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In this paper, we discuss the Hyers–Ulam stability of closable (unbounded) operators with some examples. We also present results pertaining to the Hyers–Ulam stability of the sum and product of closable operators to have the Hyers–Ulam stability and the necessary and sufficient conditions of the Schur complement and the quadratic complement of 2×2$2 \times 2$ block matrix A$\mathcal {A}$ in order to have the Hyers–Ulam stability. [ABSTRACT FROM AUTHOR]

Rehabilitation exoskeleton robots play a crucial role in restoring functional lower limb movements for individuals with locomotor disorders. Numerous research studies have concentrated on adapting the control of these rehabilitation robotic systems. In this study, we investigate an affine state-feedback control law for robust position control of a knee exoskeleton robot, taking into account its nonlinear dynamic model that includes solid and viscous frictions. To ensure robust stabilization, we employ the Lyapunov approach and propose three methods to establish stability conditions using the Schur complement, the Young inequality, the matrix inversion lemma, and the S-procedure lemma. These conditions are formulated as Linear Matrix Inequalities (LMIs). Furthermore, we conduct a comprehensive comparison among these methods to determine the most efficient approach. At the end of this work, we present simulation results to validate the developed LMI conditions and demonstrate the effectiveness of the adopted control law in achieving robust position control of the knee exoskeleton robot. [ABSTRACT FROM AUTHOR]

In this paper, a Takagi–Sugeno fuzzy parallel distributed compensation control (TS-PDCC) is proposed for low-frequency oscillation (LFO) suppression in wind energy-penetrated power systems. Firstly, the fuzzy C-mean algorithm (FCMA) is applied to cluster the daily average wind speed of the wind farm, and the obtained wind speed clustering center is used as the premise variable of TS-PDCC, which increases the freedom of parameter setting of the TS fuzzy model and is closer to the actual working environment. Secondly, based on the TS fuzzy model, the TS-PDCC is designed to adjust the active power output of the wind turbine for LFO suppression. To facilitate the computation of controller parameters, the stability conditions are transformed into a set of Linear Matrix Inequalities (LMIs) via the Schur complement. Subsequently, a Lyapunov function is designed to verify the stability of the wind energy-penetrated power system and obtain the parameter ranges. Simulation cases are conducted to verify the validity and superior performance of the proposed TS-PDCC under different operating conditions. [ABSTRACT FROM AUTHOR]

We consider the Fokker–Planck equation (FPE) for the orientation probability density of fiber suspensions. Using the continuous Galerkin method, we express the numerical solution in terms of Lagrange basis functions that are associated with N nodes of a computational mesh for a domain in the 3D physical space and M nodes of a mesh for the surface of a unit sphere representing the configuration space. The NM time-dependent unknowns of our finite element approximations are probabilities corresponding to discrete space locations and orientation angles. The framework of alternating-direction methods enables us to update the numerical solution in parallel by solving N evolution equations on the sphere and M three-dimensional advection equations in each (pseudo-)time step. To ensure positivity preservation as well as the normalization property of the probability density, we perform algebraic flux correction for each equation and synchronize the correction factors corresponding to different orientation angles. The velocity field for the spatial advection step is obtained using a Schur complement method to solve a generalized system of the incompressible Navier–Stokes equations (NSE). Fiber-induced subgrid-scale effects are taken into account using an effective stress tensor that depends on the second- and fourth-order moments of the orientation density function. Numerical studies are performed for individual subproblems and for the coupled FPE-NSE system. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a new estimate for the (product) γ -diagonally dominant degree of the Schur complement of matrices. As applications we discuss the localization of eigenvalues of the Schur complement and present several upper and lower bounds for the determinant of strictly γ -diagonally dominant matrices, which generalizes the corresponding results of Liu and Zhang (SIAM J. Matrix Anal. Appl. 27 (2005) 665-674). [ABSTRACT FROM AUTHOR]

In the context of inverse $ {{N}_{0}} $-matrices, this study focuses on the closure of generalized Perron complements by utilizing the characteristics of $ M $-matrices, nonnegative matrices, and inverse $ {{N}_{0}} $-matrices. In particular, we illustrate that the inverse $ {{N}_{0}} $-matrix and its Perron complement matrix possess the same spectral radius. Furthermore, we present certain general inequalities concerning generalized Perron complements, Perron complements, and submatrices of inverse $ {{N}_{0}} $-matrices. Finally, we provide specific examples to verify our findings.

