is employed for the resulting system. Numerical results are presented to confirm the effectiveness of the proposed numerical scheme and preconditioner.

Neural control of discrete weak formulations: Galerkin, least squares & minimal-residual methods with quasi-optimal weights

There is tremendous potential in using neural networks to optimize numerical methods. In this paper, we introduce and analyze a framework for the neural optimization of discrete weak formulations, suitable for finite element methods. The main idea of the framework is to include a neural-network function acting as a control variable in the weak form. Finding the neural control that (quasi-) minimizes a suitable cost (or loss) functional, then yields a numerical approximation with desirable attributes. In particular, the framework allows in a natural way the incorporation of known data of the exact solution, or the incorporation of stabilization mechanisms (e.g., to remove spurious oscillations).The main result of our analysis pertains to the well-posedness and convergence of the associated constrained-optimization problem. In particular, we prove under certain conditions, that the discrete weak forms are stable, and that quasi-minimizing neural controls exist, which converge quasi-optimally. We specialize the analysis results to Galerkin, least squares and minimal-residual formulations, where the neural-network dependence appears in the form of suitable weights. Elementary numerical experiments support our findings and demonstrate the potential of the framework.

MINRES: From Negative Curvature Detection to Monotonicity Properties

The conjugate gradient method (CG) has long been the workhorse for inner-iterations of second-order algorithms for large-scale nonconvex optimization. Prominent examples include line-search based algorithms, e.g., Newton-CG, and those based on a trust-region framework, e.g., CG-Steihaug. This is mainly thanks to CG's several favorable properties, including certain monotonicity properties and its inherent ability to detect negative curvature directions, which can arise in nonconvex optimization. This is despite the fact that the iterative method-of-choice when it comes to real symmetric but potentially indefinite matrices is arguably the celebrated minimal residual (MINRES) method. However, limited understanding of similar properties implied by MINRES in such settings has restricted its applicability within nonconvex optimization algorithms. We establish several such nontrivial properties of MINRES, including certain useful monotonicity as well as an inherent ability to detect negative curvature directions. These properties allow MINRES to be considered as a potentially superior alternative to CG for all Newton-type nonconvex optimization algorithms that employ CG as their subproblem solver.

An Improved GMRES Method for Solving Electromagnetic Scattering Problems by MoM

Method of moments (MoM) is a classical numerical method for analyzing electromagnetic (EM) scattering from various objects. Generalized minimal residual (GMRES) is a kind of widely used solver for linear systems of equations generated by MoM, whose calculation complexity mainly includes two parts, matrix-vector multiplications and the orthogonalization procedure. In order to accelerate the solution of matrix equations arising in MoM, an efficient strategy for constructing nonorthonormal Krylov bases is proposed in this article, which can be introduced into standard GMRES and its variants (e.g., GMRES( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> ) and quasi-GMRES) and can be easily combined with the fast approaches for MOM (e.g., FMM and ACA). The principle of the method is described in detail, and the effectiveness is verified by numerical results.

Fast short-circuit current calculation of unbalanced distribution networks with inverter-interfaced distributed generators

This study proposes an approach for the fast calculation of short-circuit currents in unbalanced distribution networks with inverter-interfaced distributed generators (IIDG). It is capable of estimating the short-circuit current in unbalanced distribution networks with several IIDGs based on the symmetric component without calculating the system bus impedance matrix. Furthermore, the effect of the load on the short-circuit current was considered. Because the IIDG fault characteristic is determined by its fault ride-through control models, the short-circuit current calculation process is formulated as an iterative linear system calculation and can be solved using the simple generalized minimal residual (SGMRES) method. Finally, the analytical results were verified by comparisons with the time-domain simulation results of several test systems in PSCAD.

A New Type of Variable Preconditioning for a Generalized Minimum Residual Scheme

Krylov linear equation solvers have been widely used on distributed-memory-type supercomputers. Krylov-type linear equation solvers are frequently applied using preconditioning techniques to achieve fast and robust computations. Flexible Krylov solvers that accept variable preconditioning have recently been developed, whereas conventional ones are invariable throughout the Krylov solver iterations. The generalized minimal residual recursive method (GMRESR) is outstandingly unique because it is combined with a preconditioning update scheme based on Eirola and Nevanlinna’s rank-one update scheme. Moreover, the Broyden–Fletcher–Goldfarb–Shanno scheme (BFGS) provides another means to update preconditioning matrices. In this study, we applied BFGS to GMRESR as an alternative means of updating the preconditioning matrices and then measured the performance. As a result, the newly developed method exhibited an approximately 10-fold better convergence than Eirola and Nevanlinna’s approach.

