Symmetric Saddle Point Research Articles

Finite element methods for multicomponent convection-diffusion

Abstract We develop finite element methods for coupling the steady-state Onsager–Stefan–Maxwell (OSM) equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations is challenging: the formulation must balance physical relevance of the variables and boundary data, regularity assumptions, tractability of the analysis, enforcement of thermodynamic constraints, ease of discretization and extensibility to the transient, anisothermal and nonideal settings. To resolve these competing goals, we employ two augmentations: the first enforces the definition of mass-average velocity in the OSM equations, while its dual modifies the Stokes momentum equation to enforce symmetry. Remarkably, with these augmentations we achieve a Picard linearization of symmetric saddle point type, despite the equations not possessing a Lagrangian structure. Exploiting structure mandated by linear irreversible thermodynamics, we prove the inf-sup condition for this linearization, and identify finite element function spaces that automatically inherit well-posedness. We verify our error estimates with a numerical example, and illustrate the application of the method to nonideal fluids with a simulation of the microfluidic mixing of hydrocarbons.

Deflation for the Off-Diagonal Block in Symmetric Saddle Point Systems

Deflation for the Off-Diagonal Block in Symmetric Saddle Point Systems

Equivalence between belief propagation instability and transition to replica symmetry breaking in perceptron learning systems

The binary perceptron is a fundamental model of supervised learning for nonconvex optimization, which is a root of the popular deep learning. The binary perceptron is able to achieve a classification of random high-dimensional data based on the marginal probabilities of binary synapses. The relationship between the belief propagation instability and the equilibrium analysis of the model remains elusive. Here, we establish the relationship by showing that the instability condition around the belief propagation fixed point is identical to the instability for breaking the replica symmetric saddle-point solution of the free-energy function. Therefore our analysis will hopefully provide insight towards other learning systems in bridging the gap between nonconvex learning dynamics and statistical mechanics properties of more complex neural networks.

A null-space approach for large-scale symmetric saddle point systems with a small and non zero (2, 2) block

Null-space methods have long been used to solve large sparse n × n symmetric saddle point systems of equations in which the (2, 2) block is zero. This paper focuses on the case where the (1, 1) block is ill conditioned or rank deficient and the k × k (2, 2) block is non zero and small (k ≪ n). Additionally, the (2, 1) block may be rank deficient. Such systems arise in a range of practical applications. A novel null-space approach is proposed that transforms the system matrix into a nicer symmetric saddle point matrix of order n that has a non zero (2, 2) block of order at most 2k and, importantly, the (1, 1) block is symmetric positive definite. Success of any null-space approach depends on constructing a suitable null-space basis. We propose methods for wide matrices having far fewer rows than columns with the aim of balancing stability of the transformed saddle point matrix with preserving sparsity in the (1, 1) block. Linear least squares problems that contain a small number of dense rows are an important motivation and are used to illustrate our ideas and to explore their potential for solving large-scale systems.

The ULT-HSS hybrid iteration method for symmetric saddle point problems

This paper proposes a hybrid iteration method for solving symmetric saddle point problem arising in CFD. It is an implicit alternative direction iteration method and named as the ULT-HSS (upper and lower triangular, Hermitian and skew- Hermitian splitting) method. The convergence analysis is provided, and the necessary and sufficient conditions are given for the convergence of the method. Some practical approaches are formulated for setting the optimal parameter of the method. Numerical experiments are given to show its efficiency.

Optimal accelerated SOR-like (ASOR) method for singular symmetric saddle point problems

In a recent paper a new iterative method for the solution of the nonsingular symmetric saddle point problem was proposed (Njeru and Guo, 2016). The ASOR method belongs to the family of the SOR-like methods and uses two parameters α and ω. Convergence intervals for the parameters involved were found. In the present work we analyze and study an extension of the above problem to the singular case, and determine optimal values for the two parameters as well as for the semi-convergence factor of the ASOR method. Numerical results are presented to show the efficiency of the optimal singular ASOR method.

Analysis of relaxed nonlinear inexact Uzawa algorithm for symmetric saddle point problems

In this paper, we propose a relaxed nonlinear inexact Uzawa algorithm (RNIU) for solving the symmetric saddle point problems. It is an inner–outer iteration method with the inner iterations using variable accuracy for solving the approximate Schur complement system. The variable relaxation parameter is introduced to improve the convergence. We give the convergence analysis of this relaxed algorithm with variable inner accuracy, based on a simple energy norm. Sufficient conditions are given for the convergence of RNIU, which slightly improve the existing convergence results for the nonlinear inexact Uzawa algorithm with uniform inner accuracy in the literature. A practical approach for setting the variable relaxed parameters is proposed, and numerical experiments are given to illustrate the efficiency and sensitivity of RNIU.

