[PPT] - Isotropic Gaussian random fields on the sphere Annika Lang Chalmers PowerPoint Presentation

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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

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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

50 100 150 200 250 300 Z 47 93 140 Y 20 40 60 80 100 120 140 X

0.006
0.004
0.002

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 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

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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

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iGRFs approximation regularity stochastic processes & SPDEs

Random fields on spheres

(Ω, A, P) probability space (S2, d) compact, metric space

S2 = {x ∈ R3, x = 1} unit sphere d(x, y) = arccosx, yR3

T : Ω × S2 → R, A ⊗ B(S2)-measurable: real-valued random field

n S2

T Gaussian random field: ∀k ∈ N, x1, . . . , xk ∈ S2, a1, . . . , ak ∈ R: k

i=1 aiT(xi) Gaussian

T isotropic, Gaussian: ∀k ∈ N, x1, . . . , xk ∈ S2, g ∈ SO(3): (T(x1), . . . , T(xk)) ∼ (T(gx1), . . . , T(gxk))

Annika Lang October 31, 2013

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iGRFs approximation regularity stochastic processes & SPDEs

Spherical harmonic functions

Legendre polynomials (Pℓ, ℓ ∈ N0): Pℓ(µ) := 2−ℓ 1 ℓ! ∂ℓ ∂µℓ (µ2 − 1)ℓ, µ ∈ [−1, 1] associated Legendre polynomials (Pℓm, ℓ ∈ N0, m = 0, . . . , ℓ): Pℓm(µ) := (−1)m(1 − µ2)m/2 ∂m ∂µm Pℓ(µ), µ ∈ [−1, 1] spherical harmonic functions (Yℓm, ℓ ∈ N0, m = −ℓ, . . . , ℓ): Yℓm(ϑ, ϕ) :=

2ℓ+1

4π (ℓ−m)! (ℓ+m)!Pℓm(cos ϑ)eimϕ

m ≥ 0 (−1)mYℓ−m(ϑ, ϕ), m < 0 (ϑ, ϕ) ∈ [0, π] × [0, 2π)

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iGRFs approximation regularity stochastic processes & SPDEs

Spherical Laplacian — Laplace–Beltrami operator

y = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ) ∈ S2 ∆S2 = (sin ϑ)−1 ∂ ∂ϑ

sin ϑ ∂

∂ϑ

+ (sin ϑ)−2 ∂2

∂ϕ2 . eigenvalues & eigenfunctions ∆S2Yℓm = −ℓ(ℓ + 1)Yℓm, ℓ ∈ N0, m = −ℓ, . . . , ℓ spaces of eigenfunctions Hℓ(S2) = span{Yℓm, m = −ℓ, . . . , ℓ} eigenbasis H := L2(S2) =

∞

ℓ=0

Hℓ(S2)

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iGRFs approximation regularity stochastic processes & SPDEs

Theorem (❬▼❛r✐♥✉❝❝✐✱ P❡❝❛tt✐ ✶✶❪)

T isotropic Gaussian random field Then: T has Karhunen–Loève expansion T =

∞

ℓ=0

ℓ

m=−ℓ

aℓmYℓm, (aℓm, ℓ ∈ N0, m = 0, . . . , ℓ) random variables, ⊥ ⊥ Re aℓm ⊥ ⊥ Im aℓm ∼ N(0, Aℓ/2) Re aℓ0 ∼ N(0, Aℓ), Im aℓ0 = 0 Re a00 ∼ N(E(T)2√π, A0) (aℓm, ℓ ∈ N0, m = −ℓ, . . . , −1) given by Re aℓm = (−1)mRe aℓ−m Im aℓm = (−1)m+1Im aℓ−m

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iGRFs approximation regularity stochastic processes & SPDEs

Lemma

T centered, isotropic Gaussian random field ℓ ∈ N, m = 1, . . . , ℓ, ϑ ∈ [0, π]: Lℓm(ϑ) :=

2ℓ + 1

4π (ℓ − m)! (ℓ + m)!Pℓm(cos ϑ) ((X1

ℓm, X2 ℓm), ℓ ∈ N0, m = 0, . . . , ℓ), ⊥

⊥ Xi

ℓm ∼ N(0, 1), i = 1, 2, m = 0

X2

ℓ0 = 0

y = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ) = ⇒ T(y) ∼

∞

ℓ=0
AℓX1

ℓ0Lℓ0(ϑ)

+

2Aℓ

ℓ

m=1

Lℓm(ϑ)(X1

ℓm cos(mϕ) + X2 ℓm sin(mϕ))

Annika Lang

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iGRFs approximation regularity stochastic processes & SPDEs

T κ(y) :=

κ

ℓ=0
AℓX1

ℓ0Lℓ0(ϑ)

