Free energy difference ∆ F(z)Free energy difference ∆ F(z)

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Monte Carlo methods in molecular dynamics.

Tony Leli `evre

CERMICS, Ecole Nationale des Ponts et Chauss ´ees, France, MicMac team, INRIA, France.

http://cermics.enpc.fr/∼lelievre

T. Leli `evre, ECODOQUI, November 2008 – p. 1

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Outline

1 Free energy and metastability, 2 Constrained dynamics,

2.1 Thermodynamic integration, 2.2 Constrained SDEs,

2.3 Non-equilibrium dynamics, 3 Adaptive methods,

3.2 Adaptive methods: algorithms, 3.2 Adaptive methods: convergence, 3.3 Multiple replicas implementations, 3.4 Application to Bayesian statistics.

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1 Free energy and metastability

We consider a molecular system with N particles with position (x₁, ...,x_N) = x ∈ R^3N interacting through the potential V (x₁, ..., x_N).

In the NVT ensemble, one wants to sample the Boltzmann-Gibbs probability measure:

dµ(x) = Z⁻¹ exp(−βV (x)) dx,

where Z = R

exp(−βV (x)) dx is the partition function and β = (k_BT)⁻¹ is proportional to the inverse of the temperature.

Aim: compute “macroscopic quantities” like the likelihood of molecular conformations, reaction rates, ...

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1 Free energy and metastability

Typically, V is a sum of potentials modelling interaction between two particles, three particles and four

particles:

V = X

i<j

V₁(x_i, x_j) + X

i<j<k

V₂(x_i, x_j, x_k) + X

i<j<k<l

V₃(x_i, x_j, x_k, x_l).

For example, V₁(x_i, x_j) = V_LJ(|x_i − x_j|) where

V_LJ(r) = 4

σ r

12

− ^σ_r 6

is the Lennard-Jones potential.

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1 Free energy and metastability

Examples of quantities of interest:

• specific heat

C ∝ hV ²i^µ − hV i²µ

• pressure

P ∝ −hq · ∇V (q)i^µ

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1 Free energy and metastability

Since this is a high-dimensional problem (N ≫ 1) Monte Carlo methods are used, typically based on Markov chains.

For example, to sample µ, one can use X_t solution to the Stochastic Differential Equation (SDE):

(GD) dX_t = −∇V (X_t) dt + p

2β⁻¹dW _t.

(gradient or over-damped Langevin dynamics).

Under suitable assumption, we have the ergodic property: for µ-a.e. X₀,

Tlim→∞

1 T

Z _T

0

φ(X_t)dt = Z

φ(x)dµ(x).

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1 Free energy and metastability

Probabilistic insert (1): discretization of SDEs.

The discretization of (GD) by the Euler scheme is (for a fixed timestep ∆t):

X_n+1 = X_n − ∇V (X_n) ∆t + p

2β⁻¹∆tG_n

where (Gⁱ_n)_{1≤i≤3N,n≥0} are i.i.d. random variables with law _N (0, 1). Indeed,

(W_(n+1)∆t − W_n∆t)_n≥0 =^L √

∆t(G_n)_n≥0.

In practice, a sequence of i.i.d. random variables with law _N (0, 1) may be obtained from a sequence of i.i.d.

random variables with law _U((0, 1)) (given by the rand() function on computers).

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1 Free energy and metastability

Proof (invariant measure): One needs to show that if the law of X₀ is µ, then the law of X_t is also µ. Let us denote X^x_t the solution to (GD) such that X₀ = x. Let us consider the function u(t, x) solution to:

( ∂_tu(t, x) = −∇V (x) · ∇u(t, x) + β⁻¹∆u(t, x), u(0, x) = φ(x)(+ assumptions on decay at infinity),

then, u(t, x) = E(φ(X^x_t )). Thus, the measure µ is invariant:

d dt

Z

E(φ(X^x_t ))dµ(x) = Z⁻¹ Z

∂_tu(t, x) exp(−βV (x))dx

= Z⁻¹ Z

−∇V · ∇u + β⁻¹∆u

exp(−βV )= 0.

Therefore,

Z

E(φ(X^x_t ))dµ(x) = Z

φ(x)dµ(x).

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1 Free energy and metastability

Probabilistic insert (2): Feynman-Kac formula.

