A continuation multilevel Monte Carlo algorithm

Abstract

We propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for weak approximation of stochastic models. The CMLMC algorithm solves the given approximation problem for a sequence of decreasing tolerances, ending when the required error tolerance is satisfied. CMLMC assumes discretization hierarchies that are defined a priori for each level and are geometrically refined across levels. The actual choice of computational work across levels is based on parametric models for the average cost per sample and the corresponding variance and weak error. These parameters are calibrated using Bayesian estimation, taking particular notice of the deepest levels of the discretization hierarchy, where only few realizations are available to produce the estimates. The resulting CMLMC estimator exhibits a non-trivial splitting between bias and statistical contributions. We also show the asymptotic normality of the statistical error in the MLMC estimator and justify in this way our error estimate that allows prescribing both required accuracy and confidence in the final result. Numerical results substantiate the above results and illustrate the corresponding computational savings in examples that are described in terms of differential equations either driven by random measures or with random coefficients.

Appendix: Normality of MLMC estimator

Theorem 7.1

([10, LindebergspsFeller Theorem, p. 114]) For each $n$, let $X_{n,m}$, for $1 \le n \le m$, be independent random variables (not necessarily identical). Denote

$$\begin{aligned} a_n&= \sum _{m=1}^n X_{n,m}, \\ Y_{n,m}&=X_{n,m} - \mathrm{E}\left[ {X_{n,m}}\right] , \\ s_n^2&= \sum _{m=1}^n \mathrm{E}\left[ {Y_{n,m}^2}\right] . \end{aligned}$$

Suppose the following Lindeberg condition is satisfied for all $\epsilon >0$:

$$\begin{aligned} \lim _{n \rightarrow \infty } s_n^{-2}\sum _{m=1}^n \mathrm{E}\left[ {Y_{n,m}^2 \mathbf {1}_{|Y_{n,m} | > \epsilon s_n}}\right] = 0. \end{aligned}$$

(7.1)

Then,

$$\begin{aligned} \lim _{n \rightarrow \infty } {{\hbox {P}}\left[ \frac{a_n - \mathrm{E}\left[ {a_n}\right] }{s_n} \le z \right] } = \varPhi (z), \end{aligned}$$

where $\varPhi (z)$ is the normal cumulative density function of a standard normal random variable.

Lemma 7.1

Consider the MLMC estimator $\mathcal {A}$ given by

$$\begin{aligned} \mathcal {A} = \sum _{\ell =0}^L \sum _{m=1}^{M_\ell } \frac{G_{\ell }(\omega _{\ell , m})}{M_\ell }, \end{aligned}$$

where $G_{\ell }(\omega _{\ell , m})$ denote as usual i.i.d. samples of the random variable $G_\ell$. The family of random variables, $(G_\ell )_{\ell \ge 0}$, is also assumed independent. Denote ${Y_\ell = |G_\ell - {\hbox {E}\left[ G_\ell \right] }|}$ and assume the following

$$\begin{aligned} C_1 \beta ^{-q_3\ell }&\le \mathrm{E}\left[ {Y_\ell ^2}\right]&\text { for all } \ell \ge 0, \end{aligned}$$

(7.2a)

$$\begin{aligned} \mathrm{E}\left[ {Y_\ell ^{2+\delta }}\right]&\le C_2 \beta ^{-\tau \ell }&\text { for all } \ell \ge 0, \end{aligned}$$

(7.2b)

for some $\beta > 1$ and strictly positive constants $C_1, C_2, q_3,\delta$ and $\tau$. Choose the number of samples on each level $M_\ell$ to satisfy, for $q_2 > 0$ and a strictly positive sequence $\{H_\ell \}_{\ell \ge 0}$

$$\begin{aligned} M_\ell \ge \beta ^{-q_2\ell } \mathrm{\mathrm{{TOL}}}^{-2} H_\ell ^{-1} \left( \sum _{\ell =0}^L H_\ell \right)&\qquad \text {for all } \ell \ge 0. \end{aligned}$$

(7.3)