The efficiency of numerical solvers of PDEs depends on the approximation properties of the discretization methods and the conditioning of the resulting linear systems. If applicable, the boundary element methods typically provide better approximation with unknowns limited to the boundary than the Schur complement of the finite element stiffness matrix with respect to the interior variables. Since both matrices correctly approximate the same object, the Steklov-Poincaré operator, it is natural to assume that the matrices corresponding to the same fine boundary discretization are similar. However, this note shows that the distribution of the spectrum of the boundary element stiffness matrix is significantly better conditioned than the finite element Schur complement. The effect of the favorable conditioning of BETI clusters is demonstrated by solving huge problems by H-TBETI-DP and H-TFETI-DP. [ABSTRACT FROM AUTHOR]

We derive an extension of the sequential homotopy method that allows for the application of inexact solvers for the linear (double) saddle‐point systems arising in the local semismooth Newton method for the homotopy subproblems. For the class of problems that exhibit (after suitable partitioning of the variables) a zero in the off‐diagonal blocks of the Hessian of the Lagrangian, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. For discretized optimal control problems with PDE constraints, this structure is often present with the canonical partitioning of the variables in states and controls. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method allows for the parallel solution of large 3D problems. [ABSTRACT FROM AUTHOR]

Numerical discretization of a regularized inverse source problem leads to a non-symmetric saddle point linear system. Interestingly, the Schur complement of the non-symmetric saddle point system is Hermitian positive definite (HPD). Then, we propose a preconditioner matching the Schur complement (MSC). Theoretically, we show that the preconditioned conjugate gradient (PCG) method for a linear system with the preconditioned Schur complement as coefficient has a linear convergence rate independent of the matrix size and value of the regularization parameter involved in the inverse problem. Fast implementations are proposed for the matrix-vector multiplication of the preconditioned Schur complement so that the PCG solver requires only quasi-linear operations. To the best of our knowledge, this is the first solver with guarantee of linear convergence for the inversion of Schur complement arising from the discrete inverse problem. Combining the PCG solver for inversion of the Schur complement and the fast solvers for the forward problem in the literature, the discrete inverse problem (the saddle point system) is solved within a quasi-linear complexity. Numerical results are reported to show the performance of the proposed matching Schur complement (MSC) preconditioning technique. • A fast solver for a regularized inverse problem. • The solver is robust w.r.t. regularization parameters and discretization parameters. • The complexity of the solver is nearly linear (optimal). [ABSTRACT FROM AUTHOR]

This paper introduces an extension of the well-known Morley element for the biharmonic equation, extending its application from triangular elements to general polytopal elements using the weak Galerkin finite element methods. By leveraging the Schur complement of the weak Galerkin method, this extension not only preserves the same degrees of freedom as the Morley element on triangular elements but also expands its applicability to general polytopal elements. The numerical scheme is devised by locally constructing weak tangential derivatives and weak second-order partial derivatives. Error estimates for the numerical approximation are established in both the energy norm and the L 2 norm. A series of numerical experiments are conducted to validate the theoretical developments. [ABSTRACT FROM AUTHOR]

As we know that, the parameterized Uzawa (PU) method can be very efficient when used to solve the standard saddle point problem, especially, when we have good and accurate estimation of preconditioned Schur complement matrix. In this paper, by taking full use of the special structure of coefficient matrix arising from complex Helmholtz equations, two types of optimized PU (OPU) methods are discussed theoretically and experimentally. Specifically, the convergence factors of these two OPU methods are less than 0.172, and the optimal result of first OPU method can reach 0.0396 with σ 1 ≥ σ 2 > 0 , which is currently the best theoretical result in the literatures. Moreover, the second OPU method has better computational advantages compared with the first OPU method as it avoids the inverse computation of W at each step, indicating the less CPU time will be costed by the second OPU method. In addition, the optimal parameters involved in our algorithms consist of constants, reducing the computational complexity associated with parameter selection. Finally, numerical results are given, not only show the effectiveness of OPU methods, but also confirm the rationality of theoretical analysis. [ABSTRACT FROM AUTHOR]

In 1934, the American statistician Samuel S. Wilks derived remarkable formulas for the joint moments of embedded principal minors of sample covariance matrices in multivariate Gaussian populations, and he used them to compute the moments of sample statistics in various applications related to multivariate linear regression. These important but little‐known moment results were extended in 1963 by the Australian statistician A. Graham Constantine using Bartlett's decomposition. In this note, a new proof of Wilks' results is derived using the concept of iterated Schur complements, thereby bypassing Bartlett's decomposition. Furthermore, Wilks' open problem of evaluating joint moments of disjoint principal minors of Wishart random matrices is related to the Gaussian product inequality conjecture. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of robust decentralized control of discrete-time nonlinear interconnected descriptor systems (IDS) under the influence of external disturbances. First, we utilize the Takagi–Sugeno fuzzy model (TSFM) to represent discrete-time nonlinear IDS. For controller design, a proportional-derivative (PD) feedback strategy is employed to formulate a decentralized fuzzy controller for the discrete-time Takagi–Sugeno IDS (DTTSIDS). Furthermore, robust and passivity constraints are incorporated into the controller design process to mitigate the effects of disturbances. Based on the Lyapunov function and free weight function methods, we can analyze the stability of the DTTSIDS. The proposed conditions are converted into Linear Matrix Inequality (LMI) conditions through Schur Complement technology so that we can solve the problem through MATLAB LMI-Toolbox. Finally, the effectiveness of the proposed methodology is demonstrated through three examples presented in the paper and a comparison with other existing studies is given in Example 3. [ABSTRACT FROM AUTHOR]