An efficient method based on least-squares technique for interface problems

In this paper, we propose and analyze a new method based on the least-squares technique in a piecewise kth order (k=3,4,5) polynomial function space for one-dimensional interface problems. By defining a residual functional, we derive a new bilinear form and the minimum residual method can be established in an alternative way. The stability of the proposed method is proven. For error estimation, we discuss the optimal convergence orders under the ‖⋅‖a-norm for general non-uniform meshes. The superconvergence behaviors of the presented method have been also discovered: the convergence rate of the element averages under the L2 and ‖⋅‖a-norms can achieve 2k−2. Our theoretical findings are verified by several numerical experiments.

Structure preserving subspace methods for the general coupled discrete-time periodic matrix equation and its application in antilinear periodic system

In this paper, we establish some efficient iteration methods for solving the general coupled discrete-time periodic matrix equation. More concretely, we propose some structure preserving matrix versions of subspace methods, such as bi-conjugate gradient, bi-conjugate residual, conjugate gradient squared, bi-conjugate gradient stabilized, generalized minimal residual and restarted generalized minimal residual methods. We demonstrate experimentally that the proposed iteration methods are feasible and efficient for the periodic matrix equation.

Recycling Newton–Krylov algorithm for efficient solution of large scale power systems

Power flow calculations are crucial for the study of power systems, as they can be used to calculate bus voltage magnitudes and phase angles, as well as active and reactive power flows on lines. In this paper, a new approach, the Recycling Newton–Krylov (ReNK) algorithm, is proposed to solve the linear systems of equations in Newton–Raphson iterations. The proposed method uses the Generalized Conjugate Residuals with inner orthogonalization and deflated restarting (GCRO-DR) method within the Newton–Raphson algorithm and reuses the Krylov subspace information generated in previous Newton runs. We evaluate the performance of the proposed method over the traditional direct solver (LU) and iterative solvers (Generalized Minimal Residual Method (GMRES), the Biconjugate Gradient Stabilized Method (Bi-CGSTAB) and Quasi-Minimal Residual Method (QMR)) as the inner linear solver of the Newton–Raphson method. We use different test systems with a number of busses ranging from 300 to 70000 and compare the number of iterations of the inner linear solver (for iterative solvers) and the CPU times (for both direct and iterative solvers). We also test the performance of the ReNK algorithm for contingency analysis and for different load conditions to simulate optimization problems and observe possible performance gains.

A Complete Assessment of the Moment‐Curvature Relationship of Compressed Steel‐concrete Composite Cross‐sections

AbstractIn the non‐linear evaluation of the behavior of steel, concrete and steel‐concrete composite cross‐sections, increment and iteration strategies are necessary for the complete construction of the moment‐curvature relationship (M‐Φ). For the correct simulation of sections composed of strain‐softening materials, a numerical strategy based on constant force increments does not capture the downward branch of the M‐Φ relationship. Therefore, the present study aims to couple the strain compatibility method to the path‐following strategies to go beyond the critical bending moment points in the construction of the relations that describe the complete cross‐section mechanical behavior. In this context, the generalized displacement technique, which depends on a generalized stiffness parameter, and the minimum residual displacement method, are adapted to the cross‐sectional non‐linear problem. To validate the proposed numerical formulation, the results obtained here are compared with the numerical data available in the literature. Additionally, the softening effect on the concrete was increased to induce descending stretches in the moment‐curvature relationship, and this condition was correctly evaluated.

Dynamic modeling and simulation of rigid-flexible coupling cable system by absolute nodal coordinate formulation

The rigid-flexible coupling cable system under large deformation is studied, and the beam element from absolute node coordinate formulation is used to establish flexible cable body of the system. Different numerical integral algorithms are discussed for solving the rigid-flexible cable system and an integration strategy which combines Implicit Euler with Minimum Residual Method (MINRES) is proposed. The influence of the position and number of rigid components and different the lengths of the flexible elements on the system dynamics are analyzed. With constant total mass of the system, higher number of rigid components and their uniform distribution contribute to stabilization of the swing of the flexible cable body. When the total length of the cable is constant, increasing the number of beam elements enhances the nonlinear characteristics of the swing motion and damages the stability. The influence of different factors on the movement of large deformation flexible cable body is obtained through modeling and simulation of the rigid-flexible coupling cable system.

