The document describes a multilevel hybrid split-step implicit tau-leap method for simulating stochastic reaction networks. It begins with background on modeling biochemical reaction networks stochastically. It then discusses challenges with existing simulation methods like the chemical master equation and stochastic simulation algorithm. The document introduces the split-step implicit tau-leap method as an improvement over explicit tau-leap for stiff systems. It proposes a multilevel Monte Carlo estimator using this method to efficiently estimate expectations of observables with near-optimal computational work.

1. Multilevel Hybrid Split Step Implicit Tau-Leap
for Stochastic Reaction Networks (SRNs)
Chiheb Ben Hammouda
Alvaro Moraes Ra´ul Tempone
RWTH Aachen University
February 21, 2020
0

2. Outline
1 Introduction
2 Multilevel Hybrid Split-Step Implicit Tau-Leap (TL)
(Ben Hammouda, Moraes, and Tempone 2017)
3 Numerical Experiments
4 Conclusions and Future Research Directions
0

3. 1 Introduction
2 Multilevel Hybrid Split-Step Implicit Tau-Leap (TL)
(Ben Hammouda, Moraes, and Tempone 2017)
3 Numerical Experiments
4 Conclusions and Future Research Directions
0

4. Stochastic Modeling of Biochemical Reaction Networks
Deterministic models describe an average behavior (macroscopic
behavior) and are only valid for large populations.
Species of small population ⇒ Considerable experimental
evidence: Dynamics dominated by stochastic effects1
(Elowitz
et al. 2002; Raj et al. 2006).
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).
⇒ Discrete state-space and stochastic simulation approaches more
relevant than continuous state-space and deterministic ones ⇒ Theory
of stochastic reaction networks (SRNs).
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. 1

5. 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)
2

6. 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. 3

7. 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,
X is the solution of a nonlinear system of stochastic differential
equations driven by Poisson Random Measures.
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.

8. 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)
.
Estimation of the 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 use Monte Carlo (MC) methods for those tasks and
consequently one need to sample efficiently paths of SRNs.
5

9. Why not Chemical Master Equation (CME)?
Notation: p(x,t) = Prob(x,t;x0,t0)
CME:
∂p(x,t)
∂t
=
J
∑
j=1
(aj(x − νj)p(x − νj,t) − aj(x)p(x,t))
=Ap(x,t)
, ∀x ∈ Nd
p(x,0) = p0(x)
Numerical approximations computed on
Ωξ = {x ∈ Nd
,x1 < ξ1,...,xd < ξd} ⊂ Nd
.
Caveat: Curse of dimensionality: A ∈ RN×N
with
N = ∏d
i=1 ξi very large.
6

10. 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).
Caveat: The explicit-TL is not adequate when dealing with stiff
systems (systems characterized by having simultaneously fast and
slow timescales) ⇒ Numerical instability issues
▸ Split step implicit tau-leap (SSI-TL) (Ben Hammouda, Moraes, and
Tempone 2017).
7

11. The Stochastic Simulation Algorithm (SSA)
(Gillespie 1976)
1 Initialize x ← x0 (The initial number of molecules of each species) and
t ← 0.
2 In state x at time t, compute (aj(x))J
j=1 and the sum a0(x) = ∑J
j=1 aj(x).
3 Generate two independent uniform (0,1) random number r1 and r2.
4 Set the next time for one of the reactions to fire, τ
τ =
1
a0
ln(1/r1)
Caveat: E[τ X(t) = x] = (a0(x))−1
could be very small.
5 Find j ∈ {1,...,J} such that
a−1
0
j−1
∑
k=1
aj < r2 ≤ a−1
0
j
∑
k=1
aj
which is equivalent to choosing from reactions {1,...,J} with the jth
reaction having probability mass function (aj(x)/a0(x))J
j=1.
6 Update: t ← t + τ and x ← x + νj.
7 Record (t,x). Return to step 2 if t < T, otherwise end the simulation.

12. 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
λj
⎞
⎟
⎟
⎟
⎠
νj,
{Pj(λj)}J
j=1: independent Poisson rdvs with rate {λj}J
j=1.
Caveat: The explicit-TL is not adequate when dealing with stiff
problems (Numerical stability ⇒ τexp
threshold ≪ 1.)

