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BOUNDARY QUASIORTHOGONALITY AND SHARP INCLUSION BOUNDS FOR LARGE DIRICHLET EIGENVALUES
"... Abstract. We study eigenfunctions φj and eigenvalues Ej of the Dirichlet Laplacian on a bounded domain Ω ⊂ R n with piecewise smooth boundary. We bound the distance between an arbitrary parameter E> 0 and the spectrum {Ej} in terms of the boundary L 2norm of a normalized trial solution u of the Hel ..."
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Abstract. We study eigenfunctions φj and eigenvalues Ej of the Dirichlet Laplacian on a bounded domain Ω ⊂ R n with piecewise smooth boundary. We bound the distance between an arbitrary parameter E> 0 and the spectrum {Ej} in terms of the boundary L 2norm of a normalized trial solution u of the Helmholtz equation ( ∆ + E)u = 0. We also bound the L 2norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all E greater than a small constant, and improve upon the bestknown bounds of Moler–Payne by a factor of the wavenumber √ E. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly starshaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasiorthogonality property of the boundary normal derivatives of the eigenmodes (Theorem 1.3 below), of interest in its own right. Namely, the operator norm of the sum of rank 1 operators ∂nφj〈∂nφj, · 〉 over all Ej in a spectral window of width √ E — a sum with about E (n−1)/2 terms — is at most a constant factor (independent of E) larger than the operator norm of any one individual term.
PERTURBATIVE ANALYSIS OF THE METHOD OF PARTICULAR SOLUTIONS FOR IMPROVED INCLUSION OF HIGHLYING DIRICHLET EIGENVALUES ∗
, 1952
"... Abstract. The Dirichlet eigenvalue or “drum ” problem in a domain Ω ⊂ R 2 becomes numerically challenging at high eigenvalue (frequency) E. In this regime the method of particular solutions (MPS) gives spectral accuracy for many domain shapes. It requires a number of degrees of freedom scaling as √ ..."
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Abstract. The Dirichlet eigenvalue or “drum ” problem in a domain Ω ⊂ R 2 becomes numerically challenging at high eigenvalue (frequency) E. In this regime the method of particular solutions (MPS) gives spectral accuracy for many domain shapes. It requires a number of degrees of freedom scaling as √ E, the number of wavelengths on the boundary, in contrast to direct discretization for which this scaling is E. Our main result is an inclusion bound on eigenvalues that is a factor O ( √ E)tighter than the classical bound of Moler–Payne and that is optimal in that it reflects the true slopes of curves appearing in the MPS. We also present an MPS variant that cures a normalization problem in the original method, while evaluating basis functions only on the boundary. This method is efficient at high frequencies, where we show that, in practice, our inclusion bound can give three extra digits of eigenvalue accuracy with no extra effort.
Eignets for function approximation on manifolds
, 909
"... Let X be a compact, smooth, connected, Riemannian manifold without boundary, G: X × X → R be P a kernel. Analogous to a radial basis function network, an eignet is an expression of the form M j=1 ajG(◦, yj), where aj ∈ R, yj ∈ X, 1 ≤ j ≤ M. We describe a deterministic, universal algorithm for constr ..."
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Let X be a compact, smooth, connected, Riemannian manifold without boundary, G: X × X → R be P a kernel. Analogous to a radial basis function network, an eignet is an expression of the form M j=1 ajG(◦, yj), where aj ∈ R, yj ∈ X, 1 ≤ j ≤ M. We describe a deterministic, universal algorithm for constructing an eignet for approximating functions in L p (µ; X) for a general class of measures µ and kernels G. Our algorithm yields linear operators. Using the minimal separation amongst the centers yj as the cost of approximation, we give modulus of smoothness estimates for the degree of approximation by our eignets, and show by means of a converse theorem that these are the best possible for every individual function. We also give estimates on the coefficients aj in terms of the norm of the eignet. Finally, we demonstrate that if any sequence of eignets satisfies the optimal estimates for the degree of approximation of a smooth function, measured in terms of the minimal separation, then the derivatives of the eignets also approximate the corresponding derivatives of the target function in an optimal manner.