Global Sensitivity Analysis for Models Described by Stochastic Differential Equations

Abstract

Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest. One of the statistical tools used to quantify the influence of each input variable on the quantity of interest are the Sobol’ sensitivity indices. In this paper, we consider stochastic models described by stochastic differential equations (SDE). We focus the study on mean quantities, defined as the expectation with respect to the Wiener measure of a quantity of interest related to the solution of the SDE itself. Our approach is based on a Feynman-Kac representation of the quantity of interest, from which we get a parametrized partial differential equation (PDE) representation of our initial problem. We then handle the uncertainty on the parametrized PDE using polynomial chaos expansion and a stochastic Galerkin projection.

Appendix

1.1 A.1 Integrability of the Stochastic Quantities of Interest

Proof

(Proof of Lemma 1) Let $q\in \mathbb {N}^{*}$ and t ≥ 0. Using Problem 5.2.15 in Karatzas and Shreve (1991), we can claim that, under (A3), for any z ∈Ξ

$$ {\int}_{C} |X_{t}(\theta,z)|^{q}\mathbb{W}(d\theta)\leq C(1+|x|^{q})\exp\left( Ct\right), $$

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where C = C(t,M). As the constant M in Eq. 1 is uniform in z, the constant C does not depend on z either. Integrating Eq. 39 over Ξ and against ℙ_ξ we get the desired result. □

Proof

(Proof of Lemma 2) Here we revisit the proof of Lemma 5.7.4 in Karatzas and Shreve (1991) and give details for the sake of completeness. Thanks to (A4), we have for example a₁₁(x,z) ≥ λ for any $x=(x_{1}, {\ldots } , x_{d})\in \mathbb {R}^{d}$, z ∈Ξ. We note $q=\min _{x\in \bar {D}}x_{1}$ and b a constant s.t. |b(x,z)|≤ b < ∞,∀x ∈ D,∀z ∈Ξ.

Such a finite constant b exists thanks to Eq. 1. Set now ν > 2b/λ and consider the function h(x) = −μ exp(νx¹), x ∈ D, where the constant μ will be determined later. This function is of class C^∞(D) and satisfies for any x ∈ D and z ∈Ξ,

$$ \begin{array}{@{}rcl@{}} &&-\frac1 2 {\sum}_{i,j=1}^{d}a_{ij}(x,z)\partial^{2}_{x_{i}x_{j}}h(x)-{\sum}_{j=1}^{d}b_{j}(x,z)\partial_{x_{i}}h(x)\\ &&\quad \quad\quad= \mu e^{\nu x^{1}}\left( \frac{1}{2}\nu^{2}a_{11}(x,z)+\nu b_{1}(x,z)\right) \geq\frac{1}{2} \mu\nu \lambda e^{\nu q}\left( \nu-\frac{2b}{\lambda} \right). \end{array} $$

Choosing μ sufficiently large, we have for any x ∈ D, any z ∈Ξ:

$$\frac1 2 \sum\limits_{i,j=1}^{d}a_{ij}(x,z)\partial^{2}_{x_{i}x_{j}}h(x)+\sum\limits_{j=1}^{d}b_{j}(x,z)\partial_{x_{i}}h(x)\leq -1.$$

Let us write τ_D(ω) = τ_D(𝜃,z). As X₀ = x ∈ D, using Itô Formula and taking the expectation against $\mathbb {W}$ we get:

$$ \forall z\in{\Xi} {\int}_{C}(t\wedge\tau_{D}(\theta,z)\mathbb{W}(d\theta)\leq h(x)-{\int}_{C}h(X_{t\wedge\tau_{D}}(\theta,z))\mathbb{W}(d\theta)\leq 2\max_{y\in\bar{D}}|h(y)| .$$

Note that in the above computation, the expectation of the stochastic integral vanishes, as h is bounded with bounded derivatives. Then, integrating over Ξ and against ℙ_ξ the above inequality we get the result by monotone convergence. □