Efficient and robust iterative solvers for strongly anisotropic elliptic equations are very challenging. Indeed, the discretization of this class of problems gives rise to a linear system with a condition number increasing with anisotropic strength. This weakness is addressed clearly by adopting the asymptotic-preserving (AP) discretizations. In this paper a block preconditioning method is introduced to solve the linear algebraic systems of a class of micro–macro asymptotic-preserving (MMAP) scheme. The MMAP method was developed by Degond et al. in 2012 where its corresponding discrete matrix has a 2 × 2 block structure. Motivated by approximate Schur complements, a series of block preconditioners are constructed. We first analyze a natural approximate Schur complement that is the coefficient matrix of the original Non-AP discretization. However it tends to be singular for very small anisotropic parameters. We then improve it by using more suitable approximation for boundary rows of the exact Schur complement. With these block preconditioners, a preconditioned GMRES iterative method is developed to solve the discrete equations. Several numerical tests show that block preconditioning methods can be a practically useful strategy with respect to grid refinement and anisotropic strengths. [ABSTRACT FROM AUTHOR]

This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in if(div). The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation--histopolation basis (cf. [W. Pazner, T. Kolev, and C. R. Dohrmann, SIAM J. Sci. Comput., 45 (2023), pp. A675-A702]), efficient matrix-free preconditioners can be constructed for the (1,1)-block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The approximate Schur complement takes the form of an M-matrix graph Laplacian and therefore can be well-preconditioned by highly scalable algebraic multigrid methods. High-performance GPU-accelerated algorithms for all components of the solution algorithm are developed, discussed, and benchmarked. Numerical results are presented on a number of challenging test cases, including the "crooked pipe" grad-div problem, the SPE10 reservoir modeling benchmark problem, and a nonlinear radiation diffusion test case. [ABSTRACT FROM AUTHOR]

In this paper, we address the numerical solution of the quadratic optimal transport problem in its dynamical form, the so-called Benamou-Brenier formulation. When solved using interior point methods, the main computational bottleneck is the solution of large saddle point linear systems arising from the associated Newton-Raphson scheme. The main purpose of this paper is to design efficient preconditioners to solve these linear systems via iterative methods. Among the proposed preconditioners, we introduce one based on the partial commutation of the operators that compose the dual Schur complement of these saddle point linear systems, which we refer to as the BB-preconditioner. A series of numerical tests show that the BB-preconditioner is the most efficient among those presented, despite a performance deterioration in the last steps of the interior point method. It is in fact the only one having a CPU time that scales only slightly worse than linearly with respect to the number of unknowns used to discretize the problem. [ABSTRACT FROM AUTHOR]

Coupled multi-physics problems are encountered in countless applications and pose significant numerical challenges. In a broad sense, one can categorise the numerical solution strategies for coupled problems into two classes: monolithic approaches and sequential (also known as split, decoupled, partitioned or segregated) approaches. The monolithic approaches treat the entire problem as one, whereas the sequential approaches are iterative decoupling techniques where the different sub-problems are treated separately. Although the monolithic approaches often offer the most robust solution strategies, they tend to require ad-hoc preconditioners and numerical implementations. Sequential methods, on the other hand, offer the possibility to add and remove equations from the model flexibly and rely on existing black-box solvers for each specific equation. Furthermore, when problems are non-linear, inner iterations need to be performed even in monolithic solvers, making the sequential approaches an even more viable alternative. The cost of running inner iterations to recover the multi-physics coupling could, however, easily become prohibitive. Moreover, the sequential approaches might not converge at all. In this work, we present a general formulation of splitting schemes for continuous operators with arbitrary implicit/explicit splitting, like in standard iterative methods for linear systems. By introducing a generic relaxation operator, we find the conditions for the convergence of the iterative schemes. We show how the relaxation operator can be thought of as a preconditioner and constructed based on an approximate Schur complement. We propose a Schur-based Partial Jacobi relaxation operator to stabilise the coupling and show its effectiveness. Although we mainly focus on scalar-scalar linear problems, most results are easily extended to non-linear and higher-dimensional problems. The schemes presented are not explicitly dependent on any particular discretisation methodologies. Numerical tests (1D and 2D) for two PDE systems, namely the Dual-Porosity model and a Quad-Laplacian operator, are carried out to investigate the practical implications of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on the analysis and the solution of the saddle‐point problem arising from a three‐field formulation of Biot's model of poroelasticity, discretized in time by the implicit Euler method. A block diagonal‐preconditioner, based on the Schur complement, is analyzed on a functional level and compared with two other block‐diagonal preconditioners having a similar structure. The problem is discretized in space using mixed finite elements and solved with appropriate iterative solvers, incorporating the investigated preconditioners. The solvers are tested on numerical examples inspired by geotechnical practice, with particular attention devoted to the solvers' robustness concerning strong heterogeneity in permeability. [ABSTRACT FROM AUTHOR]

