MCM 2020 talk: Importance Sampling for a Robust and Efficient Multilevel Monte Carlo Estimator for Stochastic Reaction Networks

1. Importance Sampling for a Robust and Efficient
Multilevel Monte Carlo Estimator
for Stochastic Reaction Networks
Chiheb Ben Hammouda
Joint work with Nadhir Ben Rached (RWTH Aachen) and Raúl Tempone
(RWTH Aachen, KAUST)
13th
International Conference on Monte Carlo Methods and
Applications
Mannheim 16 - 20 August, 2021
1

2. 1 Introduction
2 Pathwise Importance Sampling for a Robust and Efficient Multilevel
Estimator (Ben Hammouda, Ben Rached, and Tempone 2020)
3 Numerical Experiments
4 Conclusions and Future Work
1

3. Stochastic Reaction Networks (SRNs): Motivation
Deterministic models describe an average (macroscopic) behavior
and are only valid for large populations.
Species of small population ⇒ Considerable experimental
evidence: Dynamics dominated by stochastic effects1
.
⇒ Discrete state-space/stochastic simulation approaches more
relevant than continuous state-space/deterministic approaches
⇒ Theory of Stochastic Reaction Networks (SRNs).
Figure 1.1: Gene expression is affected
by both extrinsic and intrinsic noise
(Elowitz et al. 2002).
Figure 1.2: Gene expression can be
very noisy (Raj et al. 2006).
1
Populations of cells exhibit substantial phenotypic variation due to i) intrinsic
noise: biochemical process of gene expression and ii) extrinsic noise: fluctuations in
other cellular components. 2

4. SRNs Applications: Epidemic Processes (Anderson and Kurtz 2015)
and Virus Kinetics (Hensel, Rawlings, and Yin 2009),. . .
Biological Models
In-vivo population control: the
expected number of proteins.
Figure 1.3: DNA transcription and
mRNA translation (Briat, Gupta,
and Khammash 2015)
Chemical reactions
Expected number of molecules.
Sudden extinction of species
Figure 1.4: Chemical reaction
network (Briat, Gupta, and
Khammash 2015)

5. Stochastic Reaction Network (SRNs)
A stochastic reaction network (SRN) is a continuous-time Markov
chain, X(t), defined on a probability space (Ω,F,P)2
X(t) = (X(1)
(t),...,X(d)
(t)) ∶ [0,T] × Ω → Zd
+
described by J reactions channels, Rj ∶= (νj,aj), where
▸ νj ∈ Zd
+: stoichiometric (state change) vector.
▸ aj ∶ Rd
+ → R+: propensity (jump intensity) function.
aj satisfies
Prob(X(t + ∆t) = x + νj ∣ X(t) = x) = aj(x)∆t + o(∆t), j = 1,...,J.
(1)
2
In this setting i-th component, X(i)
(t), may describe the abundance of the i-th
species present in the system at time t. 4

6. Kurtz Representation (Ethier and Kurtz 1986)
Kurtz’s random time-change representation
X(t) = x0 +
J
∑
j=1
Yj (∫
t
t0
aj(X(s))ds)νj, (2)
where Yj are independent unit-rate Poisson processes,
0 0.2 0.4 0.6 0.8 1
10
0
10
1
10
2
10
3
10
4
Time
Species
M
P
D
Figure 1.5: 20 exact
i.i.d. paths of
X = (M,P,D): counts
the number of particles
of each species in a
problem from
genomics.
5

7. Typical Computational Tasks in the Context of SRNs
Estimation of the expected value of a given functional, g, of the
SRN, X, at a certain time T, i.e., E[g(X(T))].
▸ Example: The expected counting number of the i-th species, where
g(x) = x(i)
.
▸ Example: probability that X(T) is in the set B (rare event
application), where g(x) = 1x∈B(x) for a set B ⊆ Nd
.
Estimation of E[g(X(τB))]. Hitting times of X: the elapsed
random time that the process X takes to reach for the first time a
certain subset B of the state space, i.e., τB ∶= inf{t ∈ R+ ∶ X(t) ∈ B}.
▸ Example: The time of the sudden extinction of one of the species.
. . .
⇒ One needs to design efficient Monte Carlo (MC) methods for these
tasks.
6

