Stable Runge-Free, Gibbs-Free Rational Interpolation Scheme: A New Hope A New Hope Qiqi Wang

Quest for an adaptive, stable, fast convergent interpolation scheme ● Would be immensely useful in engineering, optimization, uncertainty quantification, etc. ● Radial basis function interpolation (Kriging): – Adaptive, – Unstable for fixed absolute shape parameter, – Algebraically convergent for fixed relative shape parameter. ● Grid based interpolation (Chebyshev, Smolyak) – Stable and geometrically convergent, – Not adaptive.

Runge's phenomenon Platte, Trefethan and Kuijlaars Impossibility of aproximating analytic functions from equispaced samples, Platte, Trefethan and Kuijlaars, 2010 – If a scheme tries to be geometrically convergent for all analytic functions on uniform grid, it must be very ill- conditioned.

Gibbs phenomenon ● Existing methods for dealing with Gibbs oscillations either rely on knowing the location of the discontinuity (e.g. reconstruction), or sacrifices geometric rate of convergence.

Rational blending of accurate but unstable global approximations ● p k (x): participant approximations, ● q k (x): an error estimator of p k (x). ● g(x) is a weighted average of approximations; the (locally) most accurate approximation is weighted most heavily.

When is g(x) an interpolant? ● Theorem: g(x) is an interpolant if for every data points, at least one p k (x) interpolates the data point, with q k (x) at the point. ● Similar ideas from: WENO, Floater-Hormann

Special case with polynomial participants ● Participant pproximations p k (x) are polynomial interpolants on all contiguous subsets of data points. ● Error estimate: ● where

How fast does g(x) converge? ● Theorem: For every analytic function f in [0,1] there exists positive finite a and c, such that ● where is the largest grid spacing. ● In other words, the approximation g converges uniformly at a geometrical rate to any analytic function, on uniform and almost arbitrary grids. ● A adaptive, stable, fast convergent interpolation scheme.

Compatibility with “Impossibility” proven by Platte et al. ● If a scheme tries to be geometrically convergent for all analytic functions on uniform grid, it must be very ill- conditioned – Platte, Trefethan and Kuijlaars, 2010 ● Ill-conditioned: linear sensitivity of interpolant with respect to individual data points grows exponentially. – Does not directly imply instability for nonlinear schemes. – Beneficial for schemes such as WENO, which wants to switch to a lower order scheme upon detection of even small high order oscillations.

Compatibility with “Impossibility” proven by Platte et al. ● If a scheme tries to be geometrically convergent for all analytic functions on uniform grid, it must be very ill- conditioned – Platte, Trefethan and Kuijlaars, 2010 ● Ill-conditioned: linear sensitivity of interpolant with respect to individual data points grows exponentially. – Does not directly imply instability for nonlinear schemes. – Beneficial for schemes such as WENO, which wants to switch to a lower order scheme upon detection of even small high order oscillations. ● e k in our error estimator depends on f, and must be estimated.

A Bayesian way of estimating e k ● For f to be an analytic functions, there must exist a finite C, such that ● A natural model for the growth of derivatives: ● Our estimator computes the lower derivatives of f from polynomial interpolations, estimates C 0 and C, then extrapolate to higher derivatives.

Interpolating Runge's function

Convergence to Runge's function L-infinity error L-2 error Grid points

Participating sub-interpolants

Interpolating

Convergence to L-infinity error L-2 error Excluding interval [- 0.01,0.01] Grid points

Participating Subintervals

Conclusion ● Weighted average of participant approximations – For polynomial participant approximations, we can prove uniform geometric convergence for arbitrary grid, with exact function derivative estimate. – Bayesian approach of estimating high order derivatives. ● Demonstrated to be Runge free and Gibbs free ● Extension to high dimension: Need a accurate multivariate participant approximation with a reliable error estimate.