Schur Complement Research Articles

Novel order-dependent passivity conditions of fractional generalized Cohen–Grossberg neural networks with proportional delays

This work delves into the passivity of fractional generalized Cohen–Grossberg neural networks (CGNNs) with proportional delays. In particular, two forms of Lipschitz conditions for activation function are considered. Some passivity criteria are given by utilizing the Schur Complement lemma, Lyapunov–Krasovskii functional approach and Cauchy matrix inequality. The presented conditions are derived as matrix inequality forms and related to the proportional factor and order. This can clearly reflect the influence of the order and proportional factor on the input–output energy of the system. Two examples are particularized to better substantiate the obtained passivity results.

Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a large-scale and ill-conditioned linear system must be solved, that, in case of Maxwell equations, is particularly challenging, because of the presence of both H(div) and H(curl) discretization spaces. We propose here a parallel preconditioner, that exploits the Schur complement block factorization of the linear system matrix and consists of a Jacobi preconditioner for the H(div) block and an auxiliary space preconditioner for the H(curl) block. Several parallel numerical tests have been performed to study the robustness of the solver with respect to mesh refinement, shape of polyhedral elements, time step size and the VEM stabilization parameter.

Numerical stability and efficiency of response property calculations in density functional theory

Response calculations in density functional theory aim at computing the change in ground-state density induced by an external perturbation. At finite temperature, these are usually performed by computing variations of orbitals, which involve the iterative solution of potentially badly conditioned linear systems, the Sternheimer equations. Since many sets of variations of orbitals yield the same variation of density matrix, this involves a choice of gauge. Taking a numerical analysis point of view, we present the various gauge choices proposed in the literature in a common framework and study their stability. Beyond existing methods, we propose a new approach, based on a Schur complement using extra orbitals from the self-consistent field calculations, to improve the stability and efficiency of the iterative solution of Sternheimer equations. We show the success of this strategy on nontrivial examples of practical interest, such as Heusler transition metal alloy compounds, where savings of around 40% in the number of required cost-determining Hamiltonian applications have been achieved.

On Symmetric Gauss–Seidel ADMM Algorithm for H∞ Guaranteed Cost Control With Convex Parameterization

This article involves the innovative development of a symmetric Gauss–Seidel ADMM algorithm to solve the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{\infty }$ </tex-math></inline-formula> guaranteed cost control problem. In the presence of parametric uncertainties, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{\infty }$ </tex-math></inline-formula> guaranteed cost control problem generally leads to the large-scale optimization. This is due to the exponential growth of the number of the extreme systems involved with respect to the number of parametric uncertainties. In this work, through a variant of the Youla–Kucera parameterization, the stabilizing controllers are parameterized in a convex set; yielding the outcome that the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{\infty }$ </tex-math></inline-formula> guaranteed cost control problem is converted to a convex optimization problem. Based on an appropriate reformulation using the Schur complement, it then renders possible the use of the ADMM algorithm with symmetric Gauss–Seidel backward and forward sweeps. Significantly, this approach alleviates the often-times prohibitively heavy computational burden typical in many <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal H_{\infty }$ </tex-math></inline-formula> optimization problems while exhibiting good convergence guarantees, which is particularly essential for the related large-scale optimization procedures involved. With this approach, the desired robust stability is ensured, and the disturbance attenuation is maintained at the minimum level in the presence of parametric uncertainties. Rather importantly too, with the attained effectiveness, the methodology thus evidently possesses extensive applicability in various important controller synthesis problems, such as decentralized control, sparse control, and output feedback control problems.

A smooth control allocation method for a distributed electric propulsion VTOL aircraft test platform

AbstractDistributed electronic propulsion (DEP) aircraft with vertical take‐off and landing (VTOL) ability mostly equipped with multiple redundant tiltable propulsors, which are used to provide not only thrusts, but also control torque in all flight mode. The features make the control allocation problem of these aircraft both complex and important. Therefore, this paper proposes: (і) a prototype platform for DEP VTOL aircraft that allows researchers to design and analyse control allocation strategies, and (іі) a uniform hierarchical control allocation method that can be adapted to both MC and FW modes. The platform contains ten fully actuated thrusters, which are distributed in the scaled geometry of a typical DEP‐VTOL aircraft. The proposed control allocation strategy uses hierarchical structure to avoid high‐dimensional matrix operations for the redundant actuators. The additional constraints on the thrust and tilt angles of the actuators are addressed by adopting cascade generalized inversion (CGI) to force decompensation (FD)‐based methods by a coordinate transformation. A novel smooth strategy through Schur Complement is also presented to achieve steady and smooth transition from MC to FW mode. Finally, the proposed method is implemented in the authors’ novel platform in both simulations and flight experiments to demonstrate its effectiveness and smoothness.

An SQP-based multiple shooting algorithm for large-scale PDE-constrained optimal control problems

Multiple shooting methods for solving optimal control problems have been developed rapidly in the past decades and are widely considered a promising direction to speed up the optimization process. Here we propose and analyze a new multiple shooting algorithm based on a sequential quadratic programming (SQP) method that is suitable for optimal control problems governed by large-scale time-dependent partial-differential equations (PDEs). We investigate the structure of the KKT matrix and solve the large-scale KKT system by a preconditioned conjugate gradient algorithm. A simplified block Schur complement preconditioner is proposed, that allows for the parallelization of the method in the time domain. The proposed algorithm is first validated for an optimal control problem constrained by the Nagumo equation. The results indicate that considerable accelerations can be achieved for multiple shooting approaches with appropriate starting guesses and scaling of the matching conditions. We further apply the proposed algorithm to a two-dimensional velocity tracking problem governed by the Navier–Stokes equations. We find algorithmic speed-ups of up to 12 versus single shooting on up to 50 shooting windows. We also compare results with earlier work that uses an augmented Lagrangian algorithm instead of SQP, showing better performance of the SQP method for most of the cases.

Smoothing Analysis of Two Robust Multigrid Methods for Elliptic Optimal Control Problems

In this paper we study and compare two multigrid relaxation schemes with coarsening by two, three, and four for solving elliptic sparse optimal control problems with control constraints and combined and cost functional. First, we perform a detailed local Fourier analysis (LFA) of a well-known collective Jacobi relaxation (CJR) scheme for the unconstrained case with only cost functional, where the optimal smoothing factors are derived. This insightful analysis reveals that the optimal relaxation parameters depend on both the mesh step size and the regularization parameter , which was not investigated in literature. Second, we propose and analyze a new mass-based Braess--Sarazin relaxation (BSR) scheme, which is proven to provide smaller smoothing factors than the CJR scheme when for a small constant . Finally, these schemes are successfully extended to control-constrained cases through the semismooth Newton method, where the corresponding Jacobian systems are treated by the proposed multigrid schemes. The nonstandard coarsening by three or four with BSR is competitive with the standard coarsening by two. Numerical examples are presented to validate our theoretical outcomes. The proposed inexact BSR (IBSR) scheme, where two preconditioned conjugate gradients (PCG) iterations are applied to solve the Schur complement system, yields a better computational efficiency than the CJR scheme in the conducted numerical comparison.

Robot stiffness theory reconsideration based on Schur complement eigenvalues: Extension to GSP dynamic stiffness evaluation

The known robot stiffness evaluation algorithm based on the Schur complements’ eigenvalues (SCE) is refined and its application is extended to the dynamic stiffness problem. For the refinement, the Schur complements synthesis is reduced to standard transformations of the stiffness matrix into the triangular block matrices. In these terms, the Schur complements of the robot dynamic stiffness matrix are formulated for steady-state forced sinusoidal vibrations of a damping-free system. The interrelationships between the forced-frequency-dependent SCE and natural frequencies are established: the frequencies resulting in the SCE zero values are proven to be identical to the natural frequencies, and vice versa. The SCE of the dynamic stiffness matrix defines the range and extremes of the dynamic stiffness values that allow the building of the translational and rotational stiffness ellipsoids and formulation of the stiffness-related robot performance indices. In four application examples, the SCE-based dynamic stiffness values of the Gough-Stewart platform (GSP) on weightless supports are defined. The definitions allow the formulation of the dynamic stiffness-related performance indices which are used for quantitative evaluation and comparison of the GSP designs.

Weight learning for ℋ∞$$ {\mathscr{H}}_{\infty } $$ stabilization of uncertain switched neural networks with external disturbance and reaction‐diffusion

SummaryThis paper considers the stabilization of uncertain switched neural networks with external disturbance and reaction‐diffusion. A weight learning rule that ensures the stability of the network is proposed utilizing the Lyapunov–Krasovskii functional method. Then, a useful lemma concerning the equivalence of matrix inequities is derived. With the aid of the lemma and Schur complement, a more concise existence condition on the learning rule is developed. Finally, theoretical comparisons and a numerical example are given, which show that the obtained results are extensions and improvements of an existing result.

