Polyharmonic approximation on the sphere

The purpose of this article is to provide new error estimates for a popular type of SBF approximation on the sphere: approximating by linear combinations of Green's functions of polyharmonic differential operators. We show that the $L_p$ approximation order for this kind of approximation is $\sigma$ for functions having $L_p$ smoothness $\sigma$ (for $\sigma$ up to the order of the underlying differential operator, just as in univariate spline theory). This is an improvement over previous error estimates, which penalized the approximation order when measuring error in $L_p$, p>2 and held only in a restrictive setting when measuring error in $L_p$, p<2.Comment: 16 pages; revised version; to appear in Constr. Appro

Localized linear polynomial operators and quadrature formulas on the sphere

The purpose of this paper is to construct universal, auto--adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper--)sphere \SS^q ($q\ge 2$). The approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings. Numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete Fourier projection polynomial approximations. An essential ingredient in our construction is the construction of quadrature formulas based on scattered data, exact for integrating spherical polynomials of (moderately) high degree. Our formulas are based on scattered sites; i.e., in contrast to such well known formulas as Driscoll--Healy formulas, we need not choose the location of the sites in any particular manner. While the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18, we are able to construct formulas exact for spherical polynomials of degree 178.Comment: 24 pages 2 figures, accepted for publication in SIAM J. Numer. Ana

Computer model calibration with large non-stationary spatial outputs: application to the calibration of a climate model

Bayesian calibration of computer models tunes unknown input parameters by comparing outputs with observations. For model outputs that are distributed over space, this becomes computationally expensive because of the output size. To overcome this challenge, we employ a basis representation of the model outputs and observations: we match these decompositions to carry out the calibration efficiently. In the second step, we incorporate the non-stationary behaviour, in terms of spatial variations of both variance and correlations, in the calibration. We insert two integrated nested Laplace approximation-stochastic partial differential equation parameters into the calibration. A synthetic example and a climate model illustration highlight the benefits of our approach

Real-time smoke rendering using compensated ray marching

We present a real-time algorithm called compensated ray march-ing for rendering of smoke under dynamic low-frequency environ-ment lighting. Our approach is based on a decomposition of the input smoke animation, represented as a sequence of volumetric density fields, into a set of radial basis functions (RBFs) and a se-quence of residual fields. To expedite rendering, the source radi-ance distribution within the smoke is computed from only the low-frequency RBF approximation of the density fields, since the high-frequency residuals have little impact on global illumination under low-frequency environment lighting. Furthermore, in computing source radiances the contributions from single and multiple scatter-ing are evaluated at only the RBF centers and then approximated at other points in the volume using an RBF-based interpolation. A slice-based integration of these source radiances along each view ray is then performed to render the final image. The high-frequency residual fields, which are a critical component in the local appear-ance of smoke, are compensated back into the radiance integral dur-ing this ray march to generate images of high detail. The runtime algorithm, which includes both light transfer simula-tion and ray marching, can be easily implemented on the GPU, and thus allows for real-time manipulation of viewpoint and lighting, as well as interactive editing of smoke attributes such as extinction cross section, scattering albedo, and phase function. Only moderate preprocessing time and storage is needed. This approach provides the first method for real-time smoke rendering that includes sin-gle and multiple scattering while generating results comparable in quality to offline algorithms like ray tracing