Positive-Definite Operator-Valued Kernels and Integral Representations

The equation , where , with being a K-positive definite operator and being a linear operator, is solved in a Banach space. Our scheme provides a generalization to the so-called method of moments studied in a Hilbert space by Petryshyn (1962), as well as Lax and Milgram (1954). Furthermore, an application of the inverse function theorem provides simultaneously a general solution to this equation in some neighborhood of a point , where is Fréchet differentiable and an iterative scheme which converges strongly to the unique solution of this equation.

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Iterative solutions ofK-positive definite operator equations in real uniformly smooth Banach spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201005919 ◽

2001 ◽

Vol 27 (3) ◽

pp. 155-160

Author(s):

Zeqing Liu ◽

Shin Min Kang ◽

Jeong Sheok Ume

Keyword(s):

Banach Space ◽

Iterative Method ◽

Banach Spaces ◽

Positive Definite ◽

Operator Equations ◽

Uniformly Smooth ◽

Definite Operator ◽

Iterative Solutions ◽

Uniformly Smooth Banach Spaces ◽

Positive Definite Operator

LetXbe a real uniformly smooth Banach space and letT:D(T)⫅X→Xbe aK-positive definite operator. Under suitable conditions we establish that the iterative method by Bai (1999) converges strongly to the unique solution of the equationTx=f,f∈X. The results presented in this paper generalize the corresponding results of Bai (1999), Chidume and Aneke (1993), and Chidume and Osilike (1997).

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A rapid monotonically convergent iteration algorithm for quantum optimal control over the expectation value of a positive definite operator

The Journal of Chemical Physics ◽

10.1063/1.476575 ◽

1998 ◽

Vol 109 (2) ◽

pp. 385-391 ◽

Cited By ~ 222

Author(s):

Wusheng Zhu ◽

Herschel Rabitz

Keyword(s):

Optimal Control ◽

Positive Definite ◽

Iteration Algorithm ◽

Expectation Value ◽

Definite Operator ◽

Quantum Optimal Control ◽

Positive Definite Operator

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A survey on numerical methods for spectral Space-Fractional diffusion problems

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0080 ◽

2020 ◽

Vol 23 (6) ◽

pp. 1605-1646

Author(s):

Stanislav Harizanov ◽

Raytcho Lazarov ◽

Svetozar Margenov

Keyword(s):

Finite Element ◽

Numerical Solution ◽

Rational Approximation ◽

Positive Definite ◽

Second Order ◽

Local Operator ◽

Uniform Mesh ◽

Element Approximation ◽

Definite Operator ◽

Positive Definite Operator

AbstractThe survey is devoted to numerical solution of the equation $ {\mathcal A}^\alpha u=f $, 0 < α<1, where $ {\mathcal A} $ is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω in ℝd. The fractional power $ {\mathcal A}^\alpha $ is a non-local operator and is defined though the spectrum of $ {\mathcal A} $. Due to growing interest and demand in applications of sub-diffusion models to physics and engineering, in the last decade, several numerical approaches have been proposed, studied, and tested. We consider discretizations of the elliptic operator $ {\mathcal A} $ by using an N-dimensional finite element space Vh or finite differences over a uniform mesh with N points. In the case of finite element approximation we get a symmetric and positive definite operator $ {\mathcal A}_h: V_h \to V_h $, which results in an operator equation $ {\mathcal A}_h^{\alpha} u_h = f_h $ for uh ∈ Vh.The numerical solution of this equation is based on the following three equivalent representations of the solution: (1) Dunford-Taylor integral formula (or its equivalent Balakrishnan formula, (2.5), (2) extension of the a second order elliptic problem in Ω × (0, ∞)⊂ ℝd+1 [17,55] (with a local operator) or as a pseudo-parabolic equation in the cylinder (x, t) ∈ Ω × (0, 1), [70, 29], (3) spectral representation (2.6) and the best uniform rational approximation (BURA) of zα on [0, 1], [37,40].Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of $ {\mathcal A}_h^{-\alpha} $. In this paper we present the main ideas of these methods and the corresponding algorithms, discuss their accuracy, computational complexity and compare their efficiency and robustness.

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