Fast iterative solvers for boundary value problems on a local spherical region

Boundary value problems on local spherical regions arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Meshless methods using radial basis functions provide a simple way to construct numerical solutions with high accuracy. However, the linear systems arising from these methods are usually ill-conditioned, which poses a challenge for iterative solvers. We construct preconditioners based on an additive Schwarz method to accelerate the solution process for solving boundary value problems on local spherical regions. References D. Crowdy. Point vortex motion on the surface of a sphere with impenetrable boundaries. Physics of Fluids, 18:036602 (2006). doi:10.1063/1.2183627. A. E. Gill. Atmosphere-Ocean Dynamics, International Geophysics Series Volume 30. Academic, New York (1982). R. Kidambi and P. K. Newton. Point vortex motion on a sphere with solid boundaries. Physics of Fluids, 12:581 (2000). doi:10.1063/1.870263. Q. T. Le Gia, I. H. Sloan, and T. Tran. Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere. Math. Comp., 78:79--101 (2009). doi:10.1090/S0025-5718-08-02150-9. C. Muller. Spherical Harmonics, Vol. 17 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1966). M. V. Nezlin. Some remarks on coherent structures out of chaos in planetary atmospheres and oceans. Chaos, 4:109--111 (1994). doi:10.1063/1.165997. T. Tran, Q. T. Le Gia, I. H. Sloan, and E. P. Stephan. Preconditioners for pseudodifferential equations on the sphere with radial basis functions. Numer. Math., 115:141--163 (2009). doi:10.1007/s00211-009-0269-8

Matrix probing: a randomized preconditioner for the wave-equation Hessian

This paper considers the problem of approximating the inverse of the wave-equation Hessian, also called normal operator, in seismology and other types of wave-based imaging. An expansion scheme for the pseudodifferential symbol of the inverse Hessian is set up. The coefficients in this expansion are found via least-squares fitting from a certain number of applications of the normal operator on adequate randomized trial functions built in curvelet space. It is found that the number of parameters that can be fitted increases with the amount of information present in the trial functions, with high probability. Once an approximate inverse Hessian is available, application to an image of the model can be done in very low complexity. Numerical experiments show that randomized operator fitting offers a compelling preconditioner for the linearized seismic inversion problem.Comment: 21 pages, 6 figure

Localized bases for kernel spaces on the unit sphere

Approximation/interpolation from spaces of positive definite or conditionally positive definite kernels is an increasingly popular tool for the analysis and synthesis of scattered data, and is central to many meshless methods. For a set of $N$ scattered sites, the standard basis for such a space utilizes $N$ \emph{globally} supported kernels; computing with it is prohibitively expensive for large $N$. Easily computable, well-localized bases, with "small-footprint" basis elements - i.e., elements using only a small number of kernels -- have been unavailable. Working on \sphere, with focus on the restricted surface spline kernels (e.g. the thin-plate splines restricted to the sphere), we construct easily computable, spatially well-localized, small-footprint, robust bases for the associated kernel spaces. Our theory predicts that each element of the local basis is constructed by using a combination of only $\mathcal{O}((\log N)^2)$ kernels, which makes the construction computationally cheap. We prove that the new basis is $L_p$ stable and satisfies polynomial decay estimates that are stationary with respect to the density of the data sites, and we present a quasi-interpolation scheme that provides optimal $L_p$ approximation orders. Although our focus is on $\mathbb{S}^2$, much of the theory applies to other manifolds - $\mathbb{S}^d$, the rotation group, and so on. Finally, we construct algorithms to implement these schemes and use them to conduct numerical experiments, which validate our theory for interpolation problems on $\mathbb{S}^2$ involving over one hundred fifty thousand data sites.Comment: This article supersedes arXiv:1111.1013 "Better bases for kernel spaces," which proved existence of better bases for various kernel spaces. This article treats a smaller class of kernels, but presents an algorithm for constructing better bases and demonstrates its effectiveness with more elaborate examples. A quasi-interpolation scheme is introduced that provides optimal linear convergence rate

Eigenvalues of the truncated Helmholtz solution operator under strong trapping

For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that a) the finite-element method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and b) the location of eigenvalues, and especially near-zero ones, plays a key role in understanding how iterative solvers such as the generalised minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finite-element method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretisations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [Marchand, Galkowski, Spence, Spence, 2021])

Mathematical modelling and numerical simulations in physical geodesy

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