Optimal operator preconditioning for pseudodifferential boundary problems

Gimperlein, Heiko; Stocek, Jakub; Urzúa-Torres, Carolina

doi:10.1007/s00211-021-01193-9

Cited by 9 publications

(3 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recall that the theory of standard pseudodifferential operators on a smooth surface Γ can be viewed as a generalisation of Fourier analysis on the circle. The use of pseudodifferential properties in both the analysis and numerical analysis of boundary integral equations is both well established, see, e.g., [38,128,39,37,123,91,119,92,3,74], and current [40,17,67,6,2,4,64,23].…”

Section: Discussion Of the Ideas Behind The Proof Of Theorem 21mentioning

confidence: 99%

Does the Helmholtz boundary element method suffer from the pollution effect?

Galkowski¹,

Spence²

2022

Preprint

View full text Add to dashboard Cite

In d dimensions, approximating an arbitrary function oscillating with frequency k requires ∼ k d degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber k and in d dimensions) suffers from the pollution effect if, as k → ∞, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold (i.e., faster than k d for domain-based formulations, such as finite element methods, and k d−1 for boundary-based formulations, such as boundary element methods).It is well known that the h-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth h and keeping the polynomial degree p fixed) suffers from the pollution effect, and research over the last ∼ 30 years has resulted in a near-complete rigorous understanding of how quickly the number of degrees of freedom must grow with k (and how this depends on both p and properties of the scatterer).In contrast to the h-FEM, at least empirically, the h-version of the boundary element method (BEM) does not suffer from the pollution effect (recall that in the boundary element method the scattering problem is reformulated as an integral equation on the boundary of the scatterer, with this integral equation then solved numerically using a finite-element-type approximation space). However, the current best results in the literature on how quickly the number of degrees of freedom for the h-BEM must grow with k fall short of proving this.In this paper, we prove that the h-version of the Galerkin method applied to the standard second-kind boundary integral equations for solving the Helmholtz exterior Dirichlet problem does not suffer from the pollution effect when the obstacle is nontrapping (i.e., does not trap geometric-optic rays). While the proof of this result relies on using information about the large-k behaviour of Helmholtz solution operators, we show in the case when the obstacle is a 2-d ball how this result can be proved using only Fourier series and asymptotics of Hankel and Bessel functions.

show abstract

Section: Discussion Of the Ideas Behind The Proof Of Theorem 21mentioning

confidence: 99%

Does the Helmholtz boundary element method suffer from the pollution effect?

Galkowski¹,

Spence²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Fortunately, this is also needed in operator preconditioning [32]. Therefore, investigations of such operators have already taken place in the literature, see for example [47,26,27,20] and the references therein. Building on this wellestablished knowledge about operator preconditioning, it remains to come up with a suitable discretization.…”

Section: Discussionmentioning

confidence: 99%

Full operator preconditioning and the accuracy of solving linear systems

Mohr,

Nakatsukasa,

Urzúa-Torres

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error. Often, such systems arise from the discretization of operator equations with a large number of discrete variables. In this paper we show that the accuracy can be improved significantly if the equation is transformed before discretization, a process we call full operator preconditioning (FOP). It bears many similarities with traditional preconditioning for iterative methods but, crucially, transformations are applied at the operator level. We show that while condition-number improvements from traditional preconditioning generally do not improve the accuracy of the solution, FOP can. A number of topics in numerical analysis can be interpreted as implicitly employing FOP; we highlight (i) Chebyshev interpolation in polynomial approximation, and (ii) Olver-Townsend's spectral method, both of which produce solutions of dramatically improved accuracy over a naive problem formulation. In addition, we propose a FOP preconditioner based on integration for the solution of fourth-order differential equations with the finite-element method, showing the resulting linear system is well-conditioned regardless of the discretization size, and demonstrate its error-reduction capabilities on several examples. This work shows that FOP can improve accuracy beyond the standard limit for both direct and iterative methods.

show abstract

“…Indeed, preconditioning for fractional differential operators has attracted attention recently. We mention multigrid preconditioners, [AG17] based on uniformly refined mesh hierarchies and operator preconditioning, [Hip06,GSUT19,SvV19], which requires one to realize an operator of the opposite order. Another, classical technique is the framework of additive Schwarz preconditioners, analyzed in a BPX-setting with Fourier techniques in [BLN19].…”

Section: Introductionmentioning

confidence: 99%

On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

Faustmann¹,

Melenk²,

Parvizi³

2019

Preprint

View full text Add to dashboard Cite

We provide an endpoint stability result for Scott-Zhang type operators in Besov spaces. For globally continuous piecewise polynomials these are bounded from H 3/2 into B 3/2 2,∞ ; for elementwise polynomials these are bounded from H 1/2 into B 1/2 2,∞ . As an application, we obtain a multilevel decomposition based on Scott-Zhang operators on a hierarchy of meshes generated by newest vertex bisection with equivalent norms up to (but excluding) the endpoint case. A local multilevel diagonal preconditioner for the fractional Laplacian on locally refined meshes with optimal eigenvalue bounds is presented.

show abstract

Optimal operator preconditioning for pseudodifferential boundary problems

Cited by 9 publications

References 42 publications

Does the Helmholtz boundary element method suffer from the pollution effect?

Does the Helmholtz boundary element method suffer from the pollution effect?

Full operator preconditioning and the accuracy of solving linear systems

On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

Contact Info

Product

Resources

About