Proof

(Proof of Lemma 3) According to Theorem 1, the function $\mathfrak {U}(\cdot ,\xi )$ solves Eq. 16 (for a.e. fixed value of ξ). We now use a maximum principle argument to get an a priori bound on $\mathfrak {U}(\cdot ,\xi )$. Without loss of generality we assume that D lies in the slab 0 < x¹ < δ for a certain 0 < δ < ∞ (the general case can be recovered by translation arguments). According to Theorem 3.7 in Gilbarg and Trudinger (1983) and its proof we have

$$ \sup_{x\in\bar{D}}|\mathfrak{U}(x,\xi)|\leq \frac{C^{\prime}(\xi)}{\lambda} $$

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where C^′(ξ) = e^α(ξ)δ − 1 with α(ξ) a quantity chosen s.t. $\alpha (\xi )\geq 1+\frac {1}{\lambda }\sup _{x\in D}|b(x,\xi )|$ (note that the uniform ellipticity constant λ in (A4) does not depend on z). Then, thanks to the uniformity in z of the constant M in Eq. 1 and to the boundedness of D, the quantity α(ξ) may be chosen independent on ξ. Therefore C^′(ξ) = C^′ does not depend on ξ, and thus we get the result with C = C^′/λ. □

1.2 A.2 Results on Stochastic Partial Differential Equations

Proof

(Proof of Lemma 5) As $\tilde {f}$ is uniformly bounded in x and z, it is easy to prove that the form F is continuous on $\mathcal {V}$ (with the help of Cauchy-Schwarz’s inequality). Thanks to (A7) and (A9), the bilinear form $\mathcal {A}$ is continuous on $\mathcal {V}$. Thanks to (A8)–(A9) one may check the coercivity of $\mathcal {A}$. Indeed, using Lemma 8.4 in Gilbarg and Trudinger (1983), one gets for any z ∈Ξ,

$$ \begin{array}{@{}rcl@{}} {\int}_{D}(\nabla v(\cdot,z))^{T}\tilde{a}(\cdot,z)\nabla v(\cdot,z) + {\int}_{D}(\nabla v(\cdot,z))^{T}\tilde{b}(\cdot,z)v(\cdot,z)& \\ \quad \quad \quad \geq\frac{\tilde{\lambda}}{2}{\int}_{D}|\nabla v(\cdot,z)|^{2}-\tilde{\lambda}\nu^{2}{\int}_{D}v^{2}(\cdot,z)& \end{array} $$

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where $\nu =\frac {\tilde {M}}{\tilde {\lambda }}$. Using now the definition of $||\cdot ||_{H^{1}(D)}$, Inequality Eq. 22, and integrating Eq. 41 over Ξ against ℙ_ξ we get

$$ \mathcal{A}(v,v)\geq \left( \frac{\tilde{\lambda}}{2C(d,|D|)}-\tilde{\lambda}\nu^{2}\right)||v||^{2}_{\mathcal{V}}$$

(we recall that C(d,|D|) is defined by Eq. 23). As $\tilde {M}<\frac {\tilde {\lambda }}{\sqrt {2C(d,|D|)}}$, we have that $\frac {\tilde {\lambda }}{2C(d,|D|)}-\tilde {\lambda }\nu ^{2}>0$. The existence of a unique weak solution at the stochastic level then follows from the Lax-Milgram theorem.

Now, as F and $\mathcal {A}$ are continuous, as $\mathcal {A}$ is coercive and as $\mathcal {V}^{N,P}\subset \mathcal {V}$, the existence of a unique solution to Eq. 24 follows from Lax-Milgram theorem again, but applied this time on $\mathcal {V}^{N,P}$. In order to prove the convergence result, we first notice that thanks to the Céa lemma we have for any $N,P\in \mathbb {N}^{*}$

$$ ||u-u^{N,P}||_{\mathcal{V}}\leq \tilde{C}\min_{v\in\mathcal{V}^{N,P}}||u-v||_{\mathcal{V}}, $$

where the constant $\tilde {C}$ depends on $\tilde {\lambda }$, $\tilde {{\Lambda } }$, $\tilde {M}$ and C(d,|D|). Second, we recall that $\mathcal {V}^{N,P}\subset \mathcal {V}^{N+1,P+1}$ for any N,P, and that $\bigcup _{N,P}\mathcal {V}^{N,P}$ is dense in $\mathcal {V}$. This is sufficient in order to prove that $\min _{v\in \mathcal {V}^{N,P}}||u-v||_{\mathcal {V}}\to 0$, as N,P →∞, and the result follows. □

