Continuous Data Assimilation Using General Interpolant Observables

Abstract

We present a new continuous data assimilation algorithm based on ideas that have been developed for designing finite-dimensional feedback controls for dissipative dynamical systems, in particular, in the context of the incompressible two-dimensional Navier–Stokes equations. These ideas are motivated by the fact that dissipative dynamical systems possess finite numbers of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages which govern their long-term behavior. Therefore, our algorithm allows the use of any type of measurement data for which a general type of approximation interpolation operator exists. Under the assumption that the observational measurements are free of noise, our main result provides conditions, on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, obtained by our algorithm from the measurement data, converges to the unknown reference solution over time. Our algorithm is also applicable in the context of signal synchronization in which one can recover, asymptotically in time, the solution (signal) of the underlying dissipative system that is corresponding to a continuously transmitted partial data.

Appendix A: Estimates for Nodal Interpolants

This appendix contains inequalities and estimates for interpolant operators that will be used for observables obtained from nodal measurements of the velocity field.

Consider a function $u \in H^{2}_{\rm{per}}(\varOmega)$, where Ω=[0,L]² is a basic domain of periodicity. Let $\sqrt{N}$ be a positive integer and partition Ω into N squares with sides of length $h=L/{\sqrt{N}}$. Let $\mathcal{J}=\{1,2,\ldots, \sqrt{N}\}^{2}$ and for each $\alpha\in \mathcal{J}$ define the semi-open square

$$\begin{aligned} Q_\alpha=\big[jh,(j+1)h\big)\times\big[kh,(k+1)h\big),\quad\hbox{where } \alpha=(i,j)\in \mathcal{J}. \end{aligned}$$

Moreover, for φ∈L ¹(Ω) we denote

$$\begin{aligned} \langle\varphi\rangle= \frac{1}{L^2}\int_{\varOmega}\varphi(x) \, \mathrm{d}x. \end{aligned}$$

Fix nodal points x _α ∈Q _α , and suppose we are given the nodal values u(x _α ), for every $\alpha\in \mathcal{J}$. Based on these nodal values, we define two interpolant operators, I _h and $\tilde{I}_{h}$, which we will show that they satisfy the approximation estimate (7). Specifically, define

$$ \mathcal{I}_h(u) (x)=\sum_{\alpha\in \mathcal{J}} u(x_\alpha) \psi_\alpha(x), $$

(41)

and

$$ I_h(u) (x)= \mathcal{I}_h(u) (x) - \bigl\langle \mathcal{I}_h(u)\bigr\rangle = \sum_{\alpha\in \mathcal{J}} u(x_\alpha) \bigl(\psi_\alpha(x)- \langle\psi_\alpha \rangle \bigr), $$

(42)

where

$$\begin{aligned} \psi_\alpha(x)=\sum_{(j,k)\in \mathbf{Z}^2} \chi_{Q_\alpha}(x_1+jL,x_2+kL), \end{aligned}$$

is the L-periodic characteristic function of the semi-open square Q _α . Next, we define

$$ {\tilde{\mathcal{I}}}_h(u) (x)=\sum_{\alpha\in \mathcal{J}} u(x_\alpha) {\tilde{\psi}}_\alpha(x), $$

(43)

and

$$ {\tilde{I}}_h(u) (x)= {\tilde{\mathcal{I}}}_h(u) (x)- \bigl\langle {\tilde{\mathcal{I}}}_h(u)\bigr\rangle =\sum _{\alpha\in \mathcal{J}} u(x_\alpha) \bigl({\tilde{\psi}}_\alpha (x) - \langle{\tilde{\psi}}_\alpha\rangle \bigr), $$

(44)

where

$$\begin{aligned} \tilde{\psi}_\alpha(x)= (\rho_\epsilon *\psi_\alpha) (x) \end{aligned}$$

is a mollified version of ψ _α by the mollifier ρ _ϵ (x)=ϵ ⁻² ρ(x/ϵ). Here we take

$$\begin{aligned} \rho(\xi)= \begin{cases} K_0 \displaystyle \exp \biggl({1\over1-|\xi|^2} \biggr)& \mbox{for } |\xi|<1\\ 0 & \mbox{for } |\xi|\ge1, \end{cases} \end{aligned}$$

and

$$\begin{aligned} (K_0)^{-1}=\int_{|\xi|<1} \exp \biggl( {1\over1-|\xi|^2} \biggr)\,\mathrm{d}\xi. \end{aligned}$$

The mollification parameter ϵ will be chosen $\epsilon= \frac{h}{10}$.

Observe that $\langle I_{h}\rangle= \langle\tilde{I}_{h}\rangle=0$, and that $\tilde{\mathcal{I}}_{h}(x)$ and $\tilde{I}_{h}(x)$ are C ^∞ periodic functions.

We now state as Proposition 4 the estimate that was proved by Jones and Titi as inequality (6.2) in Jones and Titi (1993).