Why u(t, x) = E(φ(X^x_t )) ? For 0 < s < t, we have (characteristic method):

du(t − s, X^x_s ) = −∂_tu(t − s, X^x_s ) ds + ∇u(t − s, X^x_s ) · dX^x_s +β⁻¹∆u(t − s, X^x_s ) ds,

=

− ∂_tu(t − s, X^x_s ) − ∇V (X^x_s ) · ∇u(t − s, X^x_s ) + β⁻¹∆u(t − s, X^x_s )

ds + p

2β⁻¹∇u(t − s, X^x_s ) · dW _s.

Thus, integrating over s ∈ (0, t) and taking the expectation:

E(u(0, X^x_t )) − E(u(t, X^x₀ )) = p

2β⁻¹E

Z _t

0 ∇u(t − s, X^x_s ) · dW_s

= 0.

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1 Free energy and metastability

Probabilistic insert (3): It ˆo’s calculus. ^{(in 1d.)}

Where does the term ∆u come from ? Starting from the discretization:

X_n+1 = X_n − V ^′(X_n) ∆t + p

2β⁻¹∆tG_n,

we have (for a time-independent function u):

u(X_n+1) = u

X_n − V ^′(X_n) ∆t + p

2β⁻¹∆tG_n ,

= u(X_n) − u^′(X_n)V ^′(X_n) ∆t + p

2β⁻¹∆tu^′(X_n)G_n +β⁻¹(G_n)²u^′′(X_n)∆t + o(∆t).

Thus, summing over n ∈ [0...t/∆t] and taking the limit ∆t → 0,

u(X_t) = u(X₀) −

Z t 0

V ^′(X_s)u^′(X_s) ds + p

2β⁻¹

Z t 0

u^′(X_s)dW_s

+β⁻¹

Z _t

u^′′(X_s) ds. T. Leli `evre, ECODOQUI, November 2008 – p. 10

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1 Free energy and metastability

In practice, (GD) is discretized in time, and Cesaro means are computed: lim_N_T_→∞ _N¹

T

PNT

n=1 φ(X_n). Remark: Practitioners do not use over-damped

Langevin dynamics but rather Langevin dynamics:

( dX_t = M⁻¹P _t dt,

dP_t = −∇V (X_t) dt − γM⁻¹P _t dt + p

2γβ⁻¹dW _t,

where M is the mass tensor and γ is the friction coefficient. In the following, we only consider

over-damped Langevin dynamics.

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1 Free energy and metastability

We therefore have a method to compute (an

approximation of) ^R φ(x)dµ(x), using X_t. But, in practice, X_t is a metastable process, so that the convergence of the ergodic limit is very slow.

A bi-dimensional example: X_t¹ is a slow variable of the system.

x₁ x₂

V (x₁, x₂)

X_t¹

t

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1 Free energy and metastability

A more realistic example : (Dellago, Geissler): Influence of the solvation on a dimer conformation. The

interaction potentials are (J.D. Weeks, D. Chandler et H.C.

Andersen):

V_WCA(r) =

( 4ǫ h

σ r

12

− ^σ_r 6i

+ ǫ if r ≤ r₀, 0 if r > r₀,

V_S(r) = h

1 − (r − r₀ − w)² w²

² ,

where ǫ, σ and w are positive constants and r₀ = 2^1/6σ.

V_S is a double-well potential.

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1 Free energy and metastability

00000000 00000000 00000000 0000

11111111 11111111 11111111 1111

000000 000000 000000 111111 111111 111111

00000000 00000000 00000000 0000

11111111 11111111 11111111 1111 00000000 00000000 00000000 0000

11111111 11111111 11111111 1111

00000000 00000000 00000000 11111111 11111111 11111111

000000 000000 000000 000

111111 111111 111111 111

00000000 00000000 00000000 11111111 11111111 11111111

000000 000000 000000 111111 111111

111111 000000000000000000000000 11111111 11111111 11111111 000000 000000 000000 111111 111111 111111 000000 000000 000000 111111 111111 111111

000000 000000 000000 111111 111111 111111

00000000 00000000 00000000 0000

11111111 11111111 11111111 1111

000000 000000 000000 111111 111111 111111

000000 000000 000000 000

111111 111111 111111 111

000000 000000 000000 111111 111111 111111

000000 000000 000000 000

111111 111111 111111 111

000000 000000 000000 000

111111 111111 111111 111

00000000 00000000 00000000 11111111 11111111 11111111 000000 000000 000000 000

111111 111111 111111 111

00000000 00000000 00000000 11111111 11111111 11111111

000000 000000 000000 000

111111 111111 111111 111

000000 000000 000000 111111 111111 111111

00000000 00000000 00000000 0000

11111111 11111111 11111111 1111 00000000 00000000 00000000 11111111 11111111 11111111 000000 000000 000000 000

111111 111111 111111 111

Left: compact state (ξ = 0). Right: stretched state (ξ = 1).