13. Monte Carlo (MC): Idea
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 Zh(T) as an
approximate path of X(T).
Let µM be a classical Monte Carlo estimator of E[g(Zh(T)]
defined by
µN =
1
N
M
∑
n=1
g(Zh,[m](T)),
where Zh,[m] are independent paths generated via the approximate
algorithm with a step-size of h.
10

14. Monte Carlo (MC): Complexity
We can notice that
E[g(X(T))] − µM
= E[g(X(T))] − E[g(Zh(T))]
bias (weak error)
+ E[g(Zh(T))] − µM
statistical error=O( 1√
M
)
.
If we have an order one method (bias = O (h) ≈ O (TOL))
Total computational complexity = (cost per path)
≈ T
h
=TOL−1
× (#paths)
=M=TOL−2
= O (TOL−1
TOL−2
) = O (TOL−3
).
The Multilevel Monte Carlo (MLMC) estimator introduced in
(Giles 2008), in the context of SDEs, reduces the total work to
O (TOL−2
log(TOL)2
), and even to O (TOL−2
) in some cases (See
(Cliffe et al. 2011)).
11

15. Multilevel Monte Carlo (MLMC) (Giles 2008)
A hierarchy of nested meshes of the time interval [0,T], indexed by { }L
=0.
h = K−
h0: The size of the subsequent time steps for levels ≥ 1 , where
K>1 is a given integer constant and h0 the step size used at level = 0.
Z : The approximate process generated using a step size of h .
E[g(ZL(T))] = E[g(Z0(T))]+
L
∑
=1
E[g(Z (T)) − g(Z −1(T))]
Var[g(Z0(T))] ≫ Var[g(Z (T)) − g(Z −1(T))] as
M0 ≫ M as
By defining
⎧⎪⎪⎪⎪⎪
⎨
⎪⎪⎪⎪⎪⎩
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))),
we arrive at the unbiased MLMC estimator, Q, of E[g(ZL(T))]
Q =
L
∑
=0
Q .
12

16. Coupling Idea in the Context of SRNs
(Kurtz 1982; Anderson and Higham 2012)
To couple two Poisson rdvs, P1(λ1), P2(λ2), with rates λ1 and λ2,
respectively, we define λ⋆
= min{λ1,λ2} and we consider the decomposition
⎧⎪⎪
⎨
⎪⎪⎩
P1(λ1) = Q(λ⋆
) + Q1(λ1 − λ⋆
)
P2(λ2) = Q(λ⋆
) + Q2(λ2 − λ⋆
)
Q(λ⋆
), Q1(λ1 − λ⋆
) and Q2(λ2 − λ⋆
) are three independent Poisson rdvs.
We have small variance between the coupled rdvs
Var[P1(λ1) − P2(λ2)] = Var[Q1(λ1 − λ⋆
) − Q2(λ2 − λ⋆
)]
= λ1 − λ2 .
Observe: If P1(λ1) and P2(λ2) are independent, then, we have a larger
variance Var[P1(λ1) − P2(λ2)] = λ1 + λ2.

17. 1 Introduction
2 Multilevel Hybrid Split-Step Implicit Tau-Leap (TL)
(Ben Hammouda, Moraes, and Tempone 2017)
3 Numerical Experiments
4 Conclusions and Future Research Directions
13

18. Multilevel Hybrid Split-Step Implicit Tau-Leap: Goal
Given
1 an initial state X0 = x0,
2 a smooth scalar observable of X, g Rd
→ R,
3 a user-selected tolerance, TOL, and
4 a confidence level 1 − α close to 1.
Goal: Provide accurate estimator Q of E[g(X(T))] such that
P( E[g(X(T))] − Q < TOL) > 1 − α, (3)
With near-optimal expected computational work.
For a class of systems characterized by having simultaneously fast
and slow time scales (Stiff systems).
14

19. Split Step Implicit Tau-Leap (SSI-TL) Method I
The explicit-TL scheme, where z = Zexp
(t), can be rewritten as follows:
Zexp
(t + τ) = z +
J
∑
j=1
Pj (aj(z)τ)νj
= z +
J
∑
j=1
(Pj (aj(z)τ) − aj(z)τ + aj(z)τ)νj
= z +
J
∑
j=1
aj(z)τνj
drift
+
J
∑
j=1
(Pj (aj(z)τ) − aj(z)τ)νj
zero-mean noise
.
15

