numerical solutions for high dimensional kolmogorov...

Numerical solutions for high dimensional Kolmogorovequations via Gaussian analysis

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Ü�ö: Franco Flandoli, Cristiano Ricci ('ip�)

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2020.6.3

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 1 / 43

Outline

1 Motivations

2 Some theoretical worksIteration schemeRelation with Girsanov transformKolmogorov equation with unbounded B : H → H

3 Numerical experiments

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 2 / 43

Outline

1 Motivations



ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 3 / 43

Climate change

∗

It is a fact that the climate is undergoing a period of change, withsea surface temperature (SST) being arised of rougly 1◦C in thelast 150 years.

Many people think that human are responsible for the climatechange, and that policies should be adopted to mitigate or slowdown the process.

∗From internet.ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 4 / 43

Mathematics and climate change

Last year, Scuola Normale Superiore ('ip�), Scuola di StudiSuperiori Sant’Anna ('i�SA�Æ) and IUSS of Pavia createdthe Center for Climate Change Sustainable Actions (3CSA), whichgathers expertise in many fields including Politics, Economics,Engineering, Agriculture and basic sciences like Mathematics andPhysics.

Role of Mathematics: together with other disciplines (Physics,Chemistry ...), try to describe the climate processes as accurate aspossible, to reveal the consequences of human activities (thenegative and the positive ones) on climate.

One of the concrete problems is concerned with numericalsolutions of high dimensional Kolmogorov equations.

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 5 / 43

Typical geophysical equations

Quantities: ρ(t, x), p(t, x), v(t, x), T (t, x).

Conservation of momentum and mass:

∂v

∂t+ v · ∇v + fe3 × v = −∇p

ρ+ ν∆v + F + noise,

∂ρ

∂t+ div(ρv) = 0.

Temperature:

∂T

∂t+ v · ∇T = κ∆T − λT 4 + J(t) + noise.

Equations of state:

atmosphere : p = ρRT,

ocean : ρ = ρ0[1− βT (T − T0) + βp(p− p0) + βs(s− s0)

].

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 6 / 43

Sub-grid modelling

Numerical solution of very high dimensional system of SDEs!

Example. The temperature T (t) = (Tk(t))k∈I = (T (t, xk))k∈I ∈ R|I|,

dT (t) =[AT (t) +B(T (t)) + J(t)

]dt+

√QdWt, T |t=0 = T (0).

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 7 / 43

Numerical solutions

The short-term weather forecast is already quite accurate, butlong-term climate forecast requires large computational effort.

For instance, if we are interested in the probability of an extremeevent, like the temperature is greater than 40◦C at some place xkat some large time t ∈ [a, b]:

P(T (t, xk) ≥ 40

)= ?

The classical method is to solve the SDE by direct Monte Carlosimulation:

P(T (t, xk) ≥ 40

)≈ 1

Ns

Ns∑i=1

1{T (t,xk,ωi)≥40}.

Drawbacks:

it takes very long time to get many simulationsit is difficult to estimate such rare events· · · · · ·

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 8 / 43

Other numerical methods

Kolmogorov equations: T = (Tk)k∈I ∈ R|I|,

∂tu(t, T ) =⟨AT +B(T ) + J(t), Du(t, T )

⟩+

1

2Tr(QD2u(t, T )

),

u(0, T ) = 1[40,∞)(Tk).

=⇒ P(T (t, xk) ≥ 40

)= E

[1{T (t,xk)≥40}

]= E

[1[40,∞)(T (t, xk))

]= u(t, T (0)).

Numerical methods for Kolmogorov eq:

finite difference, finite element ......

One has to discretize each coordinate to get N points, but thedimension d = |I| is very high, so the total number Nd of gridpoints explodes exponentially. (Curse of dimensionality)

So far, finite difference works mainly for d ≤ 3.

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 9 / 43

Recent progresses

Nan Chen-Andrew Majda†: conditional Gaussian models.

du1 = [A0(t, u1) +A1(t, u1)u2] dt+ σ1(t, u1) dW1(t),

du2 = [a0(t, u1) + a1(t, u1)u2] dt+ σ2(t, u1) dW2(t).

