Isotropic Gaussian random fields on the sphere Annika Lang Chalmers University of Technology & University of Gothenburg, Mathematical Sciences joint work with Christoph Schwab, ETH Zürich
iGRFs approximation regularity stochastic processes & SPDEs Examples of random fields collection of random variables stochastic processes, e.g., Brownian motion solutions of stochastic (partial) differential equations solutions of random partial differential equations 300 250 200 150 Z 100 140 50 93 0 0 20 40 47 60 80 Y 100 120 0 X 140 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 -0.002 -0.004 -0.006 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Annika Lang October 31, 2013 p. 2
iGRFs approximation regularity stochastic processes & SPDEs Outline isotropic Gaussian random fields approximation of random fields sample regularity of random fields stochastic processes & stochastic partial differential equations Annika Lang October 31, 2013 p. 3
iGRFs approximation regularity stochastic processes & SPDEs Random fields on spheres (Ω , A , P ) probability space ( S 2 , d ) compact, metric space S 2 = { x ∈ R 3 , � x � = 1 } unit sphere d ( x, y ) = arccos � x, y � R 3 T : Ω × S 2 → R , A ⊗ B ( S 2 ) -measurable: real-valued random field on S 2 T Gaussian random field : ∀ k ∈ N , x 1 , . . . , x k ∈ S 2 , a 1 , . . . , a k ∈ R : � k i =1 a i T ( x i ) Gaussian T isotropic , Gaussian: ∀ k ∈ N , x 1 , . . . , x k ∈ S 2 , g ∈ SO (3) : ( T ( x 1 ) , . . . , T ( x k )) ∼ ( T ( gx 1 ) , . . . , T ( gx k )) Annika Lang October 31, 2013 p. 4
iGRFs approximation regularity stochastic processes & SPDEs Spherical harmonic functions Legendre polynomials ( P ℓ , ℓ ∈ N 0 ) : ∂ ℓ P ℓ ( µ ) := 2 − ℓ 1 ∂µ ℓ ( µ 2 − 1) ℓ , µ ∈ [ − 1 , 1] ℓ ! associated Legendre polynomials ( P ℓm , ℓ ∈ N 0 , m = 0 , . . . , ℓ ) : P ℓm ( µ ) := ( − 1) m (1 − µ 2 ) m/ 2 ∂ m ∂µ m P ℓ ( µ ) , µ ∈ [ − 1 , 1] spherical harmonic functions ( Y ℓm , ℓ ∈ N 0 , m = − ℓ, . . . , ℓ ) : �� ( ℓ − m )! 2 ℓ +1 ( ℓ + m )! P ℓm (cos ϑ ) e imϕ m ≥ 0 4 π Y ℓm ( ϑ, ϕ ) := ( − 1) m Y ℓ − m ( ϑ, ϕ ) , m < 0 ( ϑ, ϕ ) ∈ [0 , π ] × [0 , 2 π ) Annika Lang October 31, 2013 p. 5
iGRFs approximation regularity stochastic processes & SPDEs Spherical Laplacian — Laplace–Beltrami operator y = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ ) ∈ S 2 + (sin ϑ ) − 2 ∂ 2 ∆ S 2 = (sin ϑ ) − 1 ∂ � sin ϑ ∂ � ∂ϕ 2 . ∂ϑ ∂ϑ eigenvalues & eigenfunctions ∆ S 2 Y ℓm = − ℓ ( ℓ + 1) Y ℓm , ℓ ∈ N 0 , m = − ℓ, . . . , ℓ spaces of eigenfunctions H ℓ ( S 2 ) = span { Y ℓm , m = − ℓ, . . . , ℓ } eigenbasis ∞ � H := L 2 ( S 2 ) = H ℓ ( S 2 ) ℓ =0 Annika Lang October 31, 2013 p. 6