+

2Aℓ

ℓ

m=1

Lℓm(ϑ)(X1

ℓm cos(mϕ) + X2 ℓm sin(mϕ))

Theorem (❬▲✳✱ ❙❝❤✇❛❜ ✶✸❪)

T centered, isotropic Gaussian random field ∃C > 0, α > 2, ℓ0 ∈ N : ∀ℓ > ℓ0 : Aℓ ≤ C · ℓ−α Then:

1. ∀0 < p < +∞ : ∃ ˆ

Cp > 0 : ∀κ ∈ N : T − T κLp(Ω;H) ≤ ˆ Cp · κ−(α−2)/2

2. asymptotically: ∀β < (α − 2)/2 : T − T κH ≤ κ−β, P-a.s.

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iGRFs approximation regularity stochastic processes & SPDEs

L2 error, 1000 Monte Carlo samples

10 10

1

10

2

10

−2

10

−1

10 10

1

number of series elements κ L2 error α = 3 L2 error O(κ1/2)

(a) α = 3

10 10

1

10

2

10

−4

10

−3

10

−2

10

−1

10 number of series elements κ L2 error α = 5 L2 error O(κ3/2)

(b) α = 5

Figure: error depending on series truncation

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iGRFs approximation regularity stochastic processes & SPDEs

Sample error

10 10

1

10

2

10

−0.8

10

−0.6

10

−0.4

10

−0.2

10 number of series elements κ path error α = 3 error O(κ1/2)

(a) α = 3

10 10

1

10

2

10

−3

10

−2

10

−1

10 number of series elements κ path error α = 5 error O(κ3/2)

(b) α = 5

Figure: error depending on series truncation

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iGRFs approximation regularity stochastic processes & SPDEs

Second moments — covariance kernels

mixed second moments kT : S2 → R kT (x, y) := E(T(x)T(y)) =

∞

ℓ=0

Aℓ

ℓ

m=−ℓ

Yℓm(x)Yℓm(y) =

∞

ℓ=0

Aℓ 2ℓ + 1 4π Pℓ(x, yR3) k : [0, π] → R as function of distance r = d(x, y) k(r) :=

∞

ℓ=0

Aℓ 2ℓ + 1 4π Pℓ(cos r) kI : [−1, 1] → R as function of inner product µ = x, yR3 kI(µ) := k(arccos µ) = ⇒ kT (x, y) = k(d(x, y)) = kI(x, yR3), x, y ∈ S2

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iGRFs approximation regularity stochastic processes & SPDEs

Decay power spectrum ⇐ ⇒ regularity kernel

Proposition (❬▲✳✱ ❙❝❤✇❛❜ ✶✸❪)

n ∈ N0 (ℓn+1/2Aℓ, ℓ ≥ n) ∈ ℓ2(N0) ⇐ ⇒ (1 − µ2)n/2 ∂n ∂µn kI(µ) ∈ L2(−1, 1) i.e., 1 (4π)2

ℓ≥n

A2

ℓ

2ℓ + 1 2 ℓ2n < +∞ ⇐ ⇒ 1

−1

∂n

∂µn kI(µ)

2

(1−µ2)n dµ < +∞ extension to non-integers and fractional weighted Sobolev spaces

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iGRFs approximation regularity stochastic processes & SPDEs

Sample regularity

Definition

X, Y random fields on S2 Y modification of X: ∀x ∈ S2 : P(X(x) = Y (x)) = 1

Theorem (❬▲✳✱ ❙❝❤✇❛❜ ✶✸❪)

T isotropic Gaussian random field with

∞

ℓ=0

Aℓℓ1+β < +∞

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

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iGRFs approximation regularity stochastic processes & SPDEs

Idea of proof

Lemma

∞

ℓ=0

Aℓℓ1+β < +∞, β ∈ [0, 2] = ⇒ ∀r ∈ [0, π] : |k(0) − k(r)| ≤ Cβrβ = ⇒ E(|T(x) − T(y)|2p) ≤ Cβ,p d(x, y)βp

Theorem (Kolmogorov–Chentsov theorem ❬▲✳✱ ❙❝❤✇❛❜ ✶✸❪)

T random field on S2 ∃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)

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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

❬▲✳✱ ❙❝❤✇❛❜ ✶✸❪

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iGRFs approximation regularity stochastic processes & SPDEs

Random fields − → stochastic processes

Brownian motion – Wiener process W = (W(t), t ≥ 0) W(t) = (W(t) − W(tn)) + (W(tn) − W(tn−1)) + · · · + (W(t1) − W(0)) independent increments (W(tn) − W(tn−1)) ∼ N(0, tn − tn−1) = N(0, ∆t) generate N(0, ∆t)-distributed random numbers

resp. N(0, ∆tQ)-distributed random field

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iGRFs approximation regularity stochastic processes & SPDEs