Beating the curse of dimension with accurate statistics for theFokker-Planck equation in complex turbulent systems. Proc. Natl.Acad. Sci. USA 114 (2017), no. 49, 12864–12869.

Efficient statistically accurate algorithms for the Fokker-Planckequation in large dimensions. J. Comput. Phys. 354 (2018),242–268.

· · · · · ·

†Introduction to PDEs and waves for the atmosphere and ocean.Courant Lecture Notes in Mathematics 9, 2003.

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 10 / 43

Recent progresses

Weinan E-Arnulf Jentzen ... : deep learning based algorithm.

PDEs→ BSDEs→ Stoch. control problem

→ model-based reinforcement learning problems.

Deep Learning-Based Numerical Methods for High-DimensionalParabolic Partial Differential Equations and Backward StochasticDifferential Equations. Commun. Math. Stat. 5 (2017), no. 4,349–380.

Solving high-dimensional partial differential equations using deeplearning. Proc. Natl. Acad. Sci. USA 115 (2018), no. 34,8505–8510.

Machine learning approximation algorithms for high-dimensionalfully nonlinear partial differential equations and second-orderbackward stochastic differential equations. J. Nonlinear Sci. 29(2019), no. 4, 1563–1619.

· · · · · ·

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 11 / 43

Outline

1 Motivations



ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 12 / 43

SPDE and Kolmogorov equation

Let Xt be the state of the system (velocity, temperature, ...) andsatisfy the following system on a Hilbert space:

dXxt =

(AXx

t +B(Xxt ))

dt+√QdWt, Xx

0 = x

where A = A∗ < 0, Q = Q∗ > 0, B : H → H is a nonlinearmapping, and Wt cylinder B.M. on H.

The corresponding Kolmogorov equation is ∂tu(t, x) = 〈Ax+B(x), Du(t, x)〉+1

2Tr(QD2u(t, x)

),

u(0, x) = u0(x).

Formally,u(t, x) = Eu0(Xx

t ).

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 13 / 43

OU semigroup and mild formulation

Linear equation (OU process):

dZxt = AZxt dt+√QdWt, Zx0 = x.

Assume Qt =∫ t0 e

sAQesA∗

ds has finite trace, then we have themild solution

Zxt = etAx+WA(t) := etAx+

∫ t

0e(t−s)A

√QdWs.

OU semigroup (St)t≥0:

Stu0(x) = Eu0(Zxt ), (t, x) ∈ [0, T ]×H.

Mild formulation of Kolmogorov equation: solve by fixed pointargument.

u(t, x) = Stu0(x) +

∫ t

0St−s(〈B,Du(s)〉)(x) ds.

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 14 / 43

Classical result

u(t, x) = Stu0(x) +

∫ t

0

St−s(〈B,Du(s)〉)(x) ds.

Theorem

Assume

etA(H) ⊂ Q1/2t (H), t > 0; define the bounded operator on H:

Λ(t) = Q−1/2t etA;

limt→0

∫ t0 ‖Λ(s)‖L(H) ds = 0;

B ∈ Cb(H,H).

Then for any u0 ∈ Cb(H), ∃ ! solution u ∈ C([0, T ], Cb(H)).

Example

If Q = Id. and A = A∗, then ‖Λ(t)‖L(H) .1√t.

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 15 / 43

Iteration scheme

Iterative approximation: let u0(t, x) = Stu0(x) and

un+1(t, x) = Stu0(x) +

∫ t

0

St−s(〈B,Dun(s)〉)(x) ds, n ≥ 0.

Define v0(t, x) = u0(t, x) and the difference (increment)

vn+1(t, x) = un+1(t, x)− un(t, x), n ≥ 0.

Then we haveknt (x) = 〈B(x), Dvn(t, x)〉,

vn+1(t, x) =

∫ t

0

(St−sk

ns

)(x) ds,

n ≥ 0

and the series expansion

u(t, x) =

∞∑n=0

vn(t, x).