iGRFs approximation regularity stochastic processes & SPDEs Theorem ( ❬▼❛r✐♥✉❝❝✐✱ P❡❝❛tt✐ ✶✶❪ ) T isotropic Gaussian random field Then: T has Karhunen–Loève expansion ℓ ∞ � � T = a ℓm Y ℓm , ℓ =0 m = − ℓ ( a ℓm , ℓ ∈ N 0 , m = 0 , . . . , ℓ ) random variables, ⊥ ⊥ Re a ℓm ⊥ ⊥ Im a ℓm ∼ N (0 , A ℓ / 2) Re a ℓ 0 ∼ N (0 , A ℓ ) , Im a ℓ 0 = 0 Re a 00 ∼ N ( E ( T )2 √ π, A 0 ) ( a ℓm , ℓ ∈ N 0 , m = − ℓ, . . . , − 1) given by Re a ℓm = ( − 1) m Re a ℓ − m Im a ℓm = ( − 1) m +1 Im a ℓ − m Annika Lang October 31, 2013 p. 7
iGRFs approximation regularity stochastic processes & SPDEs Lemma T centered, isotropic Gaussian random field ℓ ∈ N , m = 1 , . . . , ℓ , ϑ ∈ [0 , π ] : � ( ℓ − m )! 2 ℓ + 1 L ℓm ( ϑ ) := ( ℓ + m )! P ℓm (cos ϑ ) 4 π (( X 1 ℓm , X 2 ℓm ) , ℓ ∈ N 0 , m = 0 , . . . , ℓ ) , ⊥ ⊥ X i ℓm ∼ N (0 , 1) , i = 1 , 2 , m � = 0 X 2 ℓ 0 = 0 y = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ ) ∞ �� � A ℓ X 1 = ⇒ T ( y ) ∼ ℓ 0 L ℓ 0 ( ϑ ) ℓ =0 ℓ � � � L ℓm ( ϑ )( X 1 ℓm cos( mϕ ) + X 2 + 2 A ℓ ℓm sin( mϕ )) m =1 Annika Lang October 31, 2013 p. 8
iGRFs approximation regularity stochastic processes & SPDEs κ �� T κ ( y ) := � A ℓ X 1 ℓ 0 L ℓ 0 ( ϑ ) ℓ =0 ℓ � � � L ℓm ( ϑ )( X 1 ℓm cos( mϕ ) + X 2 + 2 A ℓ ℓm sin( mϕ )) m =1 Theorem ( ❬▲✳✱ ❙❝❤✇❛❜ ✶✸❪ ) T centered, isotropic Gaussian random field ∃ C > 0 , α > 2 , ℓ 0 ∈ N : ∀ ℓ > ℓ 0 : A ℓ ≤ C · ℓ − α Then: 1. ∀ 0 < p < + ∞ : ∃ ˆ C p > 0 : ∀ κ ∈ N : � T − T κ � L p (Ω; H ) ≤ ˆ C p · κ − ( α − 2) / 2 2. asymptotically: ∀ β < ( α − 2) / 2 : � T − T κ � H ≤ κ − β , P -a.s. Annika Lang October 31, 2013 p. 9
iGRFs approximation regularity stochastic processes & SPDEs L 2 error, 1000 Monte Carlo samples α = 3 α = 5 1 0 10 10 L 2 error L 2 error O( κ 1/2 ) O( κ 3/2 ) −1 10 0 10 L 2 error L 2 error −2 10 −1 10 −3 10 −2 −4 10 10 0 1 2 0 1 2 10 10 10 10 10 10 number of series elements κ number of series elements κ (a) α = 3 (b) α = 5 Figure: error depending on series truncation Annika Lang October 31, 2013 p. 10
iGRFs approximation regularity stochastic processes & SPDEs Sample error α = 3 α = 5 0 10 error error O( κ 1/2 ) O( κ 3/2 ) 0 10 −1 −0.2 10 10 path error path error −0.4 10 −2 10 −0.6 10 −0.8 10 −3 10 0 1 2 0 1 2 10 10 10 10 10 10 number of series elements κ number of series elements κ (a) α = 3 (b) α = 5 Figure: error depending on series truncation Annika Lang October 31, 2013 p. 11
iGRFs approximation regularity stochastic processes & SPDEs Second moments — covariance kernels mixed second moments k T : S 2 → R ∞ ℓ � � k T ( x, y ) := E ( T ( x ) T ( y )) = A ℓ Y ℓm ( x ) Y ℓm ( y ) ℓ =0 m = − ℓ ∞ 2 ℓ + 1 � = A ℓ P ℓ ( � x, y � R 3 ) 4 π ℓ =0 k : [0 , π ] → R as function of distance r = d ( x, y ) ∞ 2 ℓ + 1 � k ( r ) := A ℓ P ℓ (cos r ) 4 π ℓ =0 k I : [ − 1 , 1] → R as function of inner product µ = � x, y � R 3 k I ( µ ) := k (arccos µ ) x, y ∈ S 2 = ⇒ k T ( x, y ) = k ( d ( x, y )) = k I ( � x, y � R 3 ) , Annika Lang October 31, 2013 p. 12