A slow variable is ξ(X_t) where ξ(x) = ^|^x¹⁻_2w^x²^|−r⁰ is a so-called reaction coordinate.

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1 Free energy and metastability

A “real” example: ions canal in a cell membrane.

(C. Chipot).

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1 Free energy and metastability

We suppose in the following that the slow variable is of dimension 1 and known: ξ(x), where ξ : Rⁿ → R.

This slow variable contains most of the information needed in practice so that it would be enough to compute the law of ξ(X), for X with law µ.

Lemme 1 The image of the measure µ by ξ is

Z⁻¹ exp(−βA(z)) dz, where

A(z) = −β⁻¹ ln

Z

Σz

e^−βV |∇ξ|⁻¹dσ_Σ_z

= −β⁻¹ ln Z_Σ_z,

where Σ_z = {x, ξ(x) = z} is a (smooth) submanifold of

Rⁿ, and σ_Σ_z is the Lebesgue measure on Σ_z.

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1 Free energy and metastability

Co-area formula: Let X be a random variable with law

ψ(x) dx in _Rⁿ. Then ξ(X) has law ^R_Σ

z ψ |∇ξ|⁻¹ dσ_Σ_z dz,

and the law of X conditioned to a fixed value z of ξ(X)

is dµ_z = ^ψ ^|∇ξ|

−1 dσΣz

R

Σz ψ|∇ξ|⁻¹ dσΣz .

Indeed, for any bounded functions _f and _g,

E(f(ξ(X))g(X)) = Z

Rⁿ

f(ξ(x))g(x)ψ(x) dx,

= Z

R^p

Z

Σz

f ◦ ξ g ψ |∇ξ|⁻¹dσ_Σ_z dz,

= Z

R^p

f(z) R

Σ_z g ψ |∇ξ|⁻¹dσ_Σ_z R

Σz ψ |∇ξ|⁻¹dσ_Σ_z Z

Σz

ψ |∇ξ|⁻¹dσ_Σ_z dz.

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1 Free energy and metastability

Remarks:

- The measure _|∇ξ|⁻¹dσ_Σ_z is sometimes denoted

δ_ξ(_x_)−z in the literature.

- A is the free energy associated with the reaction coordinate or collective variable ξ (angle, length, ...).

A is defined up to an additive constant, so that it is enough to compute free energy differences, or the derivative of A (the mean force).

- A(z) = −β⁻¹ ln Z_Σ_z and Z_Σ_z is the partition function associated with the conditioned probability measures:

µ_Σ_z = Z_Σ⁻¹

z e^−βV |∇ξ|⁻¹dσ_Σ_z.

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1 Free energy and metastability

Example of a free energy profile (solvation of a dimer)

(Profiles computed using TI)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Parameter z

Free energy difference ∆ F(z)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Parameter z

Free energy difference ∆ F(z)

The density of the solvent molecules is lower on the left than on the right. At high density, the compact state is more likely but (claim of physicists)

spontaneous transitions are less frequent (free energy barrier) ... to be better understood. T. Leli `evre, ECODOQUI, November 2008 – p. 19

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1 Free energy and metastability

Some direct numerical simulations...

Remark: Free energy is not energy !

−3 −2 −1 0 1 2 3

0 0.5 1 1.5 2 2.5 3

x coordinate

y coordinate

−3 −2 −1 0 1 2 3

−2.5

−2

−1.5

−1

−0.5 0 0.5 1 1.5

x coordinate

Free energy

Left: The potential is 0 in the region enclosed by the curve, and +∞ outside.

Right: Associated free energy profile when the x

coordinate is the reaction coordinate (β = 1).