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 16 / 43

The first iteration

OU process Zxt = etAx+WA(t) = etAx+∫ t0 e

(t−s)A√QdWs.

v0(t, x) = Stu0(x) = Eu0(Zxt ).

k0t (x) = 〈B(x), Dv0(t, x)〉 = 〈B(x), DStu0(x)〉.

Lemma

Under suitable conditions, ∀ t > 0 and u0 ∈ Bb(H), Stu0 ∈ C∞b (H) and∀h ∈ H,

〈h,DStu0(x)〉 = E[u0(Z

xt )⟨Λ(t)h,Q

−1/2t (Zxt − etAx)

⟩],

where Λ(t) = Q−1/2t etA is a bounded operator on H; moreover,

|DStu0(x)| ≤ ‖u0‖∞‖Λ(t)‖L(H).

Therefore,

k0t (x) = E[u0(Z

xt )⟨Λ(t)B(x), Q

−1/2t (Zxt − etAx)

⟩].

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 17 / 43

The first iteration

k0s(y) = E[u0(Z

ys )⟨Λ(s)B(y), Q−1/2s (Zys − etAy)

⟩].

By Markov property,

k0s(Zxt−s) = E

[u0(Z

ys )⟨Λ(s)B(y), Q−1/2s (Zys − etAy)

⟩]y=Zx

t−s

= E[u0(Z

xt )⟨Λ(s)B(Zxt−s), Q

−1/2s (Zxt − etAZxt−s)

⟩|Zxt−s

]= E

[u0(Z



⟩|Ft−s

].

Therefore,

v1(t, x) =

∫ t

0(St−sk

0s)(x) ds =

∫ t

0E[k0s(Z

xt−s)

]ds

=

∫ t

0E[u0(Z



⟩]ds.

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 18 / 43

Series expansion

u(t, x) = Stu0(x) +

∫ t

0

St−s(〈B,Du(s)〉)(x) ds.

Theorem (Flandoli-L.-Ricci, arXiv:1907.03332)

Assume u0 and B are bounded, and ∃ δ ∈ [1/2, 1) s.t. ‖Λ(t)‖ . 1/tδ.Then the series converge uniformly on [0, T ]×H:

u(t, x) =

∞∑n=0

vn(t, x),

vn(t, x) = E[u0(Zx

t )

∫ t

0

drn

∫ rn

0

drn−1 · · ·∫ r2

0

dr1

n∏i=1

⟨Λ(ri+1 − ri)B

(Zx

ri

), Q−1/2ri+1−ri

(Zx

ri+1− e(ri+1−ri)AZx

ri

)⟩],

‖vn(t)‖∞ ≤ ‖u0‖∞‖B‖n∞Cnt,A,Q

Γ(1− δ)n

Γ(1 + n(1− δ)) .

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 19 / 43

Outline

1 Motivations



ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 20 / 43

Relation with Girsanov transform

The series can be rewritten as

u(t, x) =

∞∑n=0

vn(t, x) = E[u0(Zx

t )ρ(t, x)],

ρ(t, x) = 1 +∞∑

n=1

∫ t

0

drn

∫ rn

0

drn−1 · · ·∫ r2

0

dr1

n∏i=1

⟨Λ(ri+1 − ri)B

(Zx

ri

), Q−1/2ri+1−ri

(Zx


ri

)⟩.

Girsanov transform: let dQ = MT dP, where

Mt = exp

(∫ t

0

〈Q−1/2B(Zxs ),dWs〉 −

1

2

∫ t

0

|Q−1/2B(Zxs )|2 ds

).

Under Q, the original OU process (Zxt )t≥0 is a weak mild solutionto SPDE. Thus,

Ptu0(x) = EQ[u0(Zxt )] = EP

[u0(Z

xt )Mt

].

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 21 / 43

Relation with Girsanov transform

Denote Lt =∫ t0 〈Q

−1/2B(Zxs ),dWs〉; then Mt = eLt− 12〈L〉t and

Mt = 1 +

∫ t

0

Ms dLs.