Proof

(Proof of Theorem 3) By Corollary 1, $\mathfrak {U}(x,\xi )$ is a classical solution of Eq. 16. Then, using Theorem 6.6 in Gilbarg and Trudinger (1983) we get that, for a.e. value of ξ

$$ \forall 1\leq i\leq d,\quad \sup_{x\in\bar{D}}|\partial_{x_{i}}\mathfrak{U}(x,\xi)|\leq C^{\prime\prime}(\sup_{x\in\bar{D}}|\mathfrak{U}(x,\xi)|+1),$$

with C^″ depending on d, M, |D| and λ, but not of ξ. But, as already noticed in the proof of Lemma 3, we have $\sup _{x\in \bar {D}}|\mathfrak {U}(x,\xi )|\leq C$ with C depending on M, |D| and λ. Thus $\mathfrak {U}(x,\xi )$ belongs to the space $\mathcal {V}$.

As noticed in Section 3.2 any classical solution of Eq. 16 is a classical solution of Eq. 20). Let $v\in {C^{1}_{c}}(D)\otimes L^{2}(\Xi ,\mathbb{P} _{\xi })$. Multiplying the first line of Eq. 20 by v, integrating over D ×Ξ against dx ⊗ℙ_ξ(dz), and performing integration by parts w.r.t the variable x, we get that $\mathcal {A}(\mathfrak {U},v)=F(v)$. Thanks to Eq. 1 in (A3), the coefficients $\tilde {a}$ and $\tilde {b}$ are obviously bounded uniformly in x and z, because D is bounded, and we have therefore (A7) and the continuity of the form $\mathcal {A}$. Using then density arguments we get $\mathcal {A}(\mathfrak {U},v)=F(v)$ for all $v\in \mathcal {V}$. (A8) is a consequence of (A4), and we are under (A9). Thus we may get the approximation result by applying Lemma 5. □

Proof

(Proof of Lemma 6) In the proof of Lemma 5 we have seen that (A7) and the boundedness of $\tilde {b}$ imply that the form $\mathcal {A}$ is continuous on $\mathcal {V}$.

We now prove that we can find μ ≥ 0 large enough such that the form $\mathcal {A}_{\mu }:(u,v)\mapsto \mathcal {A}(u,v)+\mu \langle u,v\rangle _{\mathcal {H}}$ is coercive on $\mathcal {V}$. Let us denote $\tilde {M}$ the upper bound for $\tilde {b}$ (i.e. $|\tilde {b}(x,z)|\leq \tilde {M}$ for any $x\in \bar {D}$, z ∈Ξ). Proceeding as for Eq. 41 (that we integrate over Ξ against ℙ_ξ) we get that for any $v\in \mathcal {V}$

$$ \mathcal{A}_{\mu}(v,v)\geq \frac{\tilde{\lambda}}{2}\mathbb{E}_{\xi}\left( {\int}_{D}|\nabla v(\cdot,\xi)|^{2}\right)+\left( \mu-\frac{\tilde{M}^{2}}{\tilde{\lambda}}\right) \mathbb{E}_{\xi}\left( {\int}_{D}v^{2}(\cdot,\xi)\right).$$

Note that if $\tilde {M}<\frac {\tilde {\lambda }}{\sqrt {2C(d,|D|)}}$ it suffices to take μ = 0 (see the proof of Lemma 5). If this is not the case we take $\mu >\frac {\tilde {M}^{2}}{\tilde {\lambda }}$ and set $c=\min \left (\frac {\tilde {\lambda }}{2},\mu -\frac {\tilde {M}^{2}}{\tilde {\lambda }}\right )$. We then have $\mathcal {A}_{\mu }(v,v)\geq c||v||^{2}_{\mathcal {V}}$ for any $v\in \mathcal {V}$.