In this work, preconditioning strategies are developed in the context of generalized continuum formulations used to regularize multifield models for simulating localized failure of quasi‐brittle materials. Specifically, a micropolar continuum extended by a nonlocal damage formulation is considered for regularizing both, shear dominated failure and tensile cracking. For such models, additional microrotation and nonlocal damage fields, and their interactions, increase the complexity and size of the arising linear systems. This increases the demand for specialized preconditioning strategies when iterative solvers are adopted. Herein, a block preconditioning strategy, employing algebraic multigrid methods (AMG) for approximating the application of sub‐block inverses, is developed and tested in three steps. First, a block preconditioner is introduced for linear systems resulting from micropolar models. For this case, a simple sparse Schur complement approximation, which is practical to compute, is proposed and analyzed. It is tested for three different discretizations. Second, the developed preconditioner is extended to reflect the additional nonlocal damage field. This extended three‐field preconditioner is tested on the simulation of a compression test on a sandstone sample. All numerical tests show an improved performance of the block preconditioning approach in comparison to a black‐box monolithic AMG approach. Finally, a problem‐adapted preconditioner setup strategy is proposed, which involves a reuse of the multigrid hierarchy during nonlinear iterations, and additionally accounts for the different stages occurring in the simulation of localized failure. The problem‐adapted preconditioning strategy has the potential to further reduce the total computation time. [ABSTRACT FROM AUTHOR]

We propose an efficient primal-dual interior-point relaxation algorithm based on a smoothing barrier augmented Lagrangian, called IPRSDP, for solving semidefinite programming problems in this paper. The IPRSDP algorithm has three advantages over classical interior-point methods. Firstly, IPRSDP does not require the iterative points to be positive definite. Consequently, it can easily be combined with the warm-start technique used for solving many combinatorial optimization problems, which require the solutions of a series of semidefinite programming problems. Secondly, the search direction of IPRSDP is symmetric in itself, and hence the symmetrization procedure is not required any more. Thirdly, with the introduction of the smoothing barrier augmented Lagrangian function, IPRSDP can provide the explicit form of the Schur complement matrix. This enables the complexity of forming this matrix in IPRSDP to be comparable to or lower than that of many existing search directions. The global convergence of IPRSDP is established under suitable assumptions. Numerical experiments are made on the SDPLIB set, which demonstrate the efficiency of IPRSDP. [ABSTRACT FROM AUTHOR]

Recently Oishi published a paper allowing lower bounds for the minimum singular value of coefficient matrices of linearized Galerkin equations, which in turn arise in the computation of periodic solutions of nonlinear delay differential equations with some smooth nonlinearity. The coefficient matrix of linearized Galerkin equations may be large, so the computation of a valid lower bound of the smallest singular value may be costly. Oishi's method is based on the inverse of a small upper left principal submatrix, and subsequent computations use a Schur complement with small computational cost. In this note some assumptions are removed and the bounds improved. Furthermore a technique is derived to reduce the total computationally cost significantly allowing to treat infinite dimensional matrices. [ABSTRACT FROM AUTHOR]

Summary: This work describes some first efforts to design a new Linpack‐like benchmark useful to evaluate the performance of Heterogeneous Computing Resources. The benchmark is based on the Schur Complement reformulation of the solution of a linear equation system. Details about its implementation and evaluation, mainly in terms of performance scalability, are presented for a computing environment based on multi NVIDIA GP‐GPUs nodes connected by an Infiniband network. [ABSTRACT FROM AUTHOR]

In this paper, a simple preconditioner with two parameters (SPTP) is introduced for solving a class of saddle-point problems. The SPTP preconditioner is block upper triangular and does not contain the approximate matrix of the Schur complement matrix. Theoretical analyses on the sufficient conditions under which the SPTP iterative sequence converges to the unique solution, are given in detail. Convergence conditions in some special cases are also analyzed. Moreover, detailed analyses on the theoretical optimal parameters and corresponding optimal convergence factor of the SPTP method are given. Some numerical examples show that the SPTP iterative method outperforms several SOR-type methods by using theoretical optimal parameters simultaneously. Some newly proposed methods are also compared with the SPTP iterative method by using the experimental optimal parameters simultaneously, and the SPTP iterative method is still advantageous, from the point of view of iteration step and CPU time. Moreover, we also investigate relatively practical values of the parameters since searching for the optimal parameters will waste a lot of time in practice. [ABSTRACT FROM AUTHOR]