20. Split Step Implicit Tau-Leap (SSI-TL) Method II 3
The idea of SSI-TL method is to take only the drift part as implicit
while the noise part is left explicit. Let us define z = Zimp
(t) and define
Zimp
(t + τ) through the following two steps:
y = z +
J
∑
j=1
aj (y)τνj (Drift-Implicit step)
Zimp
(t + τ) = y +
J
∑
j=1
(Pj(aj(y)τ) − aj(y)τ)νj
= z +
J
∑
j=1
Pj(aj(y)τ)νj
∈Zd
+
(Tau-leap step)
3
Chiheb Ben Hammouda, Alvaro Moraes, and Ra´ul Tempone. “Multilevel hybrid
split-step implicit tau-leap”. In: Numerical Algorithms (2017), pp. 1–34 16

21. Multilevel Hybrid SSI-TL: Idea
Our multilevel hybrid SSI-TL estimator in (Ben Hammouda, Moraes, and
Tempone 2017) is
Q = QLimp
c
+
Lint
−1
∑
=Limp
c +1
Q + QLint +
L
∑
=Lint+1
Q , (4)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
QLimp
c
= 1
M
i,L
imp
c
M
i,L
imp
c
∑
m=1
g(Zimp
Limp
c ,[m]
(T))
Q = 1
Mii,
Mii,
∑
m =1
(g(Zimp
,[m ]
(T)) − g(Zimp
−1,[m ]
(T))), Limp
c + 1 ≤ ≤ Lint
− 1
QLint = 1
Mie,Lint
Mie,Lint
∑
m=1
(g(Zexp
Lint,[m]
(T)) − g(Zimp
Lint−1,[m]
(T)))
Q = 1
Mee,
Mee,
∑
m =1
(g(Zexp
,[m ]
(T)) − g(Zexp
−1,[m ]
(T))), Lint
+ 1 ≤ ≤ L.
Limp
c : the coarsest discretization level; Lint
: the interface level; L: the
finest discretization level; the number of samples per level:
M = {Mi,Limp
c
,{Mii, }Lint−1
=Limp
c +1
,Mie,Lint ,{Mee, }L
=Lint+1
}.
17

22. Estimating Limp
c
Coarsest discretization level, Limp
c , is determined by the numerical
stability constraint of our MLMC estimator, two conditions must be
satisfied:
The stability of a single path: determined by a linearized stability
analysis of the backward Euler method applied to the
deterministic ODE model corresponding to our system.
The stability of the variance of the coupled paths of our MLMC
estimator: expressed by
Var[g(ZLimp
c +1
)−g(ZLimp
c
)] ≪ Var[g(ZLimp
c
)].
18

23. Estimating L and M
Total number of levels, L, and the set of the number of
samples per level, M, are selected to satisfy the accuracy
constraint
P( E[g(X(T))] − ˆQ < TOL) > 1 − α. (5)
Assuming Normality of MLMC estimator: The MLMC
algorithm should bound the bias and the statistical error as
follows (Collier et al. 2014):
E[g(X(T)) − ˆQ] ≤ (1 − θ) TOL, (6)
Var[ ˆQ] ≤ (
θ TOL
Cα
)
2
(7)
for some given confidence parameter, Cα, such that
Φ(Cα) = 1 − α/2 ; here, Φ is the cumulative distribution function of
a standard normal rdv.
19

24. Checking Normality of the MLMC Estimator
95.098 95.1 95.102 95.104 95.106 95.108 95.11 95.112
0
50
100
150
200
250
MLMC estimates
Density
Empirical probability mass function of multilevel SSI−TL estimator for TOL=0.005 (Example 2)
−3 −2 −1 0 1 2 3
95.1
95.102
95.104
95.106
95.108
95.11
95.112
95.114
Standard Normal Quantiles
QuantilesofQoI
QoI vs. Standard Normal for TOL=0.005 (Example 2)
Figure 2.1: Left: Empirical probability mass function for 100 multilevel
SSI-TL estimates. Right: QQ-plot for the multilevel SSI-TL estimates.
20