8. Simulation of SRNs
Pathwise-Exact methods (Exact statistical distribution of the SRN
process)
▸ Stochastic simulation algorithm (SSA) (Gillespie 1976).
▸ Modified next reaction algorithm (MNRA) (Anderson 2007).
"Caveat: Computationally expensive.
Pathwise-approximate methods
▸ Explicit tau-leap (explicit-TL) (Gillespie 2001; J. Aparicio
2001).
▸ Split step implicit tau-leap (SSI-TL) (Ben Hammouda,
Moraes, and Tempone 2017), for systems characterized by having
simultaneously fast and slow timescales.
7

9. The Explicit-TL Method
(Gillespie 2001; J. Aparicio 2001)
Kurtz’s random time-change representation (Ethier and Kurtz
1986)
X(t) = x0 +
J
∑
j=1
Yj (∫
t
t0
aj(X(s))ds)νj,
where Yj are independent unit-rate Poisson processes
The explicit-TL method (kind of forward Euler approximation):
Given Zexp
(t) = z ∈ Zd
+,
Zexp
(t + τ) = z +
J
∑
j=1
Pj
⎛
⎜
⎜
⎜
⎝
aj(z)τ
´¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶
λj
⎞
⎟
⎟
⎟
⎠
νj,
{Pj(λj)}J
j=1: independent Poisson rdvs with rate {λj}J
j=1.
8

10. Monte Carlo (MC)
Setting: Let X be a stochastic process and g ∶ Rd
→ R, a function
of the state of the system which gives a measurement of interest.
Aim: approximate E[g(X(T))] efficiently, using Z∆t(T) as an
approximate path of X(T).
MC estimator: Let µM be the MC estimator of E[g(Z∆t(T))]
µM =
1
M
M
∑
m=1
g(Z∆t,[m](T)),
where Z∆t,[m] are independent paths generated via the
approximate algorithm with a step-size of ∆t.
Complexity: Given a pre-selected tolerance, TOL
MC complexity = (cost per path)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
≈ T
∆t
=TOL−1
× (#paths)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
=M=TOL−2
= O (TOL−3
).
9

11. Multilevel Monte Carlo (MLMC)
(Kebaier et al. 2005; Giles 2008)
Aim: Improve MC complexity, when estimating E[g(Z∆tL
(T))].
Setting:
▸ A hierarchy of nested meshes of [0,T], indexed by {`}L
`=0.
▸ ∆t` = K−`
∆t0: The time steps size for levels ` ≥ 1; K>1, K ∈ N.
▸ Z` ∶= Z∆t`
: The approximate process generated using a step size of ∆t`.
MLMC idea
E[g(ZL(T))] = E[g(Z0(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
+
L
∑
`=1
E[g(Z`(T)) − g(Z`−1(T))]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
(3)
Var[g(Z0(T))] ≫ Var[g(Z`(T)) − g(Z`−1(T))] ↘ as ` ↗
M0 ≫ M` ↘ as ` ↗
MLMC estimator: ̂
Q ∶=
L
∑
`=0
̂
Q`,
̂
Q0 ∶=
1
M0
M0
∑
m0=1
g(Z0,[m0](T)); ̂
Q` ∶=
1
M`
M`
∑
m`=1
(g(Z`,[m`](T)) − g(Z`−1,[m`](T))), 1 ≤ ` ≤ L
MLMC Complexity (Cliffe et al. 2011)
O (TOL
−2−max(0, γ−β
α
)
log (TOL)2×1{β=γ}
) (4)
i) Weak rate: ∣E[g (Z`(T)) − g (X(T))]∣ ≤ c12−α`
ii) Strong rate: Var[g (Z`(T)) − g (Z`−1(T))] ≤ c22−β`
iii) Work rate: W` ≤ c32γ`
(W`: expected cost)
10

12. 1 Introduction
2 Pathwise Importance Sampling for a Robust and Efficient Multilevel
Estimator (Ben Hammouda, Ben Rached, and Tempone 2020)
3 Numerical Experiments
4 Conclusions and Future Work
10