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1 Free energy and metastability

Examples of methods to compute free energy differences A(z₂) − A(z₁):

• Thermodynamic integration (Kirkwood)

(homogeneous Markov process),

• Perturbation methods (Zwanzig) (importance sampling),

• Out of equilibrium dynamics (Jarzynski)

(non-homogeneous Markov process),

• Adaptive methods (ABF, metadynamics)

(non-homogeneous and non-linear Markov process).

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1 Free energy and metastability

(a) Thermodynamic integration. (b) Perturbation method.

(c) Out of equilibrium dynamics. (d) Adaptive dynamics.

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2 Constrained dynamics

Examples of methods to compute free energy differences A(z₂) − A(z₁):

• Thermodynamic integration (Kirkwood)

(homogeneous Markov process),

• Perturbation methods (Zwanzig) (importance sampling),

• Out of equilibrium dynamics (Jarzynski)

(non-homogeneous Markov process),

• Adaptive methods (ABF, metadynamics)

(non-homogeneous and non-linear Markov process).

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2.1 Thermodynamic integration

Thermodynamic integration is based on two remarks:

(1) The derivative A^′(z) can be obtained by sampling the conditioned probability measure µ_Σ_z (Sprik, Ciccotti, Kapral, Vanden-Eijnden, E, den Otter, ...)

A^′(z) = Z_Σ⁻¹

z

Z ∇V · ∇ξ

|∇ξ|² − β⁻¹div

∇ξ

|∇ξ|²

exp(−βV )|∇ξ|⁻¹dσ_Σ_z,

= Z_Σ⁻¹

z

Z ∇ξ

|∇ξ|² ·

∇V˜ + β⁻¹H

exp(−βV˜ )dσ_Σ_z,

= Z

f dµ_Σ_z,

where V^˜ = V + β⁻¹ ln |∇ξ|, f = ^∇V_|∇ξ|^·∇ξ₂ − β⁻¹div

∇ξ

|∇ξ|²

and H = −∇ ·

∇ξ

|∇ξ|

∇ξ

|∇ξ| is the mean curvature vector.

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2.1 Thermodynamic integration

Proof: (based on the co-area formula)

Z Z

exp(−βV˜ )dσ_Σ_z ′

φ(z) dz = −

Z Z

exp(−βV˜ )dσ_Σ_zφ^′ dz,

= −

Z Z

exp(−βV˜ )φ^′ ◦ ξ dσ_Σ_z dz,

= −

Z

exp(−βV˜ )φ^′ ◦ ξ|∇ξ|dx,

= −

Z

exp(−βV˜ )∇(φ ◦ ξ) · ∇ξ

|∇ξ|² |∇ξ|dx,

=

Z

∇ ·

exp(−βV˜ ) ∇ξ

|∇ξ|

φ ◦ ξ dx,

=

Z Z

−β ∇V˜ · ∇ξ

|∇ξ|² + |∇ξ|⁻¹∇ ·

∇ξ

|∇ξ|

!

exp(−βV˜)dσ_Σ_zφ(z) dz.

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2.1 Thermodynamic integration

(2) It is possible to sample the conditioned probability measure µ_Σ_z = Z_Σ⁻¹

z exp(−βV˜ )dσ_Σ_z by considering the following constrained dynamics:

(RCD)

( dX_t = −∇V˜(X_t) dt + p

2β⁻¹dW _t + ∇ξ(X_t)dΛ_t, dΛ_t such that ξ(X_t) = z.

Moreover, we have dΛ_t = dΛ^m_t + dΛ^f_t, with

dΛ^m_t = −p

2β⁻¹ _|∇ξ|^∇ξ₂ (X_t) · dW_t and

dΛ^f_t =_|∇ξ|^∇ξ₂ ·

∇V˜ + β⁻¹H

(X_t) dt = f(X_t) dt so that

A^′(z) = lim

T→∞

1 T

Z T 0

dΛ_t = lim

T→∞

1 T

Z T 0

dΛ^f_t.

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2.1 Thermodynamic integration

The free energy profile is then obtained by thermodynamic integration:

A(z) − A(0) =

Z z 0

A^′(z) dz ≃

XK i=0

ω_iA^′(z_i).

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2.1 Thermodynamic integration

The rigidly constrained dynamics can also be written:

(RCD) dX_t = P (X_t)

−∇V˜ (X_t) dt + p

2β⁻¹dW _t

+ β⁻¹H(X_t) dt,

where P(x) is the orthogonal projection operator:

P (x) = Id ₋ n(x) ⊗ n(x),

with n the unit normal vector: n(x) = ∇ξ

|∇ξ|(x).