Therefore,

Ptu0(x) = EP[u0(Zx

t )Mt

]= Eu0(Zx

t ) + E[u0(Zx

t )

∫ t

0

Ms dLs

]= Eu0(Zx

t ) + E[u0(Zx

t )Lt

]+ E

[u0(Zx

t )

∫ t

0

(∫ s

0

Mr dLr

)dLs

]= · · · · · ·

Indeed, let M0(t) ≡ 1 and Mn(t) =∫ t0 Mn−1(s) dLs; then we can show

that:

Ptu0(x) =

∞∑n=0

E[u0(Zx

t )Mn(t)]

=

∞∑n=0

vn(t, x).

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 22 / 43

Outline

1 Motivations



ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 23 / 43

Some known results

The existing literature are mainly concerned with Kolmogorovequations with bounded B : H → H.

G. Da Prato, F. Flandoli. Pathwise uniqueness for a class of SDE inHilbert spaces and applications. J. Funct. Anal. 259 (2010), 243–267.

G. Da Prato, F. Flandoli, E. Priola, M. Rockner. Strong uniqueness forstochastic evolution equations in Hilbert spaces perturbed by a boundedmeasurable drift. Ann. Probab. 41 (2013), no. 5, 3306–3344.

G. Da Prato, F. Flandoli, E. Priola, M. Rockner. Strong uniqueness forstochastic evolution equations with unbounded measurable drift term. J.Theoret. Probab. 28 (2015), no. 4, 1571–1600.

G. Da Prato, F. Flandoli, M. Rockner, A. Yu. Veretennikov, Strong

uniqueness for SDEs in Hilbert spaces with nonregular drift. Ann.

Probab. 44 (2016), no. 3, 1985–2023.

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 24 / 43

Iteration scheme for unbounded Bv0(t, x) = Stu0(x),

knt (x) = 〈B(x), Dvn(t, x)〉,

vn+1(t, x) =

∫ t

0

(St−sk

ns

)(x) ds,

n ≥ 0.

Problem

Assume B : H → H has sublinear growth:

|B(x)| ≤ C(1 + |x|β), x ∈ H.

Can we prove convergence of∑

n vn(t, x) in suitable sense?

If Q∞ =∫∞0 esAQesA

∗ds has finite trace, then the Gaussian measure

µ = NQ∞ is the unique invariant measure for (St)t≥0.

If u0 ∈ Lp0(µ) (p0 > 1), then v0(t) = Stu0 ∈ Lp0(µ) and(

1p1

= 1p0

+ 1q1

)‖k0t ‖Lp1 ≤ ‖B‖Lq1‖Dv0(t)‖Lp0 ≤ ‖B‖Lq1‖Λ(t)‖L(H)‖u0‖Lp0 .

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 25 / 43

Estimates in Lp-setting

Note that v1(t, x) =∫ t0

(St−sk

0s

)(x) ds, we have

‖v1(t)‖Lp1 ≤∫ t

0

∥∥St−sk0s

∥∥Lp1

ds ≤ ‖B‖Lq1 ‖u0‖Lp0

∫ t

0

‖Λ(s)‖L(H) ds,

‖Dv1(t)‖Lp1 ≤∫ t

0

∥∥DSt−sk0s

∥∥Lp1

ds ≤∫ t

0

‖Λ(t− s)‖L(H)

∥∥k0s∥∥Lp1ds

≤ ‖B‖Lq1 ‖u0‖Lp0

∫ t

0

‖Λ(t− s)‖L(H)‖Λ(s)‖L(H) ds.

=⇒ ‖k1t ‖Lp2 ≤ ‖B‖Lq2 ‖Dv1(t)‖Lp1 ≤ · · ·(

1

p2=

1

p1+

1

q2

)We can take a suitable decreasing sequence {pn}n≥0 and an increasing

sequence {qn}n≥1 satisfying

pn ↓ p ≥ 1, qn ↑ ∞,1

pn=

1

pn−1+

1

qn(∀n ≥ 1),

such that the series are convergent:∑n

‖vn(t)‖Lpn <∞,∑n

‖Dvn(t)‖Lpn <∞.