iGRFs approximation regularity stochastic processes & SPDEs Decay power spectrum ⇐ ⇒ regularity kernel Proposition ( ❬▲✳✱ ❙❝❤✇❛❜ ✶✸❪ ) n ∈ N 0 ⇒ (1 − µ 2 ) n/ 2 ∂ n ( ℓ n +1 / 2 A ℓ , ℓ ≥ n ) ∈ ℓ 2 ( N 0 ) ⇐ ∂µ n k I ( µ ) ∈ L 2 ( − 1 , 1) i.e., � 1 2 ∂ n 1 2 ℓ + 1 � � ℓ 2 n < + ∞ ⇐ (1 − µ 2 ) n dµ < + ∞ � A 2 � � ⇒ ∂µ n k I ( µ ) ℓ � � (4 π ) 2 2 − 1 � � ℓ ≥ n extension to non-integers and fractional weighted Sobolev spaces Annika Lang October 31, 2013 p. 13
iGRFs approximation regularity stochastic processes & SPDEs Sample regularity Definition X , Y random fields on S 2 Y modification of X : ∀ x ∈ S 2 : P ( X ( x ) = Y ( x )) = 1 Theorem ( ❬▲✳✱ ❙❝❤✇❛❜ ✶✸❪ ) T isotropic Gaussian random field with ∞ A ℓ ℓ 1+ β < + ∞ � ℓ =0 1. β ∈ (0 , 2] : ∀ γ < β/ 2 ∃ continuous modification with Hölder exponent γ 2. β > 0 : ∀ k < β/ 2 − 1 ∃ k -times continuously differentiable modification Annika Lang October 31, 2013 p. 14
iGRFs approximation regularity stochastic processes & SPDEs Idea of proof Lemma ∞ A ℓ ℓ 1+ β < + ∞ , β ∈ [0 , 2] � ∀ r ∈ [0 , π ] : | k (0) − k ( r ) | ≤ C β r β = ⇒ ℓ =0 E ( | T ( x ) − T ( y ) | 2 p ) ≤ C β,p d ( x, y ) βp = ⇒ Theorem (Kolmogorov–Chentsov theorem ❬▲✳✱ ❙❝❤✇❛❜ ✶✸❪ ) T random field on S 2 ∃ p > 0 , ǫ ∈ (0 , 1] , C > 0 : E ( | T ( x ) − T ( y ) | p ) ≤ Cd ( x, y ) 2+ ǫp = ⇒ ∃ continuous modification which is locally Hölder continuous with exponent γ ∈ (0 , ǫ ) Annika Lang October 31, 2013 p. 15
iGRFs approximation regularity stochastic processes & SPDEs Ice crystals & Sahara dust particles radius: lognormal random field exp( T ) same regularity as isotropic Gaussian random field T ❬▲✳✱ ❙❝❤✇❛❜ ✶✸❪ Annika Lang October 31, 2013 p. 16
iGRFs approximation regularity stochastic processes & SPDEs Random fields − → stochastic processes Brownian motion – Wiener process W = ( W ( t ) , t ≥ 0) W ( t ) = ( W ( t ) − W ( t n )) + ( W ( t n ) − W ( t n − 1 )) + · · · + ( W ( t 1 ) − W (0)) independent increments ( W ( t n ) − W ( t n − 1 )) ∼ N (0 , t n − t n − 1 ) = N (0 , ∆ t ) generate N (0 , ∆ t ) -distributed random numbers resp. N (0 , ∆ tQ ) -distributed random field Annika Lang October 31, 2013 p. 17
iGRFs approximation regularity stochastic processes & SPDEs Stochastic process probability space (Ω , A , P ) filtration F = ( F t , t ≥ 0) satisfies "‘usual conditions"’ separable Hilbert space U — e.g., R d , L 2 ( D ) , H α ( D ) stochastic process X = ( X ( t ) , t ≥ 0) with values in U , e.g., U = L 2 ( S 2 ) : X = ( X ( t, x ) , t ≥ 0 , x ∈ S 2 ) with X ( t, x, ω ) ∈ R property: (often) P -a.s. nowhere differentiable Annika Lang October 31, 2013 p. 18
Recommend
More recommend