Moreover, choose the number of levels $L$ to satisfy

$$\begin{aligned} L&\le \max \left( 0, \frac{c \log \left( \mathrm{{TOL}}^{-1}\right) }{\log {\beta }} + C\right) \end{aligned}$$

(7.4)

for some constants $C$, and $c>0$. Finally, denoting

$$\begin{aligned} p= (1+\delta /2) q_3 + (\delta /2)q_2 -\tau , \end{aligned}$$

if we have that either $p>0$ or $c < \delta /p$, then

$$\begin{aligned} \lim _{\mathrm{{TOL}}\rightarrow 0} {{\hbox {P}}\left[ \frac{\mathcal {A}- \mathrm{E}\left[ {\mathcal A}\right] }{\sqrt{{\mathrm{{Var}}\left[ \mathcal A\right] }}} \le z\right] } = \varPhi \left( z \right) . \end{aligned}$$

Proof

We prove this lemma by ensuring that the Lindeberg condition (7.1) is satisfied. The condition becomes in this case

$$\begin{aligned} \lim _{\mathrm{{TOL}}\rightarrow 0} \underbrace{\frac{1}{{\mathrm{{Var}}\left[ \mathcal A\right] }} \sum _{\ell =0}^L \sum _{m=1}^{M_\ell } {\hbox {E}\left[ \frac{Y_\ell ^2}{M_\ell ^2} \mathbf {1}_{\frac{Y_\ell }{M_\ell } > \epsilon \sqrt{{\mathrm{{Var}}\left[ \mathcal A\right] }}}\right] }}_{:= F} = 0, \end{aligned}$$

for all $\epsilon > 0$. Below we make repeated use of the following identity for non-negative sequences ${\{a_\ell \}}$ and ${\{b_\ell \}}$ and $q \ge 0$.

$$\begin{aligned} \sum _{\ell }{a_\ell ^q b_\ell } \le \left( \sum _\ell a_\ell \right) ^q \sum _\ell b_\ell . \end{aligned}$$

(7.5)

First we use the Markov inequality to bound

$$\begin{aligned} F&= \frac{1}{{\mathrm{{Var}}\left[ \mathcal A\right] }} \sum _{\ell =0}^L \sum _{m=1}^{M_\ell } {\hbox {E}\left[ \frac{Y_\ell ^2}{M_\ell ^2} \mathbf {1}_{Y_\ell > \epsilon \sqrt{{\mathrm{{Var}}\left[ \mathcal A\right] }} M_\ell }\right] } \\&\le \frac{\epsilon ^{-\delta }}{{\mathrm{{Var}}\left[ \mathcal A\right] }^{1+\delta /2}} \sum _{\ell =0}^L M_\ell ^{-1-\delta } {\hbox {E}\left[ Y_\ell ^{2+\delta }\right] }. \end{aligned}$$

Using (7.5) and substituting for the variance ${\mathrm{{Var}}\left[ \mathcal A\right] }$ where we denote ${\mathrm{{Var}}\left[ G_\ell \right] } = {\hbox {E}\left[ \left( G_\ell - {\hbox {E}\left[ G_\ell \right] } \right) ^{2}\right] }$ by $V_\ell$, we find

$$\begin{aligned} F&\le \frac{\epsilon ^{-\delta } \left( \sum _{\ell =0}^L M_\ell ^{-1} V_\ell \right) ^{1+\delta /2}}{\left( \sum _{\ell =0}^L V_\ell M_\ell ^{-1} \right) ^{1+\delta /2}} \sum _{\ell =0}^L V_\ell ^{-1-\delta /2} M_\ell ^{-\delta /2} {\hbox {E}\left[ Y_\ell ^{2+\delta }\right] } \\&\le \epsilon ^{-\delta } \sum _{\ell =0}^L V_\ell ^{-1-{\delta }/{2}} M_\ell ^{-{\delta }/{2}} {\hbox {E}\left[ Y_\ell ^{2+\delta }\right] }. \end{aligned}$$