13. Why IS for Robust MLMC?
Issue: Catastrophic coupling (characteristic of pure jump processes)
Prob{Y` ∶= X`(T) − X`−1(T) = 0}
∆t`→0
= 1 − O (∆t`), (5)
X`−1, X`: coupled TL approximations of the true process X at levels (`−1,`).
Consequences:
/ Large kurtosis problem: Kurtosis, κ`, of order O(∆t−1
` ).
⇒ / Expensive cost for reliable/robust estimates of sample statistics.
Why large kurtosis is bad?
σS2(Y`) =
Var[Y`]
√
M`
√
(κ` − 1) +
2
M` − 1
; " M` ≫ κ`.
Why accurate variance estimates, V` = Var[Y`], are important?
M∗
` ∝
√
V`W−1
`
L
∑
`=0
√
V`W`.
Goal: Efficient and robust estimation of E [g(X(T))], using a pathwise
dependent importance sampling (IS) with MLMC, with g is a given functional.
11

14. Can IS improve the complexity of MLMC?
The optimal change of measure π` satisfies (g` ∶= g (X`))
dπ0 ∝ ∣g0∣dP0, dπ` ∝ ∣g` − g`−1∣dP`. (6)
⇒ Removes the probability mass at zero, where most of P` is
concentrated
The minimum variance is given by
Varπ`
[(g` − g`−1)
dP`
dπ`
]
= (EP`
[∣g` − g`−1∣])
2
(1 − (π` ({g` − g`−1 > 0}) − π` ({g` − g`−1 < 0}))
2
).
(7)
EP`
[∣g` − g`−1∣] = O (∆t`) for any Lipschitz function g (Anderson,
Ganguly, and Kurtz 2011) ⇒ optimal IS improves the strong
convergence rate ⇒ optimal complexity for MLMC.
Goal: propose a practical IS algorithm with a sub-optimal change of
measure π̄`. 12

15. Standard Paths Coupling
(Kurtz 1982; Anderson and Higham 2012)
Notation
▸ X`−1, X`: two explicit TL approximations of the process X based on levels (` − 1,`);
▸ Nl−1: number of time steps at level ` − 1;
▸ For 0 ≤ n ≤ Nl−1 − 1
☀ {tn, tn+1}: two consecutive time-mesh points for X`−1;
☀ {tn, tn + ∆t`, tn+1}: three consecutive time-mesh points for X`;
☀ ∆a1
`−1,n ∶= a (X`(tn)) − a (X`−1(tn)) in the coupling interval [tn, tn + ∆t`];
☀ ∆a2
`−1,n ∶= a (X`(tn + ∆t`)) − a (X`−1(tn)) in the coupling interval [tn + ∆t`, tn+1];
☀ P
′
n, Q
′
n, P
′′
n , Q
′′
n: independent Poisson rdvs.
Coupling idea illustration (one species & one reaction channel)
X`(tn+1) − X`−1(tn+1) = X`(tn) − X`−1(tn)
+ ν1 (P
′
n (∆a1
`−1,n∆t`)1∆a1
`−1,n
>0 − P
′′
n (−∆a1
`−1,n∆t`)1∆a1
`−1,n
<0)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
coupling in [tn,tn+∆t`]
+ ν1 (Q
′
n (∆a2
`−1,n∆t`)1∆a2
`−1,n
>0 − Q
′′
n (−∆a2
`−1,n∆t`)1∆a2
`−1,n
<0)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
coupling in [tn+∆t`,tn+1]
.
(8)
In the following, we denote, for 0 ≤ n ≤ N`−1 − 1
⎧
⎪
⎪
⎨
⎪
⎪
⎩
∆a`,2n = ∣∆a1
`−1,n∣, in [tn,tn + ∆t`].
∆a`,2n+1 = ∣∆a2
`−1,n∣, in [tn + ∆t`,tn+1].
(9)
13

16. Pathwise IS Idea 3
For 1 ≤ j ≤ J: Instead of ∆aj
`,n∆t` in (8), we propose λj
`,n
∆t`.
The parameter λj
`,n
is determined such that our change of measure
i) ↘ the kurtosis of the MLMC estimator at the deep levels.
ii) ↗ the strong convergence rate.
Our analysis suggests that a sub-optimal choice of {λj
`,n}j∈J1 is
λj
`,n = c`∆aj
`,n = ∆t−δ
` ∆aj
`,n, 0 < δ < 1, (10)
δ: a scale parameter in our IS algorithm.
For ` = 1,...,L and 0 ≤ n ≤ N` − 1: change measure (few time
steps) when
(i) j ∈ J1 ∶= {1 ≤ j ≤ J; g(X + νj) ≠ g(X)} & (ii) ∆aj
`,n ≠ 0
& (iii) ∆g`(tn) = 0, where g` ∶= g (X`)
3
Chiheb Ben Hammouda, Nadhir Ben Rached, and Raúl Tempone. “Importance
sampling for a robust and efficient multilevel Monte Carlo estimator for stochastic
reaction networks”. In: Statistics and Computing 30.6 (2020), pp. 1665–1689