In the present paper, a new technique is presented to study the problem of invertibility of unbounded block 3 × 3 operator matrices defined with diagonal domain. Sufficient criteria are established to guarantee our interest and to prove some interaction between such a model of an operator matrix and its diagonal operator entries. The effectiveness of the proposed new technique is shown by a physical example of an integro differential equation named the neutron transport equation with partly elastic collision operators. In particular, the obtained results answer the question in [H. Zguitti, A note on Drazin invertibility for upper triangular block operators, Mediterr. J. Math. 10 2013, 3, 1497–1507] and the conjecture in [A. Bahloul and I. Walha, Generalized Drazin invertibility of operator matrices, Numer. Funct. Anal. Optim. 43 2022, 16, 1836–1847]. [ABSTRACT FROM AUTHOR]

This paper deals with Fredholm properties of the one-sided coupled operator matrix M = A B 0 D I 0 L I by means of generalized Schur factorization and the associated space decompositions. For λ ∈ C , some sufficient conditions are given for λ - M to be Fredholm (resp. left or right Fredholm), and these conclusions are further used to determine the essential spectra of a delay equation and a wave equation with acoustic boundary conditions. [ABSTRACT FROM AUTHOR]

A weak Galerkin finite element method is applied to singularly perturbed delay reaction-diffusion problems. A robust uniform convergence has been proved both in the energy and balanced norms using higher-order piecewise discontinuous polynomials on Shishkin meshes. The error analysis for singularly perturbed reaction-diffusion problems with negative or positive shift in the balanced norm has appeared for the first time. The proposed method uses piecewise polynomials of order $ k\geq ~1 $ k ≥ 1 on interior of each element and piecewise constant polynomials on the end points of each element. By the Schur complement technique, the interior degrees of foredoom (DOF) can be eliminated from the discrete system resulting from the numerical scheme, and thus the degrees of freedom of the proposed method comparable with the classical finite element methods, and it is remarkably less than that of the discontinuous Galerkin method. Finally, we give various numerical experiments to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

Incomplete LU (ILU) smoothers are effective in the algebraic multigrid (AMG) V$$ V $$‐cycle for reducing high‐frequency components of the error. However, the requisite direct triangular solves are comparatively slow on GPUs. Previous work has demonstrated the advantages of Jacobi iteration as an alternative to direct solution of these systems. Depending on the threshold and fill‐level parameters chosen, the factors can be highly nonnormal and Jacobi is unlikely to converge in a low number of iterations. We demonstrate that row scaling can reduce the departure from normality, allowing us to replace the inherently sequential solve with a rapidly converging Richardson iteration. There are several advantages beyond the lower compute time. Scaling is performed locally for a diagonal block of the global matrix because it is applied directly to the factor. Further, an ILUT Schur complement smoother maintains a constant GMRES iteration count as the number of MPI ranks increases, and thus parallel strong‐scaling is improved. Our algorithms have been incorporated into hypre, and we demonstrate improved time to solution for linear systems arising in the Nalu‐Wind and PeleLM pressure solvers. For large problem sizes, GMRES+$$ + $$AMG executes at least five times faster when using iterative triangular solves compared with direct solves on massively parallel GPUs. [ABSTRACT FROM AUTHOR]

A rational homogeneous (of degree one) positive real matrix-valued function of several variables can be represented as a Schur complement to the diagonal block of a linear homogeneous matrix-valued function with positive semidefinite real matrix coefficients (the long-resolvent representation). The numerators of the partial derivatives of a positive real function are sums of squares of polynomials. [ABSTRACT FROM AUTHOR]

The Nyström method is a convenient strategy to obtain low-rank approximations to kernel matrices in nearly linear complexity. Existing studies typically use the method to approximate positive semidefinite matrices with low or modest accuracies. In this work, we propose a series of heuristic strategies to make the Nyström method reach high accuracies for nonsymmetric and/or rectangular matrices. The resulting methods (called high-accuracy Nyström methods) treat the Nyström method and a skinny rank-revealing factorization as a fast pivoting strategy in a progressive alternating direction refinement process. Two refinement mechanisms are used: alternating the row and column pivoting starting from a small set of randomly chosen columns, and adaptively increasing the number of samples until a desired rank or accuracy is reached. A fast subset update strategy based on the progressive sampling of Schur complements is further proposed to accelerate the refinement process. Efficient randomized accuracy control is also provided. Relevant accuracy and singular value analysis is given to support some of the heuristics. Extensive tests with various kernel functions and data sets show how the methods can quickly reach prespecified high accuracies in practice, sometimes with quality close to SVDs, using only small numbers of progressive sampling steps. [ABSTRACT FROM AUTHOR]