25. Estimating L
The finest discretization level, L, is determined by satisfying
relation (6) for θ = 1
2, implying
Bias(L) = E[g(X(T)) − g(ZL(T))] <
TOL
2
.
In our numerical experiments, we use the following approximation (see
Giles 2008)
Bias(L) ≈ E[g(ZL(T)) − g(ZL−1(T))].
21

26. Estimating Lint and M
1 The first step is to solve (8), for a fixed value of the interface level, Lint
⎧⎪⎪⎪⎪⎪
⎨
⎪⎪⎪⎪⎪⎩
min
M
WLint (M)
s.t. Cα
L
∑
=Limp
c
M−1V ≤ TOL
2 ,
(8)
▸ V = Var[g(Z (T)) − g(Z −1(T)] is estimated by the extrapolation of the
sample variances obtained from the coarsest levels, due to the presence of
large kurtosis (We provide a solution to this issue in (Ben Hammouda,
Ben Rached, and Tempone 2019)).
▸ WLint is the expected computational cost of the MLMC estimator given by
WLint = Ci,Limp
c
Mi,Limp
c
h−1
Limp
c
+
Lint
−1
∑
=Limp
c +1
Cii, Mii, h−1
+ Cie,Lint Mie,Lint h−1
Lint +
L
∑
=Lint+1
Cee, Mee, h−1
, (9)
where Ci, Cii, Cie and Cee are, respectively, the expected computational
costs of simulating a single SSI-TL step, a coupled SSI-TL step, a coupled
SSI/explicit TL step and a coupled explicit-TL step.
22

27. Estimating Lint and M
2 Let us denote M∗
(Lint
) as the solution of (8). Then, the optimal value of
the switching parameter, Lint∗
, is chosen by solving
⎧⎪⎪
⎨
⎪⎪⎩
min
Lint
WLint (M∗
(Lint
))
s.t. Lexp
c ≤ Lint
≤ L.
In our numerical examples, we found that the lowest computational cost is
achieved for Lint∗
= Lexp
c , i.e., the same level in which the explicit-TL is stable.
23

28. 1 Introduction
2 Multilevel Hybrid Split-Step Implicit Tau-Leap (TL)
(Ben Hammouda, Moraes, and Tempone 2017)
3 Numerical Experiments
4 Conclusions and Future Research Directions
23

29. Example (The decaying-dimerizing reaction (Gillespie 2001))
This system of reactions consists of three species, S1, S2, and S3,
and four reaction channels:
S1
c1
→ 0, S1 + S1
c2
→ S2
S2
c3
→ S1 + S1, S2
c4
→ S3,
with c = (1,10,103
,10−1
), T = 0.2, X(t) = (S1(t),S2(t),S3(t)) and
X0 = (400,798,0). The stoichiometric matrix and and the
propensity functions are given by
ν =
⎛
⎜
⎜
⎜
⎝
−1 0 0
−2 1 0
2 −1 0
0 −1 1
⎞
⎟
⎟
⎟
⎠
, a(X) =
⎛
⎜
⎜
⎜
⎝
c1S1
c2S1(S1 − 1)
c3S2
c4S2
⎞
⎟
⎟
⎟
⎠
We are interested in approximating E[X(3)
(T)].

30. Trajectories of Dimer Species of Example 1
0 0.05 0.1 0.15 0.2 0.25
0
200
400
600
800
1000
Dimer species trajectories by the SSA at final time T=0.2
Time (sec)
Molecules
X1
X2
X3
Figure 3.1: Trajectories of dimer species simulated by SSA at the final time,
T = 0.2. This system has multiple timescales. The step size, τexp
, is therefore
taken to be extremely small to ensure the numerical stability of the explicit
TL method (τlim
exp ≈ 2.3 × 10−4
).
25

31. Why Hybrid Estimator?
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Comparison of the expected cost per sample per level for the multilevel explicit TL and SSI−TL methods
Level of discretization (l)
Expectedcostpersample
Multilevel SSI−TL
Multilevel Explicit TL
Figure 3.2: Comparison of the expected cost per sample per level for the
different methods using 104
samples, for Example 1. The first observation
corresponds to the time of a single path for the coarsest level and the other
observations correspond to the time of the coupled paths per level.
26