17. IS-MLMC for SRNs: Convergence Results
Let L` be the likelihood ratio at level `
Theorem (Kurtosis) (Ben Hammouda, Ben Rached, and Tempone
2020)
Let J ≥ 1 and let us denote Y` ∶= ∆g`(T)L`. Then, for 0 ≤ δ < 1 and
∆t` → 0, we have under some assumptions
κ` ∶=
Eπ̄`
[(Y` − Eπ̄`
[Y`])4
]
(Varπ̄`
[Y`])2
= O (∆tδ−1
` ).
Theorem (Variance) (Ben Hammouda, Ben Rached, and Tempone
2020)
Let 0 ≤ δ < 1 and J ≥ 1 and let us denote Y` ∶= ∆g`(T)L`. Then, for
∆t` → 0, we have under some assumptions
Varπ̄`
[Y`] = O (∆t1+δ
` ).
15

18. Main Results and Contributions
Quantity of Interest MLMC Without IS (standard case) MLMC With IS (0 < δ < 1)
κ` O (∆t−1
` ) O (∆tδ−1
` )
V` O (∆t`) O (∆t1+δ
` )
WorkMLMC O (TOL−2
log (TOL)
2
) O (TOL−2
)
W`,sample ≈ 2 × J × Cp × ∆t−1
` ≈ 2 × J × Cp × ∆t−1
`
Table 1: Main results for the comparison of MLMC combined with our IS algorithm, and
standard MLMC.
Notation
κ`: the kurtosis of the coupled MLMC paths at level `.
V`: the variance of the coupled MLMC paths at level `.
W`,sample: the average cost of simulating coupled MLMC paths at level `.
TOL: a pre-selected tolerance for the MLMC estimator.
Cp: the cost of generating one Poisson rdv.
Note: MLMC estimators achieving O (TOL−2
) complexity have been proposed in
(Anderson and Higham 2012; Moraes, Tempone, and Vilanova 2016) but with some of
the steps simulated with an exact scheme.
16

19. 1 Introduction
2 Pathwise Importance Sampling for a Robust and Efficient Multilevel
Estimator (Ben Hammouda, Ben Rached, and Tempone 2020)
3 Numerical Experiments
4 Conclusions and Future Work
16

20. Example (Michaelis-Menten Enzyme Kinetics (Rao and Arkin
2003))
The catalytic conversion of a substrate, S, into a product, P, via
an enzymatic reaction involving enzyme, E. This is described by
Michaelis-Menten enzyme kinetics with three reactions
E + S
θ1
→ C, C
θ2
→ E + S
C
θ3
→ E + P,
θ = (10−3
,5.10−3
,10−2
), T = 1, X(t) = (E(t),S(t),C(t),P(t)) and
X0 = (100,100,0,0).
The stoichiometric matrix and the propensity functions
ν =
⎛
⎜
⎝
−1 −1 1 0
1 1 −1 0
1 0 −1 1
⎞
⎟
⎠
, a(X) =
⎛
⎜
⎝
θ1ES
θ2C
θ3C
⎞
⎟
⎠
The QoI is E[X(3)
(T)].

21. Summary of Results of Example 3
Example α β γ κL WorkMLMC
Example 3: MLMC without IS 1.02 1.03 1 1220 O (TOL−2
log (TOL)
2
)
Example 3: MLMC with IS (δ = 1/4) 1.02 1.25 1 215 O (TOL−2
)
Example 3: MLMC with IS (δ = 1/2) 1.02 1.49 1 36.5 O (TOL−2
)
Example 3: MLMC with IS (δ = 3/4) 1.03 1.75 1 5.95 O (TOL−2
)
Table 2: Comparison of convergence rates (α, β, γ) and the kurtosis at the deepest levels
of MLMC, κL, for the different numerical examples with and without IS algorithm. α,β,γ
are the estimated rates of weak convergence, strong convergence and computational work,
respectively, with a number of samples M = 106
. 0 < δ < 1 is a parameter in our IS
algorithm.
18

