Draft:Gutleb–Olver operator
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GO operator (short for Gutleb–Olver operator) is a type of Volterra integral operator defined over the unit interval and studied in the context of operator pseudospectra. It is named after mathematicians Timon Gutleb and Sheehan Olver, who proposed a high-accuracy numerical method for computing such integral operators using sparse spectral techniques[1].
Definition
[edit]The GO operator, denoted \( V_{\text{GO}} \), is defined by:
![{\displaystyle (V_{\text{GO}}u)(s)=\int _{0}^{s}e^{-10(s-{\frac {1}{3}})^{2}-10(t-{\frac {1}{3}})^{2}}u(t)\,dt,\quad s\in [0,1],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c5e99a62cabbd1ca49b8d16fa78999b5e75e92)
This operator acts on functions in and has a smooth, non-convolution kernel that is sharply peaked near .
![{\displaystyle L^{2}([0,1])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13c2586b254365ca72d662d9e6485fdee8026912)
Properties
[edit]The GO operator is a compact, non-self-adjoint Volterra integral operator. Its pseudospectral behavior has been shown to closely resemble that of the Wiener–Hopf integral operator, which is well known for its highly non-normal structure.
In contrast to many classical examples, the ε-pseudospectra of the GO operator exhibit rich structure and highlight challenges in numerical computation due to operator non-normality. The GO operator was used as a test case for a new method of computing operator pseudospectra using ultraspherical spectral techniques with adaptive degree-of-freedom refinement.
Historical Context
[edit]The GO operator was introduced in the work of Deng, Liu, and Xu (2024)[2] as a test case for comparing pseudospectral methods. The naming acknowledges the foundational work by Timon S. Gutleb and Sheehan Olver, who developed sparse spectral methods for Volterra integral equations using orthogonal polynomials on the triangle, particularly suitable for efficient and accurate resolution of such operators.
Related Work
[edit]Gutleb and Olver's earlier work presents a method to solve both first- and second-kind Volterra integral equations by leveraging Jacobi polynomials and operator-valued Clenshaw algorithms to achieve exponential convergence and sparse representations of the integral operator.
See also
[edit]- Volterra integral equation
- Pseudospectrum
- Spectral method
- Fredholm integral equation
- Wiener–Hopf integral equation
References
[edit]- K. Deng, X. Liu, and K. Xu, “A Continuous Approach to Computing Operator Pseudospectra,” 2024. arXiv:2405.03285 "A Continuous Approach to Computing Operator Pseudospectra". arXiv preprint. 2024.. See Section 5.4.
- T. S. Gutleb and S. Olver, “A Sparse Spectral Method for Volterra Integral Equations Using Orthogonal Polynomials on the Triangle,” SIAM Journal on Numerical Analysis, vol. 58, no. 3, pp. 1993–2018, 2020. doi:10.1137/19M1267441.
- ^ Gutleb, Timon S.; Olver, Sheehan (January 2020). "A Sparse Spectral Method for Volterra Integral Equations Using Orthogonal Polynomials on the Triangle". SIAM Journal on Numerical Analysis. 58 (3): 1993–2018. doi:10.1137/19M1267441. ISSN 0036-1429.
- ^ Deng, Kuan; Liu, Xiaolin; Xu, Kuan (2025-03-12), A continuous approach to computing the pseudospectra of linear operators, arXiv:2405.03285, retrieved 2025-05-02