We consider the iterative solution of large linear systems of equations in which the coefficient matrix is the sum of two terms, a sparse matrix A and a possibly dense, rank deficient matrix of the form γUUT, where γ > 0 is a parameter which in some applications may be taken to be 1. The matrix A itself can be singular, but we assume that the symmetric part of A is positive semidefinite and that A+γUUT is nonsingular. Linear systems of this form arise frequently in fields like optimization, fluid mechanics, computational statistics, and others. We investigate preconditioning strategies based on an alternating splitting approach combined with the use of the Sherman-Morrison-Woodbury matrix identity. The potential of the proposed approach is demonstrated by means of numerical experiments on linear systems from different application areas. [ABSTRACT FROM AUTHOR]

We propose a distributed-memory parallel algorithm for computing some of the algebraically smallest eigenvalues (and corresponding eigenvectors) of a large, sparse, real symmetric positive definite matrix pencil that lie within a target interval. The algorithm is based on Chebyshev interpolation of the eigenvalues of the Schur complement (over the interface variables) of a domain decomposition reordering of the pencil and accordingly exposes two dimensions of parallelism: one derived from the reordering and one from the independence of the interpolation nodes. The new method demonstrates excellent parallel scalability, comparing favorably with PARPACK, and does not require factorization of the mass matrix, which significantly reduces memory consumption, especially for 3D problems. Our implementation is publicly available on GitHub. [ABSTRACT FROM AUTHOR]

Let N ≥ 2 be an integer, a (1, N)-periodic graph G is a periodic graph whose vertices can be partitioned into two sets V 1 = { v ∣ σ (v) = v } and V 2 = { v ∣ σ i (v) ≠ v for any 1 < i < N } , where σ is an automorphism with order N of G. The subgraph of G induced by V 1 is called a fixed subgraph. Yan and Zhang (2011) studied the enumeration of spanning trees of a special type of (1, N)-periodic graphs with V 1 = 0̸ for any non-trivial automorphism with order N. In this paper, we obtain a concise formula for the number of spanning trees of (1, N)-periodic graphs. Our result can reduce to Yan and Zhang's when V 1 is empty. As applications, we give a new closed formula for the spanning tree generating function of cobweb lattices, and obtain formulae for the number of spanning trees of circulant graphs C n (s 1 , s 2 , ... , s k) , K 1 ⋁ C n (s 1 , s 2 , ... , s k) and K 2 ⋁ C n (s 1 , s 2 , ... , s k). [ABSTRACT FROM AUTHOR]

In this paper, to solve the three-by-three block saddle-point problem, a new block triangular (NBT) preconditioner is established, which can effectively avoid the solving difficulty that the coefficient matrices of linear subsystems are Schur complement matrices when the block preconditioner is applied to the Krylov subspace method. Theoretical analysis shows that the iteration method produced by the NBT preconditioner is unconditionally convergent. Besides, some spectral properties are also discussed. Finally, numerical experiments are provided to show the effectiveness of the NBT preconditioner. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical solution of a large complex linear system with a saddle-point form obtained by the discretization of the time-harmonic eddy-current optimal control problem. A new Schur complement is proposed for this algebraic system, extending it to both the block-triangular preconditioner and the structured preconditioner. A theoretical analysis proves that the eigenvalues of block-triangular and structured preconditioned matrices are located in the interval [1/2, 1]. Numerical simulations show that two new preconditioners coupled with a Krylov subspace acceleration have good feasibility and effectiveness and are superior to some existing efficient algorithms. [ABSTRACT FROM AUTHOR]

We prove the (generalized) principal pivot transform is matrix monotone, in the sense of the Löwner ordering, under minimal hypotheses. This improves on the recent results of Pascoe and Tully-Doyle (2022) [69] in two ways. First, we use the "generalized" principal pivot transform, where matrix inverses in the classical definition of the principal pivot transform are replaced with Moore-Penrose pseudoinverses. Second, the hypotheses they used to prove the monotonicity is relaxed and, in particular, we find the weakest hypotheses possible for which it can be true. We also prove the principal pivot transform is a matrix concave function on positive semi-definite matrices that have the same kernel (and, in particular, on positive definite matrices). Our proof is a corollary of a minimization variational principle for the principal pivot transform. [ABSTRACT FROM AUTHOR]

Fluid flow control problems play a crucial role in various industrial applications. The optimal control of the Navier-Stokes equations poses a significant challenge. By employing Oseen's approximation to linearize the problem, we encounter a series of large sparse structured linear systems. These coefficient matrices exhibit a two-by-two block structure with square blocks. Leveraging this block structure, we exploit matrix splitting preconditioners. Theoretical analysis indicates that the real parts of all eigenvalues of preconditioned matrix are equal to 1/2. A large number of eigenvalues are clustered in (1 ± i) / 2. The imaginary part of the eigenvalues is bounded explicitly. The eigenvalue distribution predicts the fast convergence of Krylov subspace methods. To avoid solving the saddle point subsystems in the preconditioning procedure, we propose a practical variant of the preconditioner. The theoretical analysis demonstrates that if the approximation is sufficiently close to Schur complement, the eigenvalues of the modified preconditioned matrix will lie within a circle centered at the original eigenvalues with a radius less than 1. This implies that the modified preconditioner exhibits excellent performance in terms of preconditioning. The numerical experiments demonstrate that the generalized minimal residual method, when combined with the proposed preconditioners, proves to be an efficient and effective solution for Oseen's control optimization with a variable viscosity coefficient. The number of iterations required by the preconditioned methods remains unaffected by the mesh size used in finite element discretization and only marginally depends on the regularization parameter. • Computing time is linear dependent of the degrees of freedom. • The convergence rate is independent of the discretization mesh size, regularization parameter, and Reynolds number. • Eigenvalues of the preconditioned matrix are clustered on a line segment. • The preconditioning improves the condition number of the original linear system. [ABSTRACT FROM AUTHOR]