(RCD) can also be written using the Stratonovitch product: dX_t = −P(X_t)∇V˜ (X_t) dt+ p

2β⁻¹P (X_t)◦dW _t. It is easy to check that ξ(X_t) = ξ(X₀) = z for X_t

solution to (RCD).

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2.1 Thermodynamic integration

Assume z = 0.

µ_Σ₀ is the unique invariant measure with support in Σ₀

for (RCD).

Proposition 1 Let X_t be the solution to (RCD) such that the law of X₀ is µ_Σ₀. Then, for all smooth

function φ and for all time t > 0,

E(φ(X_t)) = Z

φ(x)dµ_Σ₀(x).

Proof: Introduce the infinitesimal generator and apply the divergence theorem on submanifolds :

∀φ ∈ C¹(R^3N, R^3N),

Z

div _Σ₀(φ) dσ_Σ₀ = − Z

H · φdσ_Σ₀,

where tr .

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2.1 Thermodynamic integration

Discretization: These two schemes are consistent with (RCD):

(S1)

( X_n+1 = X_n − ∇V˜ (X_n)∆t + p

2β⁻¹∆W _n + λ_n∇ξ(X_n+1),

with λ_n ∈ R such that ξ(X_n+1) = 0,

(S2)

( X_n+1 = X_n − ∇V˜ (X_n)∆t + p

2β⁻¹∆W _n + λ_n∇ξ(X_n),

with λ_n ∈ R such that ξ(X_n+1) = 0,

where ∆W _n = W_(n+1)∆t − W_n∆t. The constraint is

exactly satisfied (important for longtime computations).

The discretization of A^′(0) = lim_T_→∞ _T¹ R T

0 dΛ_t is:

Tlim→∞ lim

∆t→0

1 T

T /∆t

X

n=1

λ_n = A^′(0).

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2.1 Thermodynamic integration

In practice, the following variance reduction scheme may be used:

( X_n+1 = X_n − ∇V˜ (X_n)∆t+p

2β⁻¹∆W_n + λ∇ξ(X_n+1),

with λ ∈ R such that ξ(X_n+1) = 0, ( X_∗ = X_n − ∇V˜(X_n)∆t−p

2β⁻¹∆W_n + λ_∗∇ξ(X_∗),

with λ_∗ ∈ R such that ξ(X_∗) = 0,

and _λ_n _{= (λ} ₊ _λ_∗_)/2.

The martingale part _dΛ^m_t (i.e. the most fluctuating part) of the Lagrange multiplier is removed.

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2.1 Thermodynamic integration

An over-simplified illustration: in dimension 2,

V (x) = ^β₂⁻¹ |x|² and ξ(x) = ^x_a²¹2 + ^x_b2²² − 1.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

-3 -2 -1 0 1 2 3 4

mes_int mes_non_int dyn_int_proj_1 dyn_int_proj_2 dyn_non_int_proj_1 dyn_non_int_proj_2

Measures samples theoretically and numerically ^{(as a}

function of the angle θ), with β = 1, a = 2, b = 1, ∆t = 0.01, and 50 000 000 timesteps.

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2.1 Thermodynamic integration

Computation of the mean force: β = 1, a = 2, b = 1. The exact value is: 0.9868348150. The numerical result

(with ∆t = 0.001, M = 50000) is: [0.940613 ; 1.03204].

The variance reduction method reduces the variance by a factor 100. The result (with ∆t = 0.001, M = 50000) is: [0.984019 ; 0.993421].

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2.1 Thermodynamic integration

App. mean force as a function of ∆t and M = T /∆t:

1e-05

1e-04

0.001

0.01

0.1 1000

10000

100000

1e+06

1e+07 0.86

0.88 0.9 0.92 0.94 0.96 0.98 1

dt

M

A balance needs to be find between the discretization error (∆t → 0) and the convergence in the ergodic limit (T → ∞).

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2.1 Thermodynamic integration

Using classical technics (Talay-Tubaro like proof), one can check that the ergodic measure µ^∆t_Σ

0 sampled by the Markov chain (X_n) is an approximation of order one of µ_Σ₀: for all smooth functions g : Σ₀ → R,

Z

Σ0

gdµ^∆t_Σ₀ − Z

Σ0

gdµ_Σ₀

≤ C∆t.

