Positive Definite Operator Research Articles

Two-level preconditioning for $$h$$-version boundary element approximation of hypersingular operator with GenEO

In the present contribution, we consider symmetric positive definite operators stemming from boundary integral equation (BIE), and we study a two-level preconditioner where the coarse space is built using local generalized eigenproblems in the overlap. We will refer to this coarse space as the GenEO coarse space.

A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means

It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted A#H-mean introduced by Kim, Lawson, and Lim [9].

Skew compressions of positive definite operators and matrices

The paper is devoted to results connecting the eigenvalues and singular values of operators composed by $P^\ast G P$ with those composed in the same way by $QG^{−1}Q^\ast$. Here $P +Q = I$ are skew complementary projections on a finite-dimensional Hilbert space and $G$ is a positive definite linear operator on this space. Also discussed are graph theoretic interpretations of one of the results.

A Class of Preconditioners Based on Positive-Definite Operator Splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations

After discretization by the finite volume method, the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure. The theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized matrix. Moreover, we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations, which is unconditionally convergent for any positive constant. Meanwhile, the iteration methods introduce a new preconditioner for Krylov subspace methods. Numerical experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner, compared with the existing approaches.

Regularized Divergences Between Covariance Operators and Gaussian Measures on Hilbert Spaces

This work presents an infinite-dimensional generalization of the correspondence between the Kullback–Leibler and Rényi divergences between Gaussian measures on Euclidean space and the Alpha Log-Determinant divergences between symmetric, positive definite matrices. Specifically, we present the regularized Kullback–Leibler and Rényi divergences between covariance operators and Gaussian measures on an infinite-dimensional Hilbert space, which are defined using the infinite-dimensional Alpha Log-Determinant divergences between positive definite trace class operators. We show that, as the regularization parameter approaches zero, the regularized Kullback–Leibler and Rényi divergences between two equivalent Gaussian measures on a Hilbert space converge to the corresponding true divergences. The explicit formulas for the divergences involved are presented in the most general Gaussian setting.

Quantum Hellinger distances revisited

This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. [Lett. Math. Phys. 109 (2019), 1777-1804] with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences, that are of the form $\phi(A,B)=\mathrm{Tr} \left((1-c)A + c B - A \sigma B \right),$ where $\sigma$ is an arbitrary Kubo-Ando mean, and $c \in (0,1)$ is the weight of $\sigma.$ We note that these divergences belong to the family of maximal quantum $f$-divergences, and hence are jointly convex and satisfy the data processing inequality (DPI). We derive a characterization of the barycenter of finitely many positive definite operators for these generalized quantum Hellinger divergences. We note that the characterization of the barycenter as the weighted multivariate $1/2$-power mean, that was claimed in the work of Bhatia et al. mentioned above, is true in the case of commuting operators, but it is not correct in the general case.

Functional inequalities for weighted Gamma distribution on the space of finite measures

Let $\mathbb{M} $ be the space of finite measures on a locally compact Polish space, and let $\mathcal{G} $ be the Gamma distribution on $\mathbb{M} $ with intensity measure $\nu \in \mathbb{M} $. Let $\nabla ^{ext}$ be the extrinsic derivative with tangent bundle $T\mathbb{M} = \cup _{\eta \in \mathbb{M} } L^{2}(\eta )$, and let $\mathcal{A} : T\mathbb{M} \rightarrow T\mathbb{M} $ be measurable such that $\mathcal{A} _{\eta }$ is a positive definite linear operator on $L^{2}(\eta )$ for every $\eta \in \mathbb{M} $. Moreover, for a measurable function $V$ on $\mathbb{M} $, let ${\mathrm{{d}} }{\mathcal{G} }^{V}= {\mathrm{{e}} }^{V}{\mathrm{{d}} }{\mathcal{G} }$. We investigate the Poincaré, weak Poincaré and super Poincaré inequalities for the Dirichlet form \[ \mathcal{E} _{\mathcal{A} ,V}(F,G):= \int _{\mathbb{M} }\langle \mathcal{A} _{\eta }\nabla ^{ext}F(\eta ), \nabla ^{ext}G(\eta )\rangle _{L^{2}(\eta )}\, {\mathrm{{d}} }{\mathcal{G} }^{V}(\eta ), \] which characterize various properties of the associated Markov semigroup. The main results are extended to the space of finite signed measures.