The aim of this paper is to compute the inverse of the Tridiagonal Quasi-Toeplitz matrix by direct method. The proposed algorithm constructs a decomposition of the given matrix, thanks to the special structure of the considered matrix, into a sum of a band Toeplitz matrix and the rest of size n x n. Then, the inverse of the matrix turns into the inverse of the obtained band matrix using the well known Schur complement and Sherman--Morrison--Woodbury formula. Illustrative example describing the different steps of the algorithm and numerical results are performed to show the effectiveness and accuracy of the algorithm. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of interval estimation for cyber–physical systems with unknown disturbance. In order to realize the interval estimation of cyber–physical systems, two technical methods are adopted. The first one requires the observer dynamic error system to be non-negative, and the second one relaxes this limitation by coordinate transformation. The sufficient conditions are established using both Lyapunov stability and positive system theory. Furthermore, according to the Schur complement, the linear matrix inequality is solved to determine the observer gains. Finally, the effectiveness and feasibility of the designed interval observer are verified by one numerical simulation. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph of order n. In this paper, we introduce a new distance matrix called the universal distance matrix of G, denoted as UD (G) and it is defined as UD (G) = αTr (G) + βD(G) + γJ + δI, where Tr (G) is the diagonal matrix whose elements are the vertex transmissions, and D(G) is the distance matrix of G. Here J is the all-ones matrix, and I is the identity matrix and α, β, γ, δ ∈ R and β ̸= 0. This unified definition enables us to derive the spectra of different matrices associated with the distance matrix of graphs. The set of eigenvalues of the universal distance matrix namely, {ρ1, ρ2,.. ., ρn} is known as the universal distance spectrum of G. As a consequence, by taking appropriate values for α, β, γ, δ ∈ R and β ̸= 0, we obtain the eigenvalues of distance matrix, distance Laplacian matrix, distance signless Laplacian matrix, generalized distance matrix, distance Seidal matrix and distance matrices of graph complements. In this paper, we obtain the universal distance spectra of regular graph, join of two regular graphs, joined union of three regular graphs, generalized joined union of n disjoint graphs with one arbitrary graph H using the Schur complement of a block matrix. [ABSTRACT FROM AUTHOR]

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The Woodbury formula is an efficient tool in mathematics to calculate low‐rank perturbation problems and has been applied to improve the computational efficiency of nonlinear seismic response analysis (NSRA) of structures with local nonlinearity. Using the Woodbury formula for NSRA can avoid the time‐consuming recalculation and factorization of the large‐dimensional global stiffness matrix of a structure by only solving a small‐dimensional Schur complement system representing local nonlinearity per iteration. Because the dimension of the Schur complement matrix is determined by the inelastic degree of freedom (IDOF) number, which represents the scale of local nonlinear regions, a small IDOF number is helpful for achieving the high‐efficiency advantage of the Woodbury formula. However, when performing NSRA for large‐scale structures, the IDOF number is usually relatively large, which contradicts the efficiency requirement of the Woodbury formula. To solve this problem and extend the advantage of the Woodbury formula to the NSRA of large‐scale structures, this paper first proposes a two‐stage IDOF number reduction method by eliminating the IDOFs that have insignificant effects on the results, and consequently, a variant Woodbury formula is derived. Because only the principal component in the Schur complement matrix is retained, the dimension of this matrix and the cost for factorizing it can be reduced significantly without losing accuracy, thus greatly improving the efficiency of the proposed method. Moreover, to reduce the additional computational time introduced by the IDOF number reduction procedure and to further improve the computational performance of the proposed method, an OpenMP parallel computational strategy is incorporated. Finally, the validity of the proposed method is verified by implementing incremental dynamic analysis for a large‐scale reinforced concrete structure. [ABSTRACT FROM AUTHOR]