Using the lower bound on the number of samples $M_\ell$ (7.3) and (7.5) again yields

$$\begin{aligned} F&\le \epsilon ^{-\delta } \mathrm{{TOL}}^\delta \left( \sum _{\ell =0}^L V_\ell ^{-1-{\delta }/{2}} \beta ^{\frac{\delta q_2 \ell }{2}} H_\ell ^{\delta /2} {\hbox {E}\left[ Y_\ell ^{2+\delta }\right] } \right) \left( \sum _{\ell =0}^L H_\ell \right) ^{-\delta /2} \\&\le \epsilon ^{-\delta } \mathrm{{TOL}}^\delta \left( \sum _{\ell =0}^L V_\ell ^{-1-{\delta }/{2}} \beta ^{(\delta /2)q_2 \ell } {\hbox {E}\left[ Y_\ell ^{2+\delta }\right] } \right) . \end{aligned}$$

Finally using the bounds (7.2a) and (7.2b)

$$\begin{aligned} F&\le \epsilon ^{-\delta } \mathrm{{TOL}}^\delta \left( C_1^{-1-{\delta }/{2}} C_2 \sum _{\ell =0}^L \beta ^{(1+\delta /2) q_3\ell } \beta ^{(\delta /2)q_2 \ell } \beta ^{-\tau \ell } \right) \\&= \epsilon ^{-\delta }\mathrm{{TOL}}^{\delta } C_1^{-1-{\delta }/{2}} C_2 \frac{\beta ^{(L+1)p} - 1}{\beta ^p-1} , \end{aligned}$$

We distinguish two cases here, namely:

• If $p>0$ is satisfied then $\lim _{\mathrm{{TOL}}\rightarrow 0} F= 0$ for any $c>0$.

• Otherwise, substituting (7.4) gives

  $$\begin{aligned} F&\le \epsilon ^{-\delta } \mathrm{{TOL}}^{\delta } C_1^{-1-{\delta }/{2}} C_2 \frac{\mathrm{{TOL}}^{-cp } \beta ^{(C+1)p} - 1}{\beta ^p-1} = {\mathcal {O}(\mathrm{{TOL}}^{\delta - cp})} , \end{aligned}$$

  and since in this case $cp < \delta$ then $\lim _{\mathrm{{TOL}}\rightarrow 0} F = 0$. $\square$

Remark 7.1

The choice (7.3) mirrors the choice (2.7), the latter being the optimal number of samples to bound the statistical error of the estimator by $\mathrm{{TOL}}$. Specifically, $H_\ell \propto \sqrt{V_\ell W_\ell }$ where $W_\ell$ is the work per sample on level $\ell$. Moreover, the choice (2.7) uses the variances $\{ V_\ell \}_{\ell =0}^L$ or an estimate of it in the actual implementation. On the other hand, the choice (7.3) uses the upper bound of $V_\ell$ instead, if $q_2$ is the rate of variance convergence therein. Furthermore, if we assume the weak error model (2.9a) holds and $h_L = h_0 \beta ^{-L}$ then we must have

$$\begin{aligned} Q_W h_L^{q_1} = Q_W h_0^{q_1} \beta ^{-L q_1} \le (1-\theta )\mathrm{{TOL}}, \end{aligned}$$

which gives a lower bound on the number of levels $L$, namely

$$\begin{aligned} L \ge \frac{\log (\mathrm{{TOL}}^{-1})}{q_1\log (\beta )} + \frac{-\log (1-\theta ) +\log (Q_W) + q_1\log (h_0)}{q_1\log (\beta )}, \end{aligned}$$

to bound the bias by $\mathrm{{TOL}}$.

Finally, in Example 2.1 the conditions (7.2) are satisfied for $q_3=2$ and, assuming $p^*> 3$, for $\delta =1$ and $\tau =6$. Similarly, Example 2.2 satisfies the conditions (7.2) are for $q_3=1$ and $\delta =2$ and $\tau =2$, cf. [20].

Remark 7.2

The assumption (7.2a) can be relaxed. For instance, one can assume instead that

$$\begin{aligned} V_{\ell +1}&\le V_{\ell } \qquad \text { for all }\ell \ge 1,\\ 0&< \lim _{\ell \rightarrow \infty } {\mathrm{{Var}}\left[ Y_\ell \right] } \beta ^{q_3 \ell } < \infty , \end{aligned}$$

and slightly different conditions on $L$.