This is sufficient to prove the result (see Lemma 7.1.2 and Theorem 7.1.4 in Ern and Guermond 2002; see also the earlier reference Lions and Magenes 1972). □

Proof

(Proof of Lemma 7) By Corollary 1, $\mathfrak {V}(t,x,\xi )$ is a classical solution of Eq. 17. Using Theorem 5.14 in Lieberman (1996) we get that for a.e. value of ξ

$$ \sup_{(t,x)\in [0,T]\times\bar{D}}| \mathfrak{V}(t,x,\xi)|+ \sup_{(t,x)\in [0,T]\times\bar{D}}|\partial_{t} \mathfrak{V}(t,x,\xi)| +\sum\limits_{i=1}^{d}\sup_{(t,x)\in [0,T]\times\bar{D}}|\partial_{x_{i}} \mathfrak{V}(t,x,\xi)| $$

is bounded above by some constant C that depends on M, |D|, λ and d but not on ξ. We use that the coefficients a and b are Lipschitz continuous and that the initial condition f is in the H ”older space H_2+α for α = 1; see pp 46–47 of Lieberman (1996) for more details on Hölder spaces and norms in the parabolic setting. The point is that here we can write these properties of a, b and f with quantities that are uniform w.r.t. the uncertain parameter ξ. Thus the function $\mathfrak {V}(t,x,\xi )$ belongs to the space $L^{2}(0,T;\mathcal {V})\cap H^{1}(0;T,\mathcal {V}^{\prime })$. As it is continuous in time and bounded (in particular uniformly w.r.t. ξ) it is also in $C([0,T];\mathcal {H})$.

As noticed in Section 3.3.1 the function $\mathfrak {V}(t,x,\xi )$ is a classical solution of Eq. 28. Multiplying the first line of Eq. 28 by a test function in ${C^{1}_{c}}(D)\otimes L^{2}(\Xi ,\mathbb{P} _{\xi })$, integrating over D ×Ξ against dx ⊗ℙ_ξ(dz), and using density arguments we get

$$ \forall w\in\mathcal{V}, \langle \partial_{t}\mathfrak{V}(t,\cdot,\cdot),w\rangle_{\mathcal{H}}+\mathcal{A}(\mathfrak{V}(t,\cdot,\cdot),w)=0, \forall t\in[0,T] \text{ and } \mathfrak{V}(0,\cdot,\cdot)=f.$$

The result is proved. □

Proof

(Proof of Lemma 8) Remember that in the proof of Lemma 6 we have seen that we can find μ ≥ 0 large enough such that the form $\mathcal {A}_{\mu }$ is coercive. Here for simplicity we assume that μ = 0, i.e. $\mathcal {A}$ is coercive (with constant c). We claim that this is without loss of generality. Indeed if μ > 0 we consider v_μ the solution of

$$ \forall w\in \mathcal{V}, \langle \partial_{t} v_{\mu}(t,\cdot) , w\rangle_{\mathcal{H}}+\mathcal{A}_{\mu}\left( v_{\mu}(t,\cdot) , w\right)=0\text{ for a.e. } t\in[0,T]\text{ and }v_{\mu}(0,\cdot)=f $$

(42)

(the solution exists because the form $\mathcal {A}_{\mu }$ is continuous and coercive). Then it is obvious that the function v(t,⋅) = e^μtv_μ(t,⋅) solves Eq. 29 (in fact Eqs. 29 and 42 are equivalent via the change of variable). Thus one may approach v_μ by the 𝜃-scheme and get an approximation of v applying again the change of variable.

Then for any N,P the form $\mathcal {A}$ defines a scalar product on $\mathcal {V}^{N,P}$, so that for any $u\in \mathcal {V}$ there is by the Riesz theorem an element π^N,Pu of $\mathcal {V}^{N,P}$ such that

$$ \forall w\in\mathcal{V}^{N,P},\quad \mathcal{A}({\Pi}^{N,P}u,w)=\mathcal{A}(u,w).$$

The application π^N,P is linear and continuous from $\mathcal {V}$ to $\mathcal {V}^{N,P}$ and is called the elliptic projection operator. It satisfies

$$ \forall u\in\mathcal{V},\quad\forall w\in\mathcal{V}^{N,P},\quad \mathcal{A}(u-{\Pi}^{N,P}u,w)=0. $$