Proposition 4

Let Q be a square with sides of length ℓ>0, and φ∈H ²(Q). Then for every x,y∈Q one has

$$\begin{aligned} \big|\varphi(x)-\varphi(y)\big|\le2 \biggl( 4\big\| \nabla\varphi\big\| _{L^2(Q)}^2 +\ell^2 \bigg\| {\partial^2 \varphi\over\partial x_1\partial x_2} \bigg\| _{L^2(Q)}^2 \biggr)^{1/2}. \end{aligned}$$

We now use Proposition 4 to obtain estimate (7) concerning the accuracy of the interpolant operators $\mathcal{I}_{h}$ and I _h . Namely, we have

Proposition 5

Suppose $u\in H_{\rm{per}}^{2}(\varOmega)$, and let $\mathcal{I}_{h}(u)$ and I _h (u) be as in (41) and (42), respectively. Then

• (i) $\|u-\mathcal{I}_{h}(u) \|_{L^{2}(\varOmega)} \le4 h \|\nabla u \|_{L^{2}(\varOmega)} +2 h^{2} \| {\partial^{2} u\over\partial x_{1}\partial x_{2}} \|_{L^{2}(\varOmega)}$.

• (ii) $\|(u-\langle u \rangle) -I_{h}(u) \|_{L^{2}(\varOmega)} \le 8h \|\nabla u \|_{L^{2}(\varOmega)} + 4h^{2} \| {\partial^{2} u\over\partial x_{1}\partial x_{2}} \|_{L^{2}(\varOmega)}$.

Moreover, if 〈u〉=0, then there exists a constant c>0 such that we can replace the term $\| {\partial^{2} u\over\partial x_{1}\partial x_{2}} \|_{L^{2}(\varOmega)}$, in the above estimates, by $c \| \Delta u \|_{L^{2}(\varOmega)}$.

Proof

First observe that

$$\begin{aligned} \sum_{\alpha\in \mathcal{J}} \psi_\alpha(x) = 1 \quad \hbox{for every }x\in \mathbf{R}^2. \end{aligned}$$

Therefore,

$$\begin{aligned} \big\| u-\mathcal{I}_h(u)\big\| _{L^2(\varOmega)}^2 =&\int _{\varOmega} \bigg|u(x) -\sum_{\alpha\in \mathcal{J}} u(x_\alpha)\psi_\alpha(x) \bigg|^2\, \mathrm{d}x \\ =&\int_{\varOmega} \bigg| \sum_{\alpha\in \mathcal{J}} \bigl(u(x)-u(x_\alpha) \bigr) \psi_\alpha(x) \bigg|^2\, \mathrm{d}x \\ =&\int_{\varOmega} \sum_{\alpha,\beta\in \mathcal{J}} \bigl(u(x)-u(x_\alpha) \bigr) \cdot \bigl(u(x)-u(x_\beta) \bigr) \psi_\alpha(x) \psi_\beta(x)\,\mathrm{d}x. \end{aligned}$$

Since

$$\begin{aligned} \psi_\alpha(x)\psi_\beta(x)= \begin{cases} 0 & \mbox{if } \alpha\ne\beta\\ \psi_\alpha(x) & \mbox{if } \alpha=\beta, \end{cases} \end{aligned}$$

the above gives

$$\begin{aligned} \big\| u-\mathcal{I}_h(u)\big\| _{L^2(\varOmega)}^2 =\int _{\varOmega} \sum_{\alpha\in \mathcal{J}} \big|u(x)-u(x_\alpha) \big|^2 \psi^2_\alpha(x)\, \mathrm{d}x. \end{aligned}$$

Applying Proposition 4 to the square $\overline{Q}_{\alpha}$ we obtain

$$\begin{aligned} \big|u(x)-u(x_\alpha)\big|^2\le 4 \biggl( 4\|\nabla u \|_{L^2(Q_\alpha)}^2 +h^2 \bigg\| {\partial^2 u\over\partial x_1\partial x_2} \bigg\| _{L^2(Q_\alpha)}^2 \biggr),\quad {\hbox{for every }} x \in \overline{Q}_\alpha. \end{aligned}$$

Hence

$$\begin{aligned} \big\| u-\mathcal{I}_h(u)\big\| _{L^2(\varOmega)}^2 \le&\sum _{\alpha\in J} 4h^2 \biggl( 4 \|\nabla u \|_{L^2(Q_\alpha)}^2 +h^2 \bigg\| {\partial^2 u\over\partial x_1\partial x_2} \bigg\| _{L^2(Q_\alpha)}^2 \biggr) \\ \le& 16h^2\|\nabla u\|_{L^2(\varOmega)}^2 +4h^4 \bigg\| {\partial^2 u\over\partial x_1\partial x_2} \bigg\| _{L^2(\varOmega)}^2; \end{aligned}$$

which proves (i).

Next, we focus on proving (ii). By virtue of the Cauchy–Schwarz inequality we observe that

$$\begin{aligned} \big\| \langle u \rangle- \bigl\langle \mathcal{I}_h(u)\bigr\rangle \big\| _{L^2(\varOmega)} \le \big\| u - \mathcal{I}_h(u) \big\| _{L^2(\varOmega)}. \end{aligned}$$

Therefore, (ii) follows from combining the triangle inequality together with the above observation, (41), (42), and part (i).