An unconditionally convergent RSCSCS iteration method for Riesz space fractional diffusion equations with variable coefficients

In this paper, a respectively scaled circulant and skew-circulant splitting (RSCSCS) iteration method is employed to solve the Toeplitz-like linear systems arising from time-dependent Riesz space fractional diffusion equations with variable coefficients. The RSCSCS iteration method is shown to be convergent unconditionally by a novel technique, and only requires computational costs of O(NlogN) with N denoting the number of interior mesh points in space. In theory, we obtain an upper bound for the convergence factor of the RSCSCS iteration method and discuss the optimal value of its iteration parameter that minimizes the corresponding upper bound. Meanwhile, we also design a fast induced RSCSCS preconditioner to accelerate the convergence rate of the Krylov subspace iteration method likes generalized minimal residual (GMRES) method. Numerical results are presented to show that the efficiencies of our proposed RSCSCS iteration method and the preconditioned GMRES method with the RSCSCS preconditioner.

Numerical algorithms for water waves with background flow over obstacles and topography

We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

EM-WaveHoltz: A Flexible Frequency-Domain Method Built From Time-Domain Solvers

A novel approach to computing time-harmonic solutions of Maxwell’s equations by time-domain simulations is presented. The method, electromagnetic-WaveHoltz (EM-WaveHoltz), results in a positive definite system of equations, which makes it amenable to iterative solution with the conjugate gradient method or with generalized minimal residual method (GMRES). Theoretical results guaranteeing the convergence of the method away from resonances are presented. Numerical examples illustrating the properties of EM-WaveHoltz are given.

A direct RBF-PU method for simulating the infiltration of cytotoxic T-lymphocytes into the tumor microenvironment

Predicting the antitumor responses of cytotoxic T-lymphocytes (i.e., CD8+T cells) can be accomplished by numerical simulations of a mathematical model of tumor-immune cell interactions to control tumor growth in cancer therapy. Cytotoxic T lymphocytes are considering as the main effector cells to target tumor cells that are in a dormant avascular state through the chemotactic mechanism and also during the cytotoxic activity of immune cells. We study a dynamical model that is formulated based on the nonlinear partial differential equations describing the interactions between tumor-infiltrating cytotoxic lymphocytes, tumor cells, TICL-tumor cell complexes, inactivated tumor-infiltrating cytotoxic lymphocytes, dead tumor cells, and chemokine. Here, a new meshless method, i.e., the direct radial basis function partition of unity (RBF-PU) technique for the spatial discretization and a semi-implicit backward differentiation formula of order 1 for the temporal discretization are employed. The proposed meshless technique is a generalized form of the original RBF-PU method such that the computational cost of computing the derivatives could be reduced in comparison with its standard form. Besides, the method depends only on the location of nodes located on the domain of the problem for the approximation, where both types of radial functions, i.e., a conditionally positive definite function, i.e., the polyharmonic splines and a positive definite function, namely Matérn function have been used The obtained fully discrete difference scheme based on the new spatial approximation is solved using an iterative method, i.e., the generalized minimal residual method with a zero-fill incomplete lower–upper preconditioner. Finally, we present some simulation results, in which we conclude that the tumor’s growth entirely depends on CTLs activation, and this behavior can be predicted by simulating the proposed mathematical model, which may have a significant role in improving cancer immunotherapy. Besides, the developed numerical method is more accurate than the classical finite element method for solving the studied model as shown in one example.

A Nonlinear Safety Equilibrium Spacing-Based Model Predictive Control for Virtually Coupled Train Set Over Gradient Terrains

The increasing demand for capacity in railway transportation has spawned the concept of virtual coupling (VC), which can further shorten the intertrain distance by applying communication technology. In this article, a centralized model predictive control (MPC) is developed for virtually coupled train set (VCTS) with a nonlinear safety equilibrium spacing policy over gradient rolling terrains. The tracking target of the controller in cruise phase is given based on the braking distance difference and speed difference of trains. A more precise braking distance model of trains explicitly considering the gradient is introduced for safety, based on which the state space applicable to the controller for VCTS is derived. To deal with the nonlinearities of the controller, a new optimization solving algorithm based on the continuation method and generalized minimum residual (GMRES) method is presented in this article. Finally, controller performances and empirical string stability of VCTS are verified by numerical simulations. It is proven that the proposed controller has a good effect for the safety and smoothness of the train operation on the railway line over gradient terrain through the test. The actual intertrain distance under the proposed controller is about 1 m closer to the equilibrium state than that under the same controller without considering the gradient at low speed (about 10 m/s) on the 0.3% downhill line, which verifies the safety of the controller. Besides, the influence of the parameters of the controller and optimization method on controller efficiency is analyzed, and for the latter, the simulation data show that the applied algorithm for the optimal control solver can save the calculation time more than seven times than the traditional algorithm.

MINRES for Second-Order PDEs with Singular Data

Minimum residual methods such as the least-squares finite element method (FEM) or the discontinuous Petrov--Galerkin method with optimal test functions (DPG) usually exclude singular data, e.g., non square-integrable loads. We consider a DPG method and a least-squares FEM for the Poisson problem. For both methods we analyze regularization approaches that allow the use of $H^{-1}$ loads, and also study the case of point loads. For all cases we prove appropriate convergence orders. We present various numerical experiments that confirm our theoretical results. Our approach extends to general well-posed second-order problems.

Accurate modelling of the optics of high resolution liquid crystal devices: a reconfigurable liquid crystal grating

ABSTRACT An accurate method to model the optical behaviour of liquid crystal (LC) devices, particularly suited to devices where diffractive effects are present is described here. An accurate electromagnetic modelling programme that takes into account the full non-uniformity and anisotropy of the LC has been developed. This is combined with an existing in-house LC finite element modelling programme based on the Landau – De Gennes theory, that uses the order tensor representation of the LC orientation and allows an accurate descriptions of structures containing LC defects and small features. The electromagnetic model is based on the total field/scattered field (TF-SF) approach to electromagnetic scattering problems and is implemented using finite differences in the frequency domain (FDFD) in a form that can accommodate perfectly matched layers (PMLs) and periodic boundary conditions. The resultant matrix problem is solved efficiently using an especially adapted form of a sweeping preconditioner and the generalised minimum residual method (GMRes). This method has been implemented in 2D and is demonstrated here with the design and analysis of a reconfigurable blazed phase grating that utilises an LC defect to produce an abrupt fly-back, with the capability of short periods and high diffraction efficiency.

Thermoelectric model to study the cardiac action potential and arrhythmias

This paper proposes a new thermoelectric model to examine the behavior of the heart in cooling situations. A modified Karma model with temperature-dependence is exploited to describe the ion exchange dynamics at the mesoscopic scale while the propagation of the action potential is governed by a mono-domain equation at the macroscopic scale. In addition to perfusion and heat metabolism, we call the Penne equation coupled to the mono-domain equation by using the Joule effect to depict the temperature behavior in the system. Galerkin’s finite element method is utilized to start solving the partial differential equations governing the action potential and temperature propagations. The incomplete lower–upper decomposition and generalized minimal residual methods are solicited to solve the resulting nonlinear system. The cases of zero temperature and potential gradients are integrated through the scheme of Runge–Kutta, and the results obtained corroborate well with those of the literature. We analyze the contributions of the nonlinear coupling tensor and arterial temperature on the thermal and electrical responses of the system. The established results reveal that when the temperature in the medium augments, the duration of the action potential decreases and the Joule coupling tensor plays a crucial role in the propagation of the potential. Moreover, we show that temperature and action potential are in phase and that propagation of this potential generates thermal energy. Furthermore, we establish the existence of spiral waves in heart cells and show that the effect of global cooling helps to modulate or dampen these spiral waves, leading to control of the cardiac arrhythmia. This work also develops a technique to resolve conduction disorders and cancel them completely. It exhibits an increased added value to the use of hypothermia as therapy during cardiac arrest and makes it possible to anticipate and perhaps avoid this pathology.

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.