Download Heat Equation and Spherical Harmonics on a Unit Sphere and more Study notes Statistics in PDF only on Docsity! Stat 992: Lecture 21 Diffusion smoothing on sphere. Moo K. Chung mchung@stat.wisc.edu December 8, 2003 Problem 16. What is the heat kernel on a unit sphere ? Solution. Read S. Rosenberg’s The Laplacian on a Rie- mannian Manifold (1997 Cambridge University Press), I Chavel’s Eigenvalues in Riemannian geometry (Aca- demic Press, 1984), E.B. Davies’ Heat kernels and spec- tral theory (Cambridge University Press, 1989). P. Olver and C. Shakiban’s Fundamentals of Applied Mathemat- ics, Prentice-Hall, in preparation. 1. Diffusion on sphere. Following lecture 7 and 20, we show the construction of heat kernel on a unit sphere. We have already showed how to construct it in implicit numerical scheme using the iterated kernel smoothing on manifolds. We may use spher- ical coordinate system: x = sin θ cosψ, y = sin θ sinψ, z = cos θ. On a unit sphere p = (x, y, z) ∈ S2, heat kernel Kt must satisfy isotropic heat equation ∂Kt ∂t = ∆S2Kt. where ∆S2 = 1 sin θ ∂ ∂θ ( sin θ ∂ ∂θ ) + 1 sin2 θ ∂2 ∂2ψ . It should satisfy the following three conditions Kt(p, q) = Kt(q, p), limt→0 Kt(p, q) = δ(p − q) and Kt(p, q) = ∫ M Ks(p, r)Kt−s(r, q) dr. Then the following PDE ∂f ∂t = ∆S2f with initial condition f(0, p) = Y (p) has a unique solution f(t, p) = ∫ S2 Kt(p, q)Y (p) dµ(p). In lecture 20, we showed how to solve it numeri- cally. 2. Spherical Helmholtz equation. We will follow lec- ture 11. In the previous lecture, we showed the met- ric to be g11 = 1, g12 = g21 = 0, g22 = sin2 θ. Define L2(S2, g) with respect to the Riemannian metric g with inner product 〈f, g〉 = ∫ S2 f(p)g(p) dµ(p). where the Lebesgue measure on a sphere is given by dµ = √ det g dθdψ = sin θdθdψ. This is the area element you must have seen it in vector calculus course. Note that ∫ S2 dµ = 4π, the area of the unit sphere. The gradient∇ : C∞(S2) → Tp(S2) in local coor- dinates is given by ∇f = ∑ i,j gij ∂f ∂uj ∂p ∂ui = ∂f ∂θ ∂p ∂θ + 1 sin2 θ ∂f ∂ψ ∂p ∂ψ . Then it can be shown that 〈∆S2f, g〉 = −〈∇f,∇g〉 = 〈f, ∆S2g〉 So the spherical Laplacian is compact self-adjoint operator and it should have eigenvalues λj and eigenfunctions Hj such that ∆S2Hj(p) = λjHj(p) and 0 = |λ0| ≤ |λ1| ≤ |λ2| · · · . To solve for eigen- values and eigenfunctions of Laplacian, we need to solve the spherical Helmholtz equation given by 1 sin θ ∂ ∂θ ( sin θ ∂H ∂θ ) + 1 sin2 θ ∂2H ∂2ψ − λH = 0 Let us use the separation of variable technique: H(θ, ψ) = α(θ)β(ψ). Then substituting the term, we get sin2 θ α′′ α + cos θ sin θ α′ α − sin2 θλ = −β ′′ β = c