Non-perturbative probability distribution function for cosmological counts in cells

We present a non-perturbative calculation of the 1-point probability distribution function (PDF) for the spherically-averaged matter density field. The PDF is represented as a path integral and is evaluated using the saddle-point method. It factorizes into an exponent given by a spherically symmetric saddle-point solution and a prefactor produced by fluctuations. The exponent encodes the leading sensitivity of the PDF to the dynamics of gravitational clustering and statistics of the initial conditions. In contrast, the prefactor has only a weak dependence on cosmology. It splits into a monopole contribution which is evaluated exactly, and a factor corresponding to aspherical fluctuations. The latter is crucial for the consistency of the calculation: neglecting it would make the PDF incompatible with translational invariance. We compute the aspherical prefactor using a combination of analytic and numerical techniques. We demonstrate the factorization of spurious enhanced contributions of large bulk flows and their cancellation due to the equivalence principle. We also identify the sensitivity to the short-scale physics and argue that it must be properly renormalized. The uncertainty associated with the renormalization procedure gives an estimate of the theoretical error. For zero redshift, the precision varies from sub percent for moderate density contrasts to tens of percent at the tails of the distribution. It improves at higher redshifts. We compare our results with N-body simulation data and find an excellent agreement.

Convergence of Some Iterative Methods for Symmetric Saddle Point Linear Systems

We consider the iterative solution of linear systems with a symmetric saddle point system matrix. We address a family of solution techniques that exploit the knowledge of a preconditioner (or appro...

Saddle point least squares preconditioning of mixed methods

We present a simple way to discretize and precondition mixed variational formulations. Our theory connects with, and takes advantage of, the classical theory of symmetric saddle point problems and the theory of preconditioning symmetric positive definite operators. Efficient iterative processes for solving the discrete mixed formulations are proposed and choices for discrete spaces that are always compatible are provided. For the proposed discrete spaces and solvers, a basis is needed only for the test spaces and assembly of a global saddle point system is avoided. We prove sharp approximation properties for the discretization and iteration errors and also provide a sharp estimate for the convergence rate of the proposed algorithm in terms of the condition number of the elliptic preconditioner and the discrete inf−sup and sup−sup constants of the pair of discrete spaces.

A Potential-Based Integral Equation Method for Low-Frequency Electromagnetic Problems

In this paper, we propose a potential-based integral equation solver for low-frequency electromagnetic (EM) problems. In this formulation, the scalar potential ( $\Phi $ ) equation is solved in tandem with the vector potential ( $\textbf {A}$ ) equation. The resulting system is immune to low-frequency catastrophe and accurate in capturing the electrostatic and magnetostatic physics. The fast convergence of the new $\textbf {A}$ - $\Phi $ system, which is a typical symmetric saddle point problem, is made possible through the design of an appropriate left constraint preconditioner. Numerical examples validate the efficiency and stability of the novel formulation in solving both EM scattering and circuit problems over a wide frequency range up to very low frequencies.

A special parameterized inexact Uzawa algorithm for symmetric saddle point problem

In this paper, we consider a symmetric saddle point problem arising in the fluid dynamics. A special parameterized inexact Uzawa algorithm is proposed for solving the symmetric saddle point problem. The convergence of this special algorithm is considered. Sufficient conditions for the convergence are given. Numerical experiments resulting from stokes problem are presented to show the efficiency of the algorithm.

Saddle point least squares iterative solvers for the time harmonic Maxwell equations

We introduce a least squares method for discretizing the time-harmonic Maxwell equations written as a first order system. The method belongs to the recently introduced least squares approach called the Saddle Point Least Squares (SPLS) method (see Bacuta and Qirko 2015, 2017). It is related to two known methods: the Bramble and Pasciak least squares method for solving electromagnetic problems (Bramble and Pasciak, 2004; Bramble et al., 2005) and the minimum residual DPG method of Demkowicz and Gopalakrishnan (2010, 2013). The proposed discretization is based on reduction to a symmetric saddle point formulation on spaces of piecewise polynomial functions. The edge elements are avoided. A discrete test space is chosen first among standard conforming finite element spaces and the discrete trial space is chosen second as the image of the first order Maxwell differential operator on the discrete test space. For the proposed iterative processes, a nodal basis for the trial space is not required and the main inversion at each step is done only on the test space. The method is efficient for both convex and non-convex domains and for a various range of frequencies as shown by the numerical results.

A comparative study of null‐space factorizations for sparse symmetric saddle point systems

SummaryNull‐space methods for solving saddle point systems of equations have long been used to transform an indefinite system into a symmetric positive definite one of smaller dimension. A number of independent works in the literature have identified that we can interpret a null‐space method as a matrix factorization. We review these findings, highlight links between them, and bring them into a unified framework. We also investigate the suitability of using null‐space factorizations to derive sparse direct methods and present numerical results for both practical and academic problems.