Stochastic process

probability space (Ω, A, P) filtration F = (Ft, t ≥ 0) satisfies "‘usual conditions"’ separable Hilbert space U — e.g., Rd, L2(D), Hα(D) stochastic process X = (X(t), t ≥ 0) with values in U, e.g., U = L2(S2) : X = (X(t, x), t ≥ 0, x ∈ S2) with X(t, x, ω) ∈ R property: (often) P-a.s. nowhere differentiable

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iGRFs approximation regularity stochastic processes & SPDEs

Examples

Brownian motion — Q–Wiener process W = (W(t), t ≥ 0)

0.006
0.004
0.002

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

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iGRFs approximation regularity stochastic processes & SPDEs

Examples

Lévy process L = (L(t), t ≥ 0)

2.0
1.5
1.0
0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 20 40 60 80 100 120 140

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iGRFs approximation regularity stochastic processes & SPDEs

Isotropic Q-Wiener process

W(t, y) =

∞

ℓ=0

ℓ

m=−ℓ

aℓm(t)Yℓm(y) =

∞

ℓ=0
Aℓβ1

ℓ0(t)Lℓ0(ϑ)

+

2Aℓ

ℓ

m=1

Lℓm(ϑ)(β1

ℓm(t) cos(mϕ) + β2 ℓm(t) sin(mϕ))

y = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ)

Brownian motions ((β1

ℓm, β2 ℓm), ℓ ∈ N0, m = 0, . . . , ℓ), ⊥

⊥, β2

ℓ0 = 0

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iGRFs approximation regularity stochastic processes & SPDEs

Covariance operator Q

QYLM =

∞

ℓ=0

ℓ

m=−ℓ

E((W(1), YLM)H(W(1), Yℓm)H)Yℓm =

∞

ℓ=0

ℓ

m=−ℓ

E(aLM(1)aℓm(1))Yℓm = ALYLM = ⇒ Q characterized by eigenvalues: (Aℓ, ℓ ∈ N0) angular power spectrum eigenfunctions: (Yℓm, ℓ ∈ N0, m = −ℓ, . . . , ℓ) spherical harmonics

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iGRFs approximation regularity stochastic processes & SPDEs

Heat equation with additive noise

H = L2(S2) separable Hilbert space dX(t) = ∆S2X(t) dt + dW(t), X(0) = X0 stochastic integral equation X(t) = X0 + t ∆S2X(s) ds + t dW(s) = X0 + t ∆S2X(s) ds + W(t)

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iGRFs approximation regularity stochastic processes & SPDEs

Examples

Ornstein–Uhlenbeck process dXt = αXt dt + dWt

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 1 2 3 4 5 Annika Lang October 31, 2013

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iGRFs approximation regularity stochastic processes & SPDEs

Examples

parabolic differential equation (heat equation) dX(t) = 1

2∆X(t) dt + dL(t)

5 10 15 20 100 200 300 400 −0.2 0.2 0.4 0.6 0.8 1 1.2

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iGRFs approximation regularity stochastic processes & SPDEs

Spherical Laplacian — Laplace–Beltrami operator

y = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ) ∈ S2 ∆S2 = (sin ϑ)−1 ∂ ∂ϑ

sin ϑ ∂

∂ϑ

+ (sin ϑ)−2 ∂2

∂ϕ2 . eigenvalues & eigenfunctions ∆S2Yℓm = −ℓ(ℓ + 1)Yℓm, ℓ ∈ N0, m = −ℓ, . . . , ℓ spaces of eigenfunctions Hℓ(S2) = span{Yℓm, m = −ℓ, . . . , ℓ} eigenbasis H := L2(S2) =

∞

ℓ=0

Hℓ(S2)

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iGRFs approximation regularity stochastic processes & SPDEs

Karhunen–Loève expansion

X(t) = X0 + Z t ∆S2X(s) ds + W (t) ∞

ℓ=0

ℓ

m=−ℓ

(X(t), Yℓm)HYℓm =

∞

ℓ=0

ℓ

m=−ℓ
(X0, Yℓm)H +

t (X(s), Yℓm)H ds∆S2 + aℓm(t)

Yℓm

=

∞

ℓ=0

ℓ

m=−ℓ
(X0, Yℓm)H − ℓ(ℓ + 1)

t (X(s), Yℓm)H ds + aℓm(t)

Yℓm

stochastic ordinary differential equations

(X(t), Yℓm)H = (X0, Yℓm)H − ℓ(ℓ + 1) t (X(s), Yℓm)H ds + aℓm(t)

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iGRFs approximation regularity stochastic processes & SPDEs