Then, following the proof of Céa’s Lemma (Theorem 3.1-2 in Raviart and Thomas 2004) one can prove that

$$ ||u-{\Pi}^{N,P}u||_{\mathcal{V}}\leq \tilde{C}\min_{w\in\mathcal{V}^{N,P}}||u-w||_{\mathcal{V}}, $$

where again $\tilde {C}$ depends on the continuity and coercivity constants of $\mathcal {A}$. Using the monotonicity and density assumptions on the $\mathcal {V}^{N,P}$’s we thus get that

$$ \forall u\in\mathcal{V},\quad ||u-{\Pi}^{N,P}u||_{\mathcal{V}}\xrightarrow[N\to\infty,P\to\infty]{}0. $$

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In the sequel we note π for π^N,P in order to lighten notations. For any $y\in C(0,T;\mathcal {H})$, t ∈ [0,T] we denote y(t) := y(t,⋅,⋅).

Then for any 0 ≤ m ≤ M we set

$$ e^{m}_{N,P}=v^{m,N,P}-{\Pi} v(t_{m})$$

(here we follow for example Raviart and Thomas (2004) §77.5). Note that Eq. 33 is equivalent to: ∀0 ≤ m ≤ M − 1, $\forall w\in \mathcal {V}^{N,P}$,

$$ \frac{1}{{\Delta} t} \langle v^{m+1,N,P}-v^{m,N,P},w\rangle_{\mathcal{H}}+\mathcal{A}\left( \theta v^{m+1,N,P}+(1-\theta)v^{m,N,P} , w\right)=0,$$

so that by some algebraic computations we see that: ∀0 ≤ m ≤ M − 1, $\forall w\in \mathcal {V}^{N,P}$,

$$ \frac{1}{{\Delta} t} \langle e^{m+1}_{N,P}-e^{m}_{N,P},w\rangle_{\mathcal{H}}+\mathcal{A}\left( \theta e^{m+1}_{N,P}+(1-\theta)e^{m}_{N,P} , w\right) =\langle \varepsilon^{m}_{N,P} , w\rangle_{\mathcal{H}} $$

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where $\varepsilon ^{m}_{N,P}\in \mathcal {V}^{\prime }$ is defined for any 0 ≤ m ≤ M − 1 by: $\forall w\in \mathcal {V}$,

$$ \langle \varepsilon^{m}_{N,P} , w\rangle_{\mathcal{H}}= -\frac{1}{{\Delta} t} \langle {\Pi} v(t_{m+1})-{\Pi} v(t_{m}),w\rangle_{\mathcal{H}} -\mathcal{A}\left( \theta v(t_{m+1})+ (1-\theta)v(t_{m}) , w\right). $$

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In the sequel we denote e^m (resp. ε) for $e^{m}_{N,P}$ (resp. $\varepsilon ^{m}_{N,P}$).

We now aim at showing that for any $\theta \in [\frac {1}{2},1]$ we have the stability result

$$ \forall 1\leq m\leq M,\quad ||e^{m}||_{\mathcal{H}}^{2}\leq ||e^{0}||_{\mathcal{H}}^{2}+\frac{{\Delta} t}{c}\sum\limits_{k=1}^{m-1}||\varepsilon^{k}||_{\mathcal{V}^{\prime}}^{2}. $$

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Then we will aim at controlling the $||\varepsilon ^{k}||_{\mathcal {V}^{\prime }}^{2}$’s (consistency result; we will only treat the case $\theta =\frac {1}{2}$). This will allow to get convergence.

Stability. Here we adapt the energy estimate method to be found pp66–67 in Dautray and Lions (1993). Let $\theta \in [\frac {1}{2},1]$ and fix 0 ≤ k ≤ M − 1. Taking $w=\theta e^{k+1}+(1-\theta )e^{k}=:\bar {e}^{k}$ in Eq. 44 we get

$$ \frac{1}{{\Delta} t}\langle e^{k+1}-e^{k},\bar{e}^{k}\rangle_{\mathcal{H}}+\mathcal{A}(\bar{e}_{k},\bar{e}_{k})=\langle \varepsilon^{k} , \bar{e}_{k}\rangle_{\mathcal{H}}. $$