This paper investigates the decentralized fuzzy control problems for nonlinear-state-unmeasured interconnected descriptor systems (IDSs) that utilize the observer-based-feedback approach and the proportional–derivative feedback control (PDFC) method. First of all, the IDS is represented as interconnected Takagi–Sugeno (T–S) fuzzy subsystems. These subsystems can effectively capture the dynamic behavior of the system through fuzzy rules. For the stability analysis of the system, this paper uses the free-weighing Lyapunov function (FWLF), which allows the designer to set the weight matrix, to achieve the desired control performance and design the controller more easily. Furthermore, the control problem can be transformed into a set of linear matrix inequalities (LMIs) through the Schur complement, which can be solved using convex optimization methods. Simulation results confirm the effectiveness of the proposed method in achieving the desired control objectives and ensuring system stability. [ABSTRACT FROM AUTHOR]

We consider a block‐structured multigrid method based on Braess–Sarazin relaxation for solving the Stokes–Darcy Brinkman equations discretized by the marker and cell scheme. In the relaxation scheme, an element‐based additive Vanka operator is used to approximate the inverse of the corresponding shifted Laplacian operator involved in the discrete Stokes–Darcy Brinkman system. Using local Fourier analysis, we present the stencil for the additive Vanka smoother and derive an optimal smoothing factor for Vanka‐based Braess–Sarazin relaxation for the Stokes–Darcy Brinkman equations. Although the optimal damping parameter is dependent on meshsize and physical parameter, it is very close to one. In practice, we find that using three sweeps of Jacobi relaxation on the Schur complement system is sufficient. Numerical results of two‐grid and V(1,1)‐cycle are presented, which show high efficiency of the proposed relaxation scheme and its robustness to physical parameters and the meshsize. Using a damping parameter equal to one gives almost the same convergence results as these for the optimal damping parameter. [ABSTRACT FROM AUTHOR]

The simulation of fluid dynamic problems often involves solving large-scale saddle-point systems. Their numerical solution with iterative solvers requires efficient preconditioners. Low-rank updates can adapt standard preconditioners to accelerate their convergence. We consider a multiplicative low-rank correction for pressure Schur complement preconditioners that is based on a (randomized) low-rank approximation of the error between the identity and the preconditioned Schur complement. We further introduce a relaxation parameter that scales the initial preconditioner. This parameter can improve the initial preconditioner as well as the update scheme. We provide an error analysis for the described update method. Numerical results for the linearized Navier–Stokes equations in a model for atmospheric dynamics on two different geometries illustrate the action of the update scheme. We numerically analyze various parameters of the low-rank update with respect to their influence on convergence and computational time. [ABSTRACT FROM AUTHOR]

The Helmholtz equation is a reliable model for acoustics in inviscid fluids. Real fluids, however, experience viscous and thermal dissipation that impact the sound propagation dynamics. The viscothermal losses primarily arise in the boundary region between the fluid and solid, the acoustic boundary layers. To preserve model accuracy for structures housing acoustic cavities of comparable size to the boundary layer thickness, meticulous consideration of these losses is essential. Recent research efforts aim to integrate viscothermal effects into acoustic boundary element methods (BEM). While the reduced discretization of BEM is advantageous over finite element methods, it results in fully populated system matrices whose conditioning deteriorates when extended with additional degrees of freedom to account for viscothermal dissipation. Solving such a linear system of equations becomes prohibitively expensive for large-scale applications, as only direct solvers can be used. This work proposes a revised formulation for the viscothermal BEM employing the Schur complement and a change of basis for the boundary coupling. We demonstrate that static condensation significantly improves the conditioning of the coupled problem. When paired with an iterative solution scheme, the approach lowers the algorithmic complexity and thus reduces the computational costs in terms of runtime and storage requirements. The results demonstrate the favorable performance of the new method, indicating its usability for applications of practical relevance in thermoviscous acoustics. [ABSTRACT FROM AUTHOR]

A transformed primal–dual (TPD) flow is developed for a class of nonlinear smooth saddle point systemThe flow for the dual variable contains a Schur complement which is strongly convex. Exponential stability of the saddle point is obtained by showing the strong Lyapunov property. Several TPD iterations are derived by implicit Euler, explicit Euler, implicit–explicit, and Gauss–Seidel methods with accelerated overrelaxation of the TPD flow. Generalized to the symmetric TPD iterations, linear convergence rate is preserved for convex–concave saddle point systems under assumptions that the regularized functions are strongly convex. The effectiveness of augmented Lagrangian methods can be explained as a regularization of the non-strongly convexity and a preconditioning for the Schur complement. The algorithm and convergence analysis depends crucially on appropriate inner products of the spaces for the primal variable and dual variable. A clear convergence analysis with nonlinear inexact inner solvers is also developed. [ABSTRACT FROM AUTHOR]

Nonlocal models allow for the description of phenomena which cannot be captured by classical partial differential equations. The availability of efficient solvers is one of the main concerns for the use of nonlocal models in real world engineering applications. We present a domain decomposition solver that is inspired by substructuring methods for classical local equations. In numerical experiments involving finite element discretizations of scalar and vectorial nonlocal equations of integrable and fractional type, we observe improvements in solution time of up to 14.6x compared to commonly used solver strategies. [ABSTRACT FROM AUTHOR]