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2.1 Thermodynamic integration

Remarks:

- There are many ways to constrain the dynamics

(GD). We chose one which is simple to discretize. We may also have used, for example (for z = 0)

dX^η_t = −∇V (X^η_t ) dt − 1

2η∇(ξ²)(X^η_t ) dt + p

2β⁻¹dW _t,

where the constraint is penalized. One can show that

lim_η→0 X^η_t = X_t ⁽ⁱⁿ ^L^∞_t_∈_[0,T_](L²_ω)-norm) where X_t satisfies (RCD). Notice that we used V and not V^˜ in the penalized dynamics.

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2.1 Thermodynamic integration

The statistics associated with the dynamics where the constraints are rigidly imposed and the dynamics

where the constraints are softly imposed through

penalization are different: “a stiff spring ₆= a rigid rod”

(van Kampen, Hinch,...).

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2.1 Thermodynamic integration

- TI yields a way to compute ^R φ(x)dµ(x):

Z

φ(x)dµ(x) = Z⁻¹ Z

φ(x)e^−βV ⁽^x⁾dx,

= Z⁻¹ Z

z

Z

Σz

φe^−βV |∇ξ|⁻¹dσ_Σ_z dz, (co-area formula)

= Z⁻¹ Z

z

R

Σz φe^−βV |∇ξ|⁻¹dσ_Σ_z R

Σz e^−βV |∇ξ|⁻¹dσ_Σ_z Z

Σz

e^−βV |∇ξ|⁻¹dσ_Σ_z dz,

=

Z

z

e^−βA(z⁾ dz

−1 Z

z

Z

Σz

φdµ_Σ_z

e^−βA(z) dz.

with Σ_z = {x, ξ(x) = z}, A(z) = −β⁻¹ ln R

Σze^−βV |∇ξ|⁻¹dσ_Σ_z

and

µ_Σ_z = e^−βV |∇ξ|⁻¹dσ_Σ_z/ R

Σze^−βV |∇ξ|⁻¹dσ_Σ_z.

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2.2 Constrained SDEs

- For a general SDE (with a non isotropic diffusion), the following diagram does not commute:

P_cont

P_disc Projected discretized process

?

Discretized process Projected continuous process

Continuous process

Discretized projected continuous process

∆t

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2.2 Constrained SDEs

We are interested in simulating a SDE:

dX_t = b(X_t) dt + σ(X_t)dW_t

subject to the constraint

q(X_t) = 0.

X_t ∈ Rⁿ, b : Rⁿ → Rⁿ and σ : Rⁿ → R^n×m (with σσ^T > 0),

W _t is a m-dimensional standard Brownian process

with filtration _F_t. The functions b, σ and q are supposed to be smooth.

In this section, q : Rⁿ → R and we suppose that

∀x ∈ Σ = {x, q(x) = 0}, _|∇q|(x) 6= 0.

(In the MD framework, q may be the reaction coordinate or some molecular constraints.)

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2.2 Constrained SDEs: continuous level

As such, the problem is ill-posed. We want to find a

F^t-adapted process Y _t such that:

( dX_t = b(X_t) dt + σ(X_t)dW_t + dY _t, q(X_t) = 0,

where dY _t = dA_t + S_t dW_t. Additional assumption:

dA_t and S_t dW _t are colinear to D(X_t)∇q(X_t) dt, where D(x) is a n × n symmetric positive matrix.

D(x)∇q(x) is the normal to Σ at point x and can be given by some additional assumptions on the

constraining term Y _t (D’alembert’s principle for example).

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2.2 Constrained SDEs: continuous level

By Itô’s calculus on the constraint q(X_t) = 0, one then obtains:

(SDE)





dX_t = P (X_t) (b(X_t) dt + σ(X_t)dW_t)

−1 2

∇²q : (P σσ^T P^T ) D∇q

||∇q||²_D

(X_t) dt,

where the projection operator P(x) is:

P (x) = Id ₋ ^D(^x⁾^∇^q(^x⁾ ^{⊗ ∇}^q⁽^x⁾

||∇q(x)||²_D(x) ,

and, for any Y ∈ Rⁿ and any SDP matrix S,

||Y ||²S = (Y · SY ).