A method for verifying the accuracy of numerical solutions of symmetric saddle point linear systems

A fast numerical verification method is proposed for evaluating the accuracy of numerical solutions for symmetric saddle point linear systems whose diagonal blocks of the coefficient matrix are semidefinite matrices. The method is based on results of an algebraic analysis of a block diagonal preconditioning. Some numerical experiments are present to illustrate the usefulness of the method.

Sparse Adaptive Tensor Galerkin Approximations of Stochastic PDE-Constrained Control Problems

For control problems constrained by linear elliptic or parabolic PDEs (partial differential equations) depending on countably many parameters, i.e., on $\sigma_j$ with $j\in\mathbb{N}$, we proved in [SIAM J. Control Optim., 51 (2013), pp. 2442--2471] the analytic parameter dependence of the state, the co-state, and the control. Moreover, we established that these functions allow expansions in terms of sparse tensorized generalized polynomial chaos (gpc) bases. Their sparsity was quantified in terms of $\mathfrak{p}$-summability of the coefficient sequences for some $0<\mathfrak{p}\leq1$. Resulting a priori estimates established the existence of an index set $\Lambda$, allowing for concurrent approximations of state, co-state, and control for which the gpc approximations attain rates of best $N$-term approximation. The regularity and $N$-term approximation results of our previous work serve as the analytical foundation for the development of adaptive Galerkin approximation methods in the present paper. Following the ideas and their realizations in several recent papers from Gittelson, Schwab, and coworkers for a single PDE, we construct deterministic adaptive Galerkin approximations of state, co-state, and control on the entire, possibly infinite-dimensional, parameter space. The starting point for these constructions are control problems formulated as abstract symmetric saddle point problems, as in our previous paper. Specifying this to adaptive wavelet-based schemes in space and time, we prove convergence as well as optimal complexity estimates, when compared to best $N$-term approximations.

A generalization of preconditioned parameterized inexact Uzawa method for indefinite saddle point problems

The parameterized inexact Uzawa methods have been used to solve some of the symmetric saddle point problems. In this paper, a new preconditioned parameterized inexact Uzawa method is presented to solve indefinite saddle point problems. After preconditioning, theoretical analyses show that the iteration method converges under certain conditions. So we propose three new algorithms based on these conditions. Numerical experiments are provided to show the effectiveness of the proposed preconditioner and all these algorithms have fantastic convergence rates by choosing optimal parameters.

Sharp stability and approximation estimates for symmetric saddle point systems

We establish sharp well posedness and approximation estimates for variational saddle point systems at the continuous level. The main results of this note have been known to be true only in the finite dimensional case. Known spectral results from the discrete case are reformulated and proved using a functional analysis view, making the proofs in both cases, discrete and continuous, less technical than the known discrete approaches. We focus on analyzing the special case when the form is bounded, symmetric, and coercive, and the mixed form is bounded and satisfies a standard or LBB condition. We characterize the spectrum of the symmetric operators that describe the problem at the continuous level. For a particular choice of the inner product on the product space of , we prove that the spectrum of the operator representing the system at continuous level is . As consequences of the spectral description, we find the minimal length interval that contains the ratio between the norm of the data and the norm of the solution, and prove explicit approximation estimates that depend only on the continuity constant and the continuous and the discrete condition constants

Statistical mechanics of the spherical hierarchical model with random fields

We study analytically the equilibrium properties of the spherical hierarchical model in the presence of random fields. The expression for the critical line separating a paramagnetic from a ferromagnetic phase is derived. The critical exponents characterising this phase transition are computed analytically and compared with those of the corresponding D-dimensional short-range model, leading to conclude that the usual mapping between one dimensional long-range models and D-dimensional short-range models holds exactly for this system, in contrast to models with Ising spins. Moreover, the critical exponents of the pure model and those of the random field model satisfy a relationship that mimics the dimensional reduction rule. The absence of a spin-glass phase is strongly supported by the local stability analysis of the replica symmetric saddle-point as well as by an independent computation of the free-energy using a renormalization-like approach. This latter result enlarges the class of random field models for which the spin-glass phase has been recently ruled out.

Modified accelerated parameterized inexact Uzawa method for singular and nonsingular saddle point problems

Recently, Bai and Wang (2008) [9] presented an efficient accelerated parameterized inexact Uzawa (APIU) method for solving both symmetric and nonsymmetric nonsingular saddle point problems. In this work, we propose a modified APIU (MAPIU) method for solving saddle point problems, which is an extension of the APIU iteration method and covers a series of existing iterative methods. The MAPIU method can be applied not only to the nonsingular case but also to the singular case. The convergence properties for the nonsingular saddle point problems and the semi-convergence properties for the singular one are carefully discussed under suitable restrictions. We prove that the MAPIU method has a wider convergence (or semi-convergence) region than the APIU method.