22. Cost Analysis: Illustration
0 1 2 3 4 5 6 7 8 9 10 11 12
level l
10-6
10
-5
10
-4
10
-3
10-2
10
-1
W
l,sample
Without IS
With IS
2
l
Figure 3.1: Example 3: Comparison of the average cost per sample path per level (in CPU
time and estimated with 106
samples). δ = 3
4
for IS.
0 1 2 3 4 5 6 7 8 9 10
level, l
100
10
1
102
103
104
average number of IS steps
N
l
= 2l
= t
l
-1
Figure 3.2: Example 3: Average number of IS steps for different MLMC levels, with IS
(δ = 3
4
), with 105
samples.
19

23. Numerical Results without IS:
0 2 4 6 8 10
-10
-5
0
5
0 2 4 6 8 10
-10
-5
0
5
0 2 4 6 8 10
0
2
4
6
8
10
0 2 4 6 8 10
10
2
10
3
kurtosis
Figure 3.3: Convergence plots for MLMC without IS, to approximate
E [X(3)
(T)], with the number of samples M` = 106
.
Kurtosis blows up, variance decays like O(∆t).
20

24. Numerical Results with IS
0 2 4 6 8 10
-20
-15
-10
-5
0
5
10
0 2 4 6 8 10
-10
-5
0
5
0 2 4 6 8 10
0
2
4
6
8
10
0 2 4 6 8 10
1
2
3
4
5
6
kurtosis
Figure 3.4: Convergence plots for MLMC with IS (δ = 3
4
) to approximate
E [X(3)
(T)] with the number of samples M` = 106
.
Kurtosis REMAINS SMALL, variance decays like O(∆t1+δ
).
Optimal complexity achieved!
21

25. Numerical Complexity Comparison
10
-4
10
-3
10
-2
10
-1
TOL
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
E[W]
MC + SSA
TOL
-2
standard MLMC + TL
TOL
-2
log(TOL)
2
MLMC + TL + IS
TOL
-2
Figure 3.5: Comparison of the numerical complexity of the different methods
i) MC with exact scheme (SSA), ii) standard MLMC, and iii) MLMC
combined with importance sampling (δ = 3
4
).
22

26. 1 Introduction
2 Pathwise Importance Sampling for a Robust and Efficient Multilevel
Estimator (Ben Hammouda, Ben Rached, and Tempone 2020)
3 Numerical Experiments
4 Conclusions and Future Work
22

27. Conclusions
1 A novel method that combines IS with MLMC estimator for a robust
and efficient estimation of statistical quantities for SRNs.
2 Decrease dramatically the high kurtosis due to catastrophic coupling.
3 Theoretically and numerically showed that we improve the strong
convergence rate from β = 1 for the standard case, to β = 1 + δ
(0 < δ < 1), with a negligible additional cost.
4 Improvement of the complexity from O (TOL−2
log(TOL)2
) to
O (TOL−2
), without steps simulated with an exact scheme as in
(Anderson and Higham 2012; Moraes, Tempone, and Vilanova 2016).
5 More details can be found in
Chiheb Ben Hammouda, Nadhir Ben Rached, and Raúl Tempone.
“Importance sampling for a robust and efficient multilevel Monte
Carlo estimator for stochastic reaction networks”. In: Statistics and
Computing 30.6 (2020), pp. 1665–1689
6 Code can be found in https://github.com/hammouc/MLMC_IS_SRNs.
23

28. Future Work
1 Design a more optimal IS scheme to be used for MLMC, for
instance by using a hierarchy of δ`, where the parameter δ used in
our pathwise IS method would depend on the level of
discretization.
2 Design novel IS scheme for MLMC, to address the catastrophic
decoupling issue4
(Lester, Yates, and Baker 2018), the second
cause of the high kurtosis phenomenon in the context of SRNs
when using MLMC.
3 Generalize our IS approach for MLMC estimator, to diffusion
problems, for instance problems in option pricing with rare events
and low regular payoffs (see (Bayer, Ben Hammouda, and
Tempone 2020)).
4
Observed for general stochastic processes where terminal values of the sample
paths of both coarse and fine levels become very different from each other.

29. References I
D. Anderson and D. Higham. “Multilevel Monte Carlo for continuous Markov chains,
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33. Thank you for your attention
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