A FFT accelerated fourth order finite difference method for solving three-dimensional elliptic interface problems

In this paper, a fourth order augmented matched interface and boundary (AMIB) method is proposed for solving a three-dimensional elliptic interface problem which involves a smooth material interface inside a cuboid domain. On the boundary of the cuboid domain, the fourth order AMIB method can handle different types of boundary conditions, including Dirichlet, Neumann, Robin and their mix combinations in fictitious value generation. Moreover, zero-padding solutions are introduced so that the fast Fourier transform (FFT) algorithm is still valid near the boundary. In dealing with the interior interface, a fourth order ray-casting matched interface and boundary (MIB) scheme is proposed, which enforces the jump conditions along the normal direction for calculating fictitious values. Comparing with the existing MIB scheme, the ray-casting scheme naturally bypasses the corner issue and becomes more robust in handling complex geometry. Based on fictitious values generated near interface and boundary, the fourth order central difference can be corrected at various irregular points including corner points, by introducing Cartesian derivative jumps as auxiliary variables. This gives rise to an enlarged linear system, which can be efficiently solved by the Schur complement procedure together with the FFT inversion of the discrete Laplacian. Extensive numerical experiments have been carried to test the proposed ray-casting AMIB method for numerical accuracy, efficiency, and robustness in corner treatment. The numerical results demonstrate that the ray-casting AMIB scheme not only maintains a fourth order of accuracy in treating various interfaces and boundaries for both solutions and solution gradients, but also attains an overall efficiency on the order of O(n3log⁡n) for a n×n×n uniform grid.

Asynchronous multisplitting-based primal Schur method

This paper presents a new scalable asynchronous domain decomposition method which significantly outperforms the recent two-level asynchronous restricted additive Schwarz (RAS) method. The novel approach consists of considering the classical primal Schur complement framework and solving the interface problem using a multisplitting relaxation based on a weighted additive Schwarz preconditioner. The computational model does not require any significant implementation effort compared to an asynchronous RAS solver. Experiments were conducted on a linear elasticity test case, and the results clearly show the practical superiority of the new asynchronous method, which is even competitive with its conjugate gradient counterpart.

On the use of graph centralities to compute generalized inverse of singular finite element operators: Applications to the analysis of floating substructures

SummaryThis article introduces a robust and affordable method to compute nullspace and generalized inverse of finite element operators involved in dual domain decomposition methods. The methodology relies on the operator partial factorization and on the analysis of a well chosen Schur complement. The sparse linear operator is interpreted as a network and graph centrality measures are used to select the condensation variables. Eigenvector, Katz and Page Rank centralities are evaluated. An extension to deal with symmetric indefinite systems arising from mixed finite elements is also presented. The approach is assessed on highly heterogeneous problems and one industrial application is presented: the numerical homogenization of solid propellant.

Stability Analysis of State Delay Multiagent Systems with Observer-Based Control Protocols

The consistency problem of multiagent systems with the output feedback and state delay was considered in this paper. First, the reduced-order observer based on the consensus protocol of state delay is designed, and the consensus protocol is proposed by the output information of neighboring agents. Then, by constructing the Lyapunov–Krasovskii functional and combining matrix inequality and the Schur complement lemma, the asymptotic stability of the system can be obtained, and the consistency of state-delay multiagent systems is derived. Finally, the theoretical results are verified by a simulation example.

Schur complement-based infinity norm bounds for the inverse of $ S $-Sparse Ostrowski Brauer matrices

&lt;abstract&gt;&lt;p&gt;In this paper, we study the Schur complement problem of $ S $-SOB matrices, and prove that the Schur complement of $ S $-Sparse Ostrowski-Brauer ($ S $-SOB) matrices is still in the same class under certain conditions. Based on the Schur complement of $ S $-SOB matrices, some upper bound for the infinite norm of $ S $-SOB matrices is obtained. Numerical examples are given to certify the validity of the obtained results. By using the infinity norm bound, an error bound is given for the linear complementarity problems of $ S $-SOB matrices.&lt;/p&gt;&lt;/abstract&gt;

Error estimations of a weak Galerkin finite element method for a linear system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion equations in the energy and balanced norms

&lt;abstract&gt;&lt;p&gt;This paper introduces a weak Galerkin finite element method for a system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion problems. The proposed method is independent of parameter and uses piecewise discontinuous polynomials on interior of each element and constant on the boundary of each element. By the Schur complement technique, the interior unknowns can be locally efficiently eliminated from the resulting linear system, and the degrees of freedom of the proposed method are comparable with the classical FEM. It has been reported that the energy norm is not adequate for singularly perturbed reaction-diffusion problems since it can not efficiently reflect the behaviour of the boundary layer parts when the diffusion coefficient is very small. For the first time, the error estimates in the balanced norm has been presented for a system of coupled singularly perturbed problems when each equation has different parameter. Optimal and uniform error estimates have been established in the energy and balanced norm on an uniform Shishkin mesh. Finally, we carry out various numerical experiments to verify the theoretical findings.&lt;/p&gt;&lt;/abstract&gt;

Radially Extended 3D Birdcage Antenna for MRI With Loop-Type Legs and Connection to the Shield

A novel radially extended 3D birdcage (BC) coil antenna was designed, simulated, and optimized for a cylindrical dielectric phantom at 4.7 T. The 3D birdcage consists of a classical birdcage with additional protrusions toward the inside and connections to the shield. To overcome the huge computational burden of the optimization process using the particle swarm optimization (PSO) method, an in-house method of moment (MoM) code exploiting the symmetry of the structure has been developed. The code produces open-circuit embedded element patterns and makes use of a Schur complement to analyze the effects of passive loads, in the presence of a phantom. The simulation results have been verified with the use of commercial software. The proposed 3D BC coil showed a 34 % improvement in the homogeneity of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and a 37 % reduction in the peak specific absorption rate (SAR), compared with a conventional BC coil.

Drives’ stiffness of 5-axis machine designs: Evaluation and comparison by Schur complement eigenvalues

Drives’ stiffness of 5-axis machine designs: Evaluation and comparison by Schur complement eigenvalues

Использование метода декомпозиции области для распараллеливания моделирования течения вязкой несжимаемой среды методом LS-STAG и дополнительного предобуславливания

As a rule, the main part of the computational costs in the numerical solution of problems in continuum mechanics consists in the solving of large sparse systems of linear algebraic equations. For this reason, efficient parallelization of this particular procedure can significantly speed up the simulation. To solve this problem, two main approaches can be used. The simplest approach consists in parallelizing of matrix-vector operations in a usual iterative solver. It requires several synchronization points and exchanges of coefficients at each iteration of the solver, which does not significantly speed up the simulation process as a whole. Therefore, domain decomposition methods are preferable. These methods involve dividing the computational domain into subdomains, constructing and solving separate problems in them, as well as some procedure to coordinate the solution between subdomains to ensure global convergence. Subdomains can overlap, as in the Schwartz method used in OpenFOAM, or they can be separated by interface sections, on which their own interface task is built, as in the Schur complement method. The latter method is used in this research to construct a parallel algorithm for viscous incompressible flow simulation by using the immersed boundary method LS-STAG with cut-cells and level-set functions. The resulting matrix of the interface system has a block tridiagonal structure. To speed up prototyping, OpenMP parallel programming technology is used in the software implementation of the developed algorithm, so computational experiments are carried out only on systems with shared memory, in particular on individual nodes of the educational and experimental cluster of the Applied Mathematics Department, Bauman Moscow State Technical University. To verify and evaluate the effectiveness of the developed parallel algorithm, a well-studied test problem about simulation of two-dimensional flow around a stationary circular airfoil is considered. Computations on a sequence of meshes with different numbers of subdomains show that the parallel algorithm allows one to obtain the same numerical solution as the original algorithm of the LS-STAG method, and the computed values of the Strouhal number and drag coefficient are in good agreement with the experimental and computational data known in the literature. Experiments demonstrate that the developed algorithm with domain decomposition allows to accelerate simulation even in sequential mode by reducing the number of solver iterations, i.e. the domain decomposition method acts as an additional preconditioner. Due to this property, the acceleration is superlinear when simulating in parallel mode with developed algorithm. This effect persists up to a certain number of subdomains, which depends on the size of the problem.

Null vectors, Schur complements, and Parter vertices

Null vectors, Schur complements, and Parter vertices

A hybridized weak Galerkin finite element scheme for linear elasticity problem

In this paper, we present a hybridized weak Galerkin finite element method for a class of a linear elasticity model. The hybridized weak Galerkin finite element method is an improvement of the standard weak Galerkin method by introducing Lagrange multipliers, which relaxes certain constraints and inherits its flexibility. By applying Schur complement technique, the degrees of freedom is reduced and the calculation efficiency is improved. The optimal convergence orders of error estimates are achieved. Some numerical experiments are presented to illustrate the theoretical analysis.