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This study is concerned with the pointwise stabilization for a star-shaped network of variable coefficients strings connected at the common node by a point mass and subject to boundary feedback dampings at all extreme nodes. It is shown that the closed-loop system has a sequence of generalized eigenfunctions which forms a Riesz basis for the state Hilbert space. As a consequence, the spectrum-determined growth condition fulfills. In the meanwhile, the asymptotic expression of the spectrum is presented, and the exponential stability of the system is obtained by giving the optimal decay rate. We prove also that a phenomenon of lack of uniform stability occurs in the absence of damper at one extreme node. This paper reconfirmed the main stability results given by Hansen and Zuazua [SIAM J. Control Optim., 33 (1995), 1357-1391] in a very particular case. [ABSTRACT FROM AUTHOR]

In this paper we provide a generalization of the Douglas--Rachford splitting (DRS) and the primal-dual algorithm [L. Condat, J. Optim. Theory Appl., 158 (2013), pp. 460-479; B. C. V\~u, Adv. Comput. Math., 38 (2013), pp. 667-681] for solving monotone inclusions in a real Hilbert space involving a general linear operator. The proposed method allows for primal and dual nonstandard metrics and activates the linear operator separately from the monotone operators appearing in the inclusion. In the simplest case when the linear operator has full range, it reduces to classical DRS. Moreover, the weak convergence of primal-dual sequences to a Kuhn--Tucker point is guaranteed, generalizing the main result in [B. F. Svaiter, SIAM J. Control Optim., 49 (2011), pp. 280-287]. Inspired by [D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, M. Fortin and R. Glowinski, eds., Stud. Math. Appl. 15, North-Holland, Amsterdam, 1983, pp. 299-331], we also derive a new split alternating direction method of multipliers (SADMM) by applying our method to the dual of a convex optimization problem involving a linear operator which can be expressed as the composition of two linear operators. The proposed SADMM activates one linear operator implicitly and the other one explicitly, and we recover ADMM when the latter is set as the identity. Connections and comparisons of our theoretical results with respect to the literature are provided for the main algorithm and SADMM. The flexibility and efficiency of both methods is illustrated via numerical simulations in total variation image restoration and a sparse minimization problem. [ABSTRACT FROM AUTHOR]

This paper studies the J-method of [E. Jarlebring, S. Kvaal, W. Michiels. SIAM J. Sci. Comput. 36-4:A1978-A2001, 2014] for nonlinear eigenvector problems in a general Hilbert space framework. This is the basis for variational discretization techniques and a mesh-independent numerical analysis. A simple modification of the method mimics an energy-decreasing discrete gradient flow. In the case of the Gross–Pitaevskii eigenvalue problem, we prove global convergence towards an eigenfunction for a damped version of the J-method. More importantly, when the iterations are sufficiently close to an eigenfunction, the damping can be switched off and we recover a local linear convergence rate previously known from the discrete setting. This quantitative convergence analysis is closely connected to the J-method's unique feature of sensitivity with respect to spectral shifts. Contrary to classical gradient flows, this allows both the selective approximation of excited states as well as the amplification of convergence beyond linear rates in the spirit of the Rayleigh quotient iteration for linear eigenvalue problems. These advantageous convergence properties are demonstrated in a series of numerical experiments involving exponentially localized states under disorder potentials and vortex lattices in rotating traps. [ABSTRACT FROM AUTHOR]

Reduced model spaces, such as reduced bases and polynomial chaos, are linear spaces Vn of finite dimension n which are designed for the efficient approximation of certain families of parametrized PDEs in a Hilbert space V. The manifold M that gathers the solutions of the PDE for all admissible parameter values is globally approximated by the space Vn with some controlled accuracy εn, which is typically much smaller than when using standard approximation spaces of the same dimension such as finite elements. Reduced model spaces have also been proposed in [Y. Maday et al., Internat. J. Numer. Methods Ergrg., 102 (2015), pp. 933-965] as a vehicle to design a simple linear recovery algorithm of the state u ∈ M corresponding to a particular solution instance when the values of parameters are unknown but a set of data is given by m linear measurements of the state. The measurements are of the form ℓj(u), j = 1, ..., m, where the ℓj are linear functionals on V. The analysis of this approach in [P. Binev et al., SIAM/ASA J. Uncertain. Quantif., 5 (2017), pp. 1-29] shows that the recovery error is bounded by μnεn, where μn = μ(Vn, W) is the inverse of an inf-sup constant that describe the angle between Vn and the space W spanned by the Riesz representers of (ℓ1, ..., ℓm). A reduced model space which is efficient for approximation might thus be ineffective for recovery if μn is large or infinite. In this paper, we discuss the existence and effective construction of an optimal reduced model space for this recovery method. We extend our search to affine spaces which are better adapted than linear spaces for various purposes. Our basic observation is that this problem is equivalent to the search of an optimal affine algorithm for the recovery of M in the worst case error sense. This allows us to perform our search by a convex optimization procedure. Numerical tests illustrate that the reduced model spaces constructed from our approach perform better than the classical reduced basis spaces. [ABSTRACT FROM AUTHOR]

In [L. Li, M. Ni, and W. Zheng, SIAM J. Sci. Comput., 41 (2019), pp. B796-B815] a charge-conservative finite element method is proposed for solving inductionless and incompressible magnetohydrodynamic (MHD) equations. The purpose of this paper is to propose a robust solver for the discrete problem. Using the framework of field-of-values-equivalence, we first study the preconditioned Krylov space method for the continuous problem in the setting of Hilbert spaces. The algebraic preconditioner for the discrete problem is then obtained by representing the preconditioner for the continuous problem in finite element spaces. By three numerical examples, the optimality of the solver to the number of unknowns is demonstrated for both stationary and time-dependent MHD problems. [ABSTRACT FROM AUTHOR]

In this paper, to find a common element of the fixed point set of common fixed points of a countable family of nonexpansive mappings and the solution set of the variational inequality for α-inverse-strongly monotone, we introduce an iterative approximation method in a real Hilbert space. Then the strong convergence theorem is proved under some appropriate conditions imposed on the parameters. This result extended and improved the corresponding results of Yao and Yao [15] and many others. [ABSTRACT FROM AUTHOR]

We study the issue of strong convergence of inexact proximal point algorithms (introduced by Rockafellar in [SIAM J. Control Optim., 14 (1976), pp. 877-898]) for maximal monotone operators on Hilbert spaces. A unified global/local strong convergence of inexact proximal point algorithms is established under the Hölder metrically subregular condition. Furthermore, quantitative estimates on the convergence rate of inexact proximal point algorithms are also provided. Applying to the special case of the classical (exact) proximal point algorithm, our results improve the corresponding ones in [G. Li and B. S. Mordukhovich, SIAM J. Optim., 22 (2012), pp. 1655-1684]. Finally, as applications, global/local strong convergence and estimates on the convergence rate of inexact proximal point algorithms for optimization problems are presented. [ABSTRACT FROM AUTHOR]

In a two-player noncooperative game framework, we deal with the affine relaxations of the best response algorithm (where a player's strategy is a best response to the strategy of the other player that comes from the previous step), motivated by the first results obtained for convex relaxations in zero-sum games (J. Morgan, Int. J. Comput. Math., 4 (1974), pp. 143-175) and for nonconvex affine relaxations in non-zero-sum games (F. Caruso, M. C. Ceparano, and J. Morgan, SIAM J. Optim., 30 (2020), pp. 1638-1663). In order to be able to specify the convergence of any type of affine relaxation of the best response algorithm, we define a new class of games, called ratiobounded games. This class contains games broadly used in literature (such as weighted potential and zero-sum games), both in finite and infinite dimensional settings. Its definition relies on a unifying property and on three associate key parameters explicitly related to the data. Depending on how the parameters are ordered, we provide a classification of the ratio-bounded games in four subclasses such that, for each of them, the following issues are answered when the strategy sets are real Hilbert spaces: existence and uniqueness of the Nash equilibria, global convergence of the affine relaxations of the best response algorithm, estimation of the related errors, determination of the algorithm with the highest speed of convergence, and comparison with the known results. Moreover, the investigation is supplemented by illustrating numerical examples and by describing a black-box model with a first-order oracle for an implementation of the algorithm. [ABSTRACT FROM AUTHOR]

Suppose that a linear bounded operator B on a Hilbert space exhibits at least linear GMRES convergence, i.e., there exists Mp < 1 such that the GMRES residuals fulfill ||rfc|| < MB||rk-1|| f°r every initial residual ro and step k E N. We prove that GMRES with a compactly perturbed operator A = B + C admits the bound ||rkll < rij=i (Mb + (1 + Mb) IM_1II (C*)), i.e., the singular values αj(C) control the departure from the bound for the unperturbed problem. This result can be seen as an extension of [I. Moret, SIAM J. Numer. Anal., 34 (1997), pp. 513-516], where only the case B = XI is considered. In this special case Mp = 0 convergence is superlinear. [ABSTRACT FROM AUTHOR]

In our previous work [J. R. Singler, SIAM J. Numer. Anal., 52 (2014), pp. 852-876], we considered the proper orthogonal decomposition (POD) of time varying PDE solution data taking values in two different Hilbert spaces. We considered various POD projections of the data and obtained new results concerning POD projection errors and error bounds for POD reduced order models of PDEs. In this work, we improve on our earlier results concerning POD projections by extending to a more general framework that allows for nonorthogonal POD projections and seminorms. We obtain new exact error formulas and convergence results for POD data approximation errors, and also prove new pointwise convergence results and error bounds for POD projections. We consider both the discrete and continuous cases of POD. We also apply our results to several example problems and show how the new results improve on previous work. [ABSTRACT FROM AUTHOR]

We prove that a cyclic coordinate fixed point algorithm for nonexpansive mappings when the underlying Hilbert space is decomposed into a Cartesian product of finitely many block spaces is weakly convergent to a fixed point of the mapping under investigation. Our result relaxes a condition imposed on the stepsizes of Theorem 3.4 of Chow, et al [Chow, Y. T., Wu, T. and Yin, W., Cyclic coordinate-update algorithms for fixed-point problems: analysis and applcations, SIAM J. Sci. Comput., 39 (2017), No. 4, A1280-A1300]. [ABSTRACT FROM AUTHOR]