Concavity of some operator functions and Fréchet differential mappings

By using operator perspectives of regular operator mappings we obtain some operator concave (convex) functions and derive the concavity (convexity) of some trace functions associated with the Fréchet differential mapping of certain power functions. Especially, we explore the concavity of the Fréchet differential mapping x→dg(x)⁎df(x)−1 in positive definite invertible operators with f(t)=tp for 0<p≤1 and g(t)=tq for p≤q≤p+1, and also with f(t)=log⁡t and g(t)=tq for 0<q≤1.

Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations

In this paper, we study the initial value problem for a semilinear delay hyperbolic equation in Hilbert spaces with a self-adjoint positive definite operator. The mean theorem on the existence and uniqueness of a bounded solution of this differential problem for a semilinear hyperbolic equation with unbounded time delay term is established. In applications, the existence and uniqueness of bounded solutions of four problems for semilinear hyperbolic equations with time delay in unbounded term are obtained. For the approximate solution of this abstract differential problem, the two-step difference scheme of a first order of accuracy is presented. The mean theorem on the existence and uniqueness of a uniformly bounded solution of this difference scheme with respect to time stepsize is established. In applications, the existence and uniqueness of a uniformly bounded solutions with respect to time and space stepsizes of difference schemes for four semilinear partial differential equations with time delay in unbounded term are obtained. In general, it is not possible to get the exact solution of semilinear hyperbolic equations with unbounded time delay term. Therefore, numerical results for the solution of difference schemes for one and two dimensional semilinear hyperbolic equation with time delay are presented. Finally, some numerical examples are given to confirm the theoretical analysis.

Efficient representation of the linear density-density response function.

We present a thorough derivation of the mathematical foundations of the representation of the molecular linear electronic density-density response function in terms of a computationally highly efficient moment expansion. Our new representation avoids the necessities of computing and storing numerous eigenfunctions of the response kernel by means of a considerable dimensionality reduction about from 103 to 101 . As the scheme is applicable to any compact, self-adjoint, and positive definite linear operator, we present a general formulation, which can be transferred to other applications with little effort. We also present an explicit application, which illustrates the actual procedure for applying the moment expansion of the linear density-density response function to a water molecule that is subject to a varying external perturbation potential. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.

Alpha-Beta Log-Determinant Divergences Between Positive Definite Trace Class Operators

This work presents a parametrized family of divergences, namely Alpha-Beta Log-Determinant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Alpha-Beta Log-Determinant divergences between symmetric, positive definite matrices to the infinite-dimensional setting. The family of Alpha-Beta Log-Det divergences is highly general and contains many divergences as special cases, including the recently formulated infinite-dimensional affine-invariant Riemannian distance and the infinite-dimensional Alpha Log-Det divergences between positive definite unitized trace class operators. In particular, it includes a parametrized family of metrics between positive definite trace class operators, with the affine-invariant Riemannian distance and the square root of the symmetric Stein divergence being special cases. For the Alpha-Beta Log-Det divergences between covariance operators on a Reproducing Kernel Hilbert Space (RKHS), we obtain closed form formulas via the corresponding Gram matrices.

Canonical Divergence for Flat α-Connections: Classical and Quantum

A recent canonical divergence, which is introduced on a smooth manifold endowed with a general dualistic structure , is considered for flat -connections. In the classical setting, we compute such a canonical divergence on the manifold of positive measures and prove that it coincides with the classical -divergence. In the quantum framework, the recent canonical divergence is evaluated for the quantum -connections on the manifold of all positive definite Hermitian operators. In this case as well, we obtain that the recent canonical divergence is the quantum -divergence.

Infinite-dimensional Log-Determinant divergences between positive definite Hilbert–Schmidt operators

The current work generalizes the author’s previous work on the infinite-dimensional Alpha Log-Determinant (Log-Det) divergences and Alpha-Beta Log-Det divergences, defined on the set of positive definite unitized trace class operators on a Hilbert space, to the entire Hilbert manifold of positive definite unitized Hilbert–Schmidt operators. This generalization is carried out via the introduction of the extended Hilbert–Carleman determinant for unitized Hilbert–Schmidt operators, in addition to the previously introduced extended Fredholm determinant for unitized trace class operators. The resulting parametrized family of Alpha-Beta Log-Det divergences is general and contains many divergences between positive definite unitized Hilbert–Schmidt operators as special cases, including the infinite-dimensional affine-invariant Riemannian distance and the infinite-dimensional generalization of the symmetric Stein divergence.