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Using then the algebraic equality

$$ \langle e^{k+1}-e^{k}, \theta e^{k+1}+(1-\theta)e^{k}\rangle_{\mathcal{H}}=\frac{1}{2} ||e^{k+1}||_{\mathcal{H}}^{2}-\frac{1}{2} ||e^{k}||_{\mathcal{H}}^{2} +(\theta-\frac{1}{2}) ||e^{k+1}-e^{k}||_{\mathcal{H}}^{2} $$

in Eq. 47 we get

$$ \frac{1}{2} ||e^{k+1}||_{\mathcal{H}}^{2}-\frac{1}{2} ||e^{k}||_{\mathcal{H}}^{2} +\left( \theta-\frac{1}{2}\right) ||e^{k+1}-e^{k}||_{\mathcal{H}}^{2}+{\Delta} t\mathcal{A}(\bar{e}_{k},\bar{e}_{k})={\Delta} t \langle \varepsilon^{k} , \bar{e}_{k}\rangle_{\mathcal{H}}$$

and then

$$ \frac{1}{2} ||e^{k+1}||_{\mathcal{H}}^{2}-\frac{1}{2} ||e^{k}||_{\mathcal{H}}^{2}+ {\Delta} t\mathcal{A}(\bar{e}_{k},\bar{e}_{k})\leq{\Delta} t \langle \varepsilon^{k} , \bar{e}_{k}\rangle_{\mathcal{H}}. $$

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But for any $f\in \mathcal {V}^{\prime }$ and any $w\in \mathcal {V}$ we have, using Young’s inequality and the coercivity of $\mathcal {A}$,

$$ \langle f,w\rangle_{\mathcal{H}}\leq ||f||_{\mathcal{V}^{\prime}}||w||_{\mathcal{V}}\leq \frac{c}{2}||w||_{\mathcal{V}}^{2}+\frac{1}{2c}||f||_{\mathcal{V}^{\prime}}^{2}\leq \frac{1}{2}\mathcal{A}(w,w)+\frac{1}{2c}||f||_{\mathcal{V}^{\prime}}^{2}.$$

Using this in Eq. 48 we get

$$ \frac{1}{2} ||e^{k+1}||_{\mathcal{H}}^{2}-\frac{1}{2} ||e^{k}||_{\mathcal{H}}^{2}+ {\Delta} t\mathcal{A}(\bar{e}_{k},\bar{e}_{k})\leq{\Delta} t \left( \frac{1}{2} \mathcal{A}(\bar{e}^{k},\bar{e}^{k})+\frac{1}{2c}||\varepsilon^{k}||_{\mathcal{V}^{\prime}}^{2}\right) $$

and then $||e^{k+1}||_{\mathcal {H}}^{2}- ||e^{k}||_{\mathcal {H}}^{2}\leq \frac {{\Delta } t}{c}||\varepsilon ^{k}||_{\mathcal {V}^{\prime }}^{2}$. Summing over 0 ≤ k ≤ m − 1 for any 1 ≤ m ≤ M we get Eq. 46.

Consistency. Here we follow the lines of the proof of Lemma 7.5-1 in Raviart and Thomas (2004). Using Eq. 29 and the fact that $v\in C^{1}(0,T;\mathcal {V})$ we can rewrite Eq. 45 into: $\forall w\in \mathcal {V}$,

$$ \langle \varepsilon^{m} , w\rangle_{\mathcal{H}}= \left\langle \theta \partial_{t} v(t_{m+1})+(1-\theta)\partial_{t} v(t_{m}) \right\rangle_{\mathcal{H}} -\frac{1}{{\Delta} t} \left\langle {\Pi} v(t_{m+1})-{\Pi} v(t_{m}),w\right\rangle_{\mathcal{H}} $$

and further: $\forall w\in \mathcal {V}$,

$$ \begin{array}{lll} \langle \varepsilon^{m} , w\rangle_{\mathcal{H}}&=& \left\langle \theta \partial_{t} v(t_{m+1})+(1-\theta)\partial_{t} v(t_{m}) \right\rangle_{\mathcal{H}} -\frac{1}{{\Delta} t} \left\langle v(t_{m+1})- v(t_{m}),w\right\rangle_{\mathcal{H}}\\ \\ &&\displaystyle \hspace{3cm}+\frac{1}{{\Delta} t}{\int}_{t_{m}}^{t_{m+1}} \left\langle (I-{\Pi})\partial_{t}v(s) , w \right\rangle_{\mathcal{H}} ds. \end{array} $$

From this we get that for any 0 ≤ m ≤ M − 1

$$ ||\varepsilon^{m}||_{\mathcal{V}^{\prime}}=||\varepsilon^{m}||_{\mathcal{H}}\leq ||\eta(t_{m})||_{\mathcal{H}}+\frac{1}{{\Delta} t}{\int}_{t_{m}}^{t_{m+1}} \left|\left| (I-{\Pi})\partial_{t}v(s) \right|\right|_{\mathcal{H}} ds $$

with

$$ \eta(t_{m})=\frac{1}{{\Delta} t} \left( v(t_{m+1})- v(t_{m})\right)-\theta \partial_{t} v(t_{m+1})-(1-\theta)\partial_{t} v(t_{m}). $$

We take now $\theta =\frac {1}{2}$. Using a Taylor expansion with rest in integral form and the fact that $v\in C^{3}(0,T;\mathcal {H})$ we get:

$$ \begin{array}{@{}rcl@{}} \eta(t_{m}) & = &\displaystyle \frac{1}{{\Delta} t}\left( \partial_{t}v(t_{m}){\Delta} t+\frac{1}{2} \partial^{2}_{t^{2}}v(t_{m})({\Delta} t)^{2}+ \frac{1}{2} {\int}_{t_{m}}^{t_{m+1}}\partial^{3}_{t^{3}}v(s)(t_{m+1}-s)^{2}ds \right)\\\\ &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\displaystyle -\frac{1}{2} \partial_{t} v(t_{m+1})-\frac{1}{2} \partial_{t} v(t_{m})\\ &=&\displaystyle \frac{1}{2} \partial^{2}_{t^{2}}v(t_{m}) {\Delta} t+ \frac{1}{ 2{\Delta} t} {\int}_{t_{m}}^{t_{m+1}}\partial^{3}_{t^{3}}v(s)(t_{m+1}-s)^{2}ds\\ & &\qquad\qquad\qquad\qquad\displaystyle -\frac{1}{2} \left( \partial^{2}_{t^{2}}v(t_{m}) {\Delta} t + {\int}_{t_{m}}^{t_{m+1}}\partial^{3}_{t^{3}}v(s)(t_{m+1}-s)ds\right)\\\\ & = &\displaystyle {\int}_{t_{m}}^{t_{m+1}}\partial^{3}_{t^{3}}v(s)(t_{m+1}-s) \left[-\frac{1}{2} + \frac{1}{ 2{\Delta} t}(t_{m+1}-s)\right]ds\\\\ &=& \displaystyle \frac{1}{ 2{\Delta} t} {\int}_{t_{m}}^{t_{m+1}} (t_{m+1}-s)(t_{n}-s) \partial^{3}_{t^{3}}v(s) ds. \end{array} $$

Then we have $||\eta (t_{m})||_{\mathcal {H}}\leq C{\Delta } t{\int }_{t_{m}}^{t_{m+1}}\left |\left |\partial ^{3}_{t^{3}}v(s) \right |\right |_{\mathcal {H}} ds$ where C is some universal constant. To sum up we have for any 0 ≤ m ≤ M − 1

$$ ||\varepsilon^{m}||_{\mathcal{V}^{\prime}}\leq \frac{1}{{\Delta} t}{\int}_{t_{m}}^{t_{m+1}} \left|\left| (I-{\Pi})\partial_{t}v(s) \right|\right|_{\mathcal{H}} ds +C{\Delta} t{\int}_{t_{m}}^{t_{m+1}}\left|\left|\partial^{3}_{t^{3}}v(s) \right|\right|_{\mathcal{H}} ds $$

and, using Cauchy-Schwarz inequality,

$$ ||\varepsilon^{m}||_{\mathcal{V}^{\prime}}\leq \frac{1}{\sqrt{{\Delta} t}}\left( {\int}_{t_{m}}^{t_{m+1}} \left|\left| (I-{\Pi})\partial_{t}v(s) \right|\right|_{\mathcal{H}}^{2} ds\right)^{\frac{1}{2}} +C({\Delta} t)^{\frac{3}{2}}\left( {\int}_{t_{m}}^{t_{m+1}}\left|\left|\partial^{3}_{t^{3}}v(s) \right|\right|_{\mathcal{H}}^{2} ds\right)^{\frac{1}{2}}. $$

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Convergence. Let 0 ≤ m ≤ M. Using now $||v^{m,N,P}-v(t_{m})||_{\mathcal {H}}\leq ||e^{m}||_{\mathcal {H}}+||v(t_{m})-{\Pi } v(t_{m})||_{\mathcal {H}}$ and Eqs. 46 and 49 we get

$$ \begin{array}{@{}rcl@{}} &&\displaystyle ||v^{m,N,P}-v(t_{m})||_{\mathcal{H}}\leq\displaystyle ||v(t_{m})-{\Pi} v(t_{m})||_{\mathcal{H}}+\left\{\vphantom{{{\int}_{t_{j}}^{t_{j+1}}}^{\frac{1}{2}}} ||f^{N,P}-{\Pi} f||_{\mathcal{H}}^{2}\right.\\ &&\displaystyle \quad \quad + \frac{1}{c}{\int}_{0}^{t_{m}} \left|\left| (I-{\Pi})\partial_{t}v(s) \right|\right|_{\mathcal{H}}^{2} ds+\frac{C}{c}({\Delta} t)^{4}{\int}_{0}^{t_{m}}\left|\left|\partial^{3}_{t^{3}}v(s) \right|\right|_{\mathcal{H}}^{2} ds\\ &&\left.\displaystyle \quad\quad+ \frac{C}{c}({\Delta} t)^{2} \displaystyle{\sum}_{k=1}^{m-1}{\sum}_{j=1}^{m-1}\left( {\int}_{t_{k}}^{t_{k+1}} \left|\left| (I-{\Pi})\partial_{t}v(s) \right|\right|_{\mathcal{H}}^{2} ds \times {\int}_{t_{j}}^{t_{j+1}} \left|\left|\partial^{3}_{t^{3}}v(s) \right|\right|_{\mathcal{H}}^{2} ds \right)^{\frac{1}{2}} \right\}^{\frac{1}{2}}\\ &&\leq \displaystyle \sup_{s\in[0,T]}||(I-{\Pi})v(s)||_{\mathcal{H}}+ \left\{ \left( ||f^{N,P}-f||_{\mathcal{H}}+||f-{\Pi} f||_{\mathcal{H}}\right)^{2}\right.\\ &&\displaystyle\quad \quad \quad+ \frac{T}{c} \sup_{s\in[0,T]}||(I-{\Pi})\partial_{t} v(s)||_{\mathcal{H}}^{2}+\frac{C}{c} ||\partial^{3}_{t^{3}}v||_{L^{2}(0,T;\mathcal{H})}^{2} ({\Delta} t)^{4}\\ &&\left.+\frac{C}{c} T^{2}\sup_{s\in[0,T]}||\partial^{3}_{t^{3}}v(s)||_{\mathcal{H}} {\Delta} t \sup_{s\in[0,T]}||(I-{\Pi})\partial_{t} v(s)||_{\mathcal{H}} \right\}^{\frac{1}{2}} \end{array} $$

(50)

Recall now that we have $||f^{N,P}-f||_{\mathcal {H}}\to 0$ as N,P →∞ and Eq. 43. If $y\in C(0,T;\mathcal {V})$ the family (I −π)y(s), s ∈ [0,T], is equicontinuous, and we know thanks to Eq. 43 that for any s ∈ [0,T] we have $||(I-{\Pi })y(s)||_{\mathcal {H}} \to 0$ as N,P →∞. With the help of Ascoli theorem we can see that we have then

$$ \sup_{s\in[0,T]}||(I-{\Pi})y(s)||_{\mathcal{H}}\xrightarrow[N\to\infty,P\to\infty]{}0. $$

Using this with y = v,∂_tv, in Eq. 50 we get the announced convergence (note that it is in order (Δt)² is time; cf Remark 8). □