32. Multilevel Hybrid SSI-TL: Results I
Method / TOL 0.02 0.01 0.005
Multilevel explicit-TL 890 (9) 5300 (45) 2.2e+04 (96)
Multilevel SSI-TL 24 (0.8) 110 (3) 5.3e+02 (7)
Wexp
MLMC/WSSI
MLMC 37.04 47.33 41.27
Table 1: Comparison of the expected total work for the different methods (in
seconds) using 100 multilevel runs, for Example 1. (⋅) refers to the standard
deviation. The last row shows that the multilevel SSI-TL estimator
outperforms the multilevel explicit-TL estimator by about 40 times in terms
of computational work.
27

33. Multilevel Hybrid SSI-TL: Results II
10
−3
10
−2
10
−1
10
0
10
2
10
4
10
6
10
8
Comparison of the expected total work for the different methods for different values of tolerance TOL
TOL
Expectedtotalwork(W)(seconds)
Drift−implicit MLMC tau leap (Lc
imp
=0)
Linear reference with slope −2.25
Explicit MLMC tau leap (Lc
exp
=11)
Linear reference with slope −2.37
Drift−implicit MC tau leap
Linear reference with slope −2.77
y=TOL
−2
(log(TOL))
2
y=TOL−3
Figure 3.3: Comparison of the expected total work for the different methods
with different values of tolerance (TOL) for Example 1 using 100 multilevel
runs.
The computational work, W, of both multilevel SSI-TL and
multilevel explicit-TL methods is O (TOL−2
(log(TOL))2
)
compared to O (TOL−3
) for the MC SSI-TL.
The multilevel SSI-TL estimator significantly outperforms the
multilevel explicit-TL estimator, in terms of computational work.
28

34. Multilevel Hybrid SSI-TL: Results III
10
−3
10
−2
10
−1
10
−4
10
−3
10
−2
10
−1
TOL
Totalerror
TOL vs. Global error
1 %
1 %
2 %
5 %
Figure 3.4: TOL versus the actual computational error. The numbers above
the straight line show the percentage of runs that had errors larger than the
required tolerance.
29

35. 1 Introduction
2 Multilevel Hybrid Split-Step Implicit Tau-Leap (TL)
(Ben Hammouda, Moraes, and Tempone 2017)
3 Numerical Experiments
4 Conclusions and Future Research Directions
29

36. Conclusions
1 We propose a novel scheme to simulate single paths of SRNs, for
systems with the presence of slow and fast timescales (stiff
systems), where the explicit TL scheme suffers from numerical
instability.
2 We propose a novel hybrid multilevel estimator that uses a novel
implicit scheme only at the coarser levels and then, starting from a
certain interface level, it switches to the explicit scheme.
3 Our analysis and numerical results show that our proposed
estimator achieve a computational complexity of order
O (TOL−2
log(TOL)2
) with a significantly smaller constant than
MLMC explicit Tau-leap (Anderson and Higham 2012).
4 More details can be found in
Chiheb Ben Hammouda, Alvaro Moraes, and Ra´ul Tempone.
“Multilevel hybrid split-step implicit tau-leap”. In: Numerical
Algorithms (2017), pp. 1–34.
30

37. Future Work
1 Implement Hybridization techniques involving adaptive methods
introduced in (Karlsson and Tempone 2011; Moraes, Tempone,
and Vilanova 2015; Lester et al. 2015) ⇒ to construct adaptive
hybrid multilevel estimators by switching between SSI-TL, explicit
TL, and exact SSA, not only within levels of MLMC, but also
within the course of a single sample.
2 Addressing spatial inhomogeneities described, for instance, by
graphs and/or continuum volumes through the use of Multi-Index
Monte Carlo technology (Haji-Ali, Nobile, and Tempone 2015).
3 Investigating connections between the different scales and
approaches. For instance: connecting the theory and tools of
SRNs (microscopic scale) to population balance equations
(macroscopic scale).
31

38. References I
D. Anderson and D. Higham. “Multilevel Monte Carlo for
continuous Markov chains, with applications in biochemical
kinetics”. In: SIAM Multiscal Model. Simul. 10.1 (2012).
David F. Anderson. “A modified next reaction method for
simulating chemical systems with time dependent propensities
and delays”. In: The Journal of Chemical Physics 127.21 (2007),
p. 214107.
David F Anderson and Thomas G Kurtz. Stochastic analysis of
biochemical systems. Springer, 2015.
Chiheb Ben Hammouda, Nadhir Ben Rached, and
Ra´ul Tempone. “Importance sampling for a robust and efficient
multilevel Monte Carlo estimator for stochastic reaction
networks”. In: arXiv preprint arXiv:1911.06286 (2019).
32

39. References II
Chiheb Ben Hammouda, Alvaro Moraes, and Ra´ul Tempone.
“Multilevel hybrid split-step implicit tau-leap”. In: Numerical
Algorithms (2017), pp. 1–34.
Corentin Briat, Ankit Gupta, and Mustafa Khammash. “A
Control Theory for Stochastic Biomolecular Regulation”. In:
SIAM Conference on Control Theory and its Applications. SIAM.
2015.
K Andrew Cliffe et al. “Multilevel Monte Carlo methods and
applications to elliptic PDEs with random coefficients”. In:
Computing and Visualization in Science 14.1 (2011), p. 3.
Nathan Collier et al. “A continuation multilevel Monte Carlo
algorithm”. In: BIT Numerical Mathematics 55.2 (2014),
pp. 399–432.
Michael B Elowitz et al. “Stochastic gene expression in a single
cell”. In: Science 297.5584 (2002), pp. 1183–1186.
33

40. References III
Stewart N. Ethier and Thomas G. Kurtz. Markov processes :
characterization and convergence. Wiley series in probability and
mathematical statistics. J. Wiley & Sons, 1986. isbn:
0-471-08186-8.
Michael Giles. “Multi-level Monte Carlo path simulation”. In:
Operations Research 53.3 (2008), pp. 607–617.
D. T. Gillespie. “Approximate accelerated stochastic simulation
of chemically reacting systems”. In: Journal of Chemical Physics
115 (July 2001), pp. 1716–1733. doi: 10.1063/1.1378322.
Daniel T. Gillespie. “A General Method for Numerically
Simulating the Stochastic Time Evolution of Coupled Chemical
Reactions”. In: Journal of Computational Physics 22 (1976),
pp. 403–434.
34

41. References IV
Abdul-Lateef Haji-Ali, Fabio Nobile, and Ra´ul Tempone.
“Multi-index Monte Carlo: when sparsity meets sampling”. In:
Numerische Mathematik (2015), pp. 1–40.
SebastianC. Hensel, JamesB. Rawlings, and John Yin.
“Stochastic Kinetic Modeling of Vesicular Stomatitis Virus
Intracellular Growth”. English. In: Bulletin of Mathematical
Biology 71.7 (2009), pp. 1671–1692. issn: 0092-8240.
H. Solari J. Aparicio. “Population dynamics: Poisson
approximation and its relation to the langevin process”. In:
Physical Review Letters (2001), p. 4183.
Jesper Karlsson and Raul Tempone. “Towards Automatic Global
Error Control: Computable Weak Error Expansion for the
Tau-Leap Method”. In: Monte Carlo Methods and Applications
17.3 (2011), pp. 233–278. doi: 10.1515/MCMA.2011.011.
35

42. References V
Thomas G. Kurtz. “Representation and approximation of
counting processes”. In: Advances in Filtering and Optimal
Stochastic Control. Vol. 42. Lecture Notes in Control and
Information Sciences. Springer Berlin Heidelberg, 1982,
pp. 177–191.
Christopher Lester et al. “An adaptive multi-level simulation
algorithm for stochastic biological systems”. In: The Journal of
chemical physics 142.2 (2015), p. 024113.
Alvaro Moraes, Ra´ul Tempone, and Pedro Vilanova. “Multilevel
adaptive reaction-splitting simulation method for stochastic
reaction networks.” In: preprint arXiv:1406.1989 (2015).
Arjun Raj et al. “Stochastic mRNA synthesis in mammalian
cells”. In: PLoS biology 4.10 (2006), e309.
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43. Thank you for your attention
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