Theory of Integral Equations for Axisymmetric Scattering by a Disk

A theory of integral equations for radial currents in the axisymmetric problem of scattering by a disk is constructed. The theory relies on the extraction of the principal part of a continuously invertible operator and on the proof of its positive definiteness. Existences and uniqueness theorems are obtained for the problem. An orthonormal basis is constructed for the energy space of the positive definite operator. Each element of the basis on the boundary behaves in the same manner as the unknown function. The structure of the matrix of the integral operator in this basis is studied. It is found that the principal part has an identity matrix, while the matrix of the next operator is tridiagonal.

The Elastic Polarization Matrix for a Junction of Isotropic Half-Strips

We introduce an elastic polarization matrix M for a junction Ξ of isotropic homogeneous half-strips Π1, . . . ,ΠJ. Thematrix M is used to describe the boundary layer phenomenon near nodes of elastic lattices and is formed by coefficients in expansions at infinity of special solutions to the problem of elasticity theory for the body Ξ. It is shown that the 3J × 3J-matrix M is symmetric and degenerate on a subspace of dimension three, but, under the “correct” choice of local coordinates in the half-strips, it corrsponds to a positive definite operator on the orthogonal complement of this subspace.

Operator-adapted wavelets for finite-element differential forms

We introduce in this paper an operator-adapted multiresolution analysis for finite-element differential forms. From a given continuous, linear, bijective, and self-adjoint positive-definite operator L, a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a fine-to-coarse fashion and in quasilinear time. The resulting wavelets are L-orthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that its stiffness matrix becomes block-diagonal, with uniformly well-conditioned and sparse blocks. Because our approach applies to arbitrary differential p-forms, we can derive both scalar-valued and vector-valued wavelets block-diagonalizing a prescribed operator. We also discuss the generality of the construction by pointing out that it applies to various types of computational grids, offers arbitrary smoothness orders of basis functions and wavelets, and can accommodate linear differential constraints such as divergence-freeness. Finally, we demonstrate the benefits of the corresponding operator-adapted multiresolution decomposition for coarse-graining and model reduction of linear and non-linear partial differential equations.

Least squares preconditioning for mixed methods with nonconforming trial spaces

ABSTRACT We consider a preconditioning technique for mixed methods with a conforming test space and a nonconforming trial space. Our method is based on the classical saddle point disccretization theory for mixed methods and the theory of preconditioning symmetric positive definite operators. Efficient iterative processes for solving the discrete mixed formulations are proposed and choices for discrete compatible spaces are provided. For discretization, a basis is needed only for the test spaces and assembly of a global saddle point system is avoided. We provide 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 and constants of the pair of discrete spaces. We focus on applications to elliptic PDEs with discontinuous coefficients. Numerical results for two- and three-dimensional domains are included to support the proposed method.

Some upper bounds for the Berezin number of Hilbert space operators

In this paper, we obtain some Berezin number inequalities based on the definition of Berezin symbol. Among other inequalities, we show that if A,B be positive definite operators in B(H), and A#B is the geometric mean of them, then ber2(A#B) ? ber (A2+B2/2)- 1/2 inf ????(?k); where ?(?k?) = ?(A-B)?k?,?k??2, and ?k? is the normalized reproducing kernel of the space H for ? belong to some set.

Positive definite indistinguishability operators

Two geometric characterizations of positive definite reflexive and symmetric fuzzy relations are provided based on the embeddability of their naturally associated pseudodistances with respect to the continuous Archimedean t-norms with additive generators t(x)=arccos⁡x and t(x)=1−x.Due to the important role of the t-norm Tarccos⁡x with additive generator t(x)=arccos⁡x in these characterizations, different characterizations of it are given. In particular, a left-continuous t-norm T is Tarccos⁡x if and only if the rank of the matrices of all one-dimensional indistinguishability operators with respect to T are less than or equal to 2.

A Stable Difference Scheme for a Third-Order Partial Differential Equation

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. In applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary value problems for third order partial differential equations are obtained. Numerical results for oneand two-dimensional third order partial differential equations are provided.