Stochastic ordinary differential equations

(X(t), Yℓm)H = (X0, Yℓm)H − ℓ(ℓ + 1) t (X(s), Yℓm)H ds + aℓm(t) variation of constants formula (X(t), Yℓm)H = e−ℓ(ℓ+1)t(X0, Yℓm)H + t e−ℓ(ℓ+1)(t−s) daℓm(s) SPDE X(t) =

∞

ℓ=0

ℓ

m=−ℓ
e−ℓ(ℓ+1)t(X0, Yℓm)H +

t e−ℓ(ℓ+1)(t−s) daℓm(s)

Yℓm

=:

∞

ℓ=0

Xℓ(t)

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iGRFs approximation regularity stochastic processes & SPDEs

X(t) =

∞

ℓ=0

Xℓ(t) recursion formula Xℓ(t + h) = e−ℓ(ℓ+1)hXℓ(t) +

Aℓ

t+h

t

e−ℓ(ℓ+1)(t+h−s) dβ1

ℓ0(s) Yℓ0

+ √ 2

ℓ

m=1

t+h

t

e−ℓ(ℓ+1)(t+h−s) dβ1

ℓm(s) Re Yℓm

+ t+h

t

e−ℓ(ℓ+1)(t+h−s) dβ2

ℓm(s) Im Yℓm

Itô formula (cf. ❬❏❡♥t③❡♥✱ ❑❧♦❡❞❡♥ ✵✾❪)

t+h

t

e−ℓ(ℓ+1)(t+h−s) dβi

ℓm(s) ∼ N(0, (2ℓ(ℓ + 1))−1(1 − e−2ℓ(ℓ+1)h))

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iGRFs approximation regularity stochastic processes & SPDEs

Spectral approximation — L2(Ω; H)

Xκ(t) =

κ

ℓ=0

Xℓ(t)

Lemma

∃ℓ0 ∈ N, α > 0, C > 0 : ∀ℓ > ℓ0 : Aℓ ≤ C · ℓ−α = ⇒ ∀t ∈ T : ∀0 = t0 < · · · < tn = t : X(t) − Xκ(t)L2(Ω;H) ≤ ˆ C · κ−α/2, κ ≥ ℓ0 where ˆ C2 = X02

L2(Ω;H) + C ·

2 α + 1 α + 1

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iGRFs approximation regularity stochastic processes & SPDEs

Spectral approximation — Lp(Ω; H) & P-a.s.

Lemma

∃ℓ0 ∈ N, α > 0, C > 0 : ∀ℓ > ℓ0 : Aℓ ≤ C · ℓ−α = ⇒ ∀t ∈ T : ∀0 = t0 < · · · < tn = t : ∀p > 0 : ∃ ˆ Cp > 0 : X(t) − Xκ(t)Lp(Ω;H) ≤ ˆ Cp · κ−α/2, κ ≥ ℓ0

Corollary

∃ℓ0 ∈ N, α > 0, C > 0 : ∀ℓ > ℓ0 : Aℓ ≤ C · ℓ−α = ⇒ ∀t ∈ T : ∀0 = t0 < · · · < tn = t : ∀β < α/2 : X(t) − Xκ(t)L2(S2) ≤ κ−β, P-a.s.

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iGRFs approximation regularity stochastic processes & SPDEs

L2 error, 100 Monte Carlo samples

10 10

1

10

2

10

−2

10

−1

10 10

1

number of series elements κ L2 error α = 1 L2 error O(κ1/2)

(a) α = 1

10 10

1

10

2

10

−4

10

−3

10

−2

10

−1

10 number of series elements κ L2 error α = 3 L2 error O(κ3/2)

(b) α = 3

10 10

1

10

2

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 number of series elements κ L2 error α = 5 L2 error O(κ5/2)

(c) α = 5

Figure: error depending on series truncation

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iGRFs approximation regularity stochastic processes & SPDEs

Sample error

10 10

1

10

2

10

−2

10

−1

10 10

1

number of series elements κ path error α = 1 error O(κ1/2)

(a) α = 1

10 10

1

10

2

10

−3

10

−2

10

−1

10 number of series elements κ path error α = 3 error O(κ3/2)

(b) α = 3

10 10

1

10

2

10

−5

10

−4

10

−3

10

−2

10

−1

10 number of series elements κ path error α = 5 error O(κ5/2)

(c) α = 5

Figure: error depending on series truncation

Annika Lang October 31, 2013

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iGRFs approximation regularity stochastic processes & SPDEs

Conclusions & outlook

isotropic Gaussian random fields on S2 approximation & regularity stochastic partial differential equations regularity of random fields on manifolds ❬❆♥❞r❡❡✈✱ ▲✳ ✶✸❪ random partial differential equations

Thank you for your attention!

Annika Lang October 31, 2013

p. 34