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 26 / 43

Outline

1 Motivations



ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 27 / 43

Recall the iteration scheme

u(t, x) = Stu0(x) +

∫ t

0

St−s(〈B,Du(s)〉)(x) ds.v0(t, x) = Stu0(x),

knt (x) = 〈B(x), Dvn(t, x)〉,

vn+1(t, x) =

∫ t

0

(St−sk

ns

)(x) ds.

Theorem (Flandoli-L.-Ricci, arXiv:1907.03332)

Assume u0 and B are bounded, and ∃ δ ∈ [1/2, 1) s.t. ‖Λ(t)‖ . 1/tδ.Then

u(t, x) =

∞∑n=0

vn(t, x),

vn(t, x) = E[u0(Zx

t )

∫ t

0

drn

∫ rn

0

drn−1 · · ·∫ r2

0

dr1

n∏i=1

⟨Λ(ri+1 − ri)B

(Zx

ri

), Q−1/2ri+1−ri

(Zx


ri

)⟩].

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 28 / 43

Strategy of numerical simulations

The above results hold on Hilbert space. The following steps areimplemented in finite dimensional case.

We compute accurately and store a large number of samples of thecentered Gaussian process, depending only on A and Q,

dZt = AZt dt+√QdWt, Z0 = 0

(note that the OU process Zxt = etAx+ Zt.)

For different choices of t, x, u0 and even B, we use the storedsamples of Z to compute

vn(t, x) = E[u0(Zx

t )

∫ t

0

drn

∫ rn

0

drn−1 · · ·∫ r2

0

dr1

n∏i=1

⟨Λ(ri+1 − ri)B

(Zx

ri

), Q−1/2ri+1−ri

(Zx


ri

)⟩].

for n = 1, 2, · · · , n0 until |vn0(t, x)| ≤ ε0.

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 29 / 43

Comparisons with Monte Carlo

The computations of samples of nonlinear SPDEs are replaced bycomputing those of a linear equation.

The same samples of the linear equation are used for differentvalues (t, x) and even for different B, u0.

If we write the original SPDE as

dXt = (AXt +B(Xt)) dt+ σ√QdWt

and keep the linear equation

dZt = AZt dt+√QdWt,

then Zxt = etAx+ σZt. Thus we can use the same samples of Z fordifferent values of σ.

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 30 / 43

Numerical experiments

dXt = (AXt +B(Xt)) dt+ σ dWt, X0 = x ∈ Rd.

Q = Id.;

A diagonal, Akk = −k2, k = 1, 2, · · · , d (Laplacian on circle);

Nonlinearity B in four cases:

sin(xk)

sine × rotation

quadratic function with cut-off

cubic function with cut-off

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 31 / 43

B = sinDimension = 10

reference:1

Ns

Ns∑i=1

u0(Xx,it ); 0-th iteration:

1

Ns

Ns∑i=1

u0(Zx,it ).

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 32 / 43

B = sine × rotationDimension = 10

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 33 / 43

B = quadratic with cut-offDimension = 10

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 34 / 43

B = cubic with cut-offDimension = 10

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 35 / 43

B = sinDimension = 50

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 36 / 43

B = quadratic with cut-offDimension = 50

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 37 / 43


ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 38 / 43


ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 39 / 43

B = sine × rotationDimension = 10Small noise analysis can be very cheap, compared to Monte Carlo

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 40 / 43

B = sine × rotationDimension = 10Changing initial condition is very cheap compared to Monte Carlo andmay allow to explore the directions of maximal variation

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 41 / 43

Problems

Degenerate noise:

dXt = (AXt +B(Xt)) dt+√QdWt.

Multiplicative transport noise:

dXt = (AXt +B(Xt)) dt+∑k

σk · ∇Xt ◦ dW kt .

· · · · · ·

ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 42 / 43

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ÛÛÛ�� (¥¥¥��êêêÆÆÆ��) 43 / 43

numerical solutions for high dimensional kolmogorov...

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