Finally, we recall the fact that for 〈u〉=0 one has $\| u \|_{H^{2}(\varOmega)}\le c \| \Delta u \|_{L^{2}(\varOmega )}$, which concludes the proof. □

We now provide similar estimates for the C ^∞ periodic interpolants $\tilde{\mathcal{I}}_{h}$ and $\tilde{I}_{h}$. In order to do this we make the assumptions that N≥9 and $\epsilon=\frac{h}{10}$. Define

$$\begin{aligned} \tilde{Q}_\alpha =\bigl[(j-1)h,(j+2)h\bigr]\times\bigl[(k-1)h,(k+2)h \bigr], \quad\hbox{where } \alpha=(k,j)\in \mathcal{J}. \end{aligned}$$

Since ϵ<h/2 we obtain

$$\begin{aligned} \mathcal{U}_\alpha=Q_\alpha+B(0,\epsilon) =\{ x+y : x\in Q_\alpha\hbox{ and }|y|<\epsilon\} \subseteq\tilde{Q}_\alpha,\quad \hbox{for } \alpha\in \mathcal{J}, \end{aligned}$$

and

$$\begin{aligned} \mathcal{C}_\alpha=\mathcal{U}_\alpha\setminus\bigcup _{\beta\ne\alpha } \mathcal{U}_\beta \ne\emptyset, \quad\hbox{for } \alpha\in \mathcal{J}. \end{aligned}$$

The following two propositions now follow immediately from the definition of $\tilde{\psi}_{\alpha}$ and the fact that $\epsilon=\frac{h}{10}$.

Proposition 6

The functions $\tilde{\psi}_{\alpha}$, for $\alpha\in \mathcal{J}$, form a smooth partition of unity satisfying

• (i) $0 \le\tilde{\psi}_{\alpha}(x)\le1$, and ${\rm supp}(\tilde{\psi}_{\alpha})\subseteq (\mathcal{U}_{\alpha}+ (L \mathbf{ Z} )^{2} )$,

• (ii) $\tilde{\psi}_{\alpha}(x)=1$, for all $x\in (\mathcal {C}_{\alpha}+ (L \mathbf{Z} )^{2} )$, and $\sum_{\alpha\in \mathcal{J}} \tilde{\psi}_{\alpha}(x)=1$, for all x∈R ²,

• (iii) $\langle\tilde{\psi}_{\alpha}\rangle= (\frac{h}{L} )^{2}$ and $\frac{4}{5} h \le \|\tilde{\psi}_{\alpha}\| _{L^{2}(\varOmega)} \le\frac{6}{5} h$,

• (iv) ${\rm supp}(\nabla\tilde{\psi}_{\alpha})\subseteq ( (\mathcal{U}_{\alpha}\setminus\mathcal{C}_{\alpha})+ L \mathbf {Z}^{2} )$,

• (v) $|\nabla\tilde{\psi}_{\alpha}(x)|\le c h^{-1}$, and $|\frac {\partial^{2}}{\partial x_{i} \partial x_{j}}\tilde{\psi}_{\alpha}(x)|\le c h^{-2}$, for all x∈R ²,

• (vi) $\|\nabla\tilde{\psi}_{\alpha}\|_{L^{2}(\varOmega)} \le c$.

Proposition 7

Let $\mathcal{K}=\{ -1,0,1\}^{2}$. The functions $\tilde{\psi}_{\alpha}$ are nearly orthogonal in the following sense:

• (i) $\int_{\varOmega} \tilde{\psi}_{\alpha}(x) \tilde{\psi}_{\beta}(x)\, \mathrm{d}x=\int_{\varOmega} (\nabla\tilde{\psi}_{\alpha}(x) ) \cdot (\nabla\tilde{\psi}_{\beta}(x) )\,\mathrm{d}x=0$ for all $\alpha,\beta\in \mathcal{J}$ with $\beta-\alpha\notin\mathcal{K}$.

• (ii) $|\int_{\varOmega} \tilde{\psi}_{\alpha}(x) \tilde{\psi}_{\beta}(x)\,\mathrm{d}x | \le(h+2\epsilon)^{2}= \frac{36}{25}h^{2}$, for all $\alpha,\beta\in \mathcal{J}$ with $\beta-\alpha\in\mathcal{K}$.

• (iii) $|\int_{\varOmega} (\nabla\tilde{\psi}_{\alpha}(x) ) \cdot (\nabla\tilde{\psi}_{\beta}(x) )\,\mathrm{d}x | \le c$, for all $\alpha,\beta\in \mathcal{J}$ with $\beta-\alpha\in\mathcal{K}$.

We are now ready to prove estimates concerning the accuracy of the interpolant operators ${\tilde{\mathcal{I}}}_{h}$ and $\tilde{I}_{h}$ that are the analog to those of Proposition 5.

Proposition 8

Suppose $u\in H_{\rm{per}}^{2}(\varOmega)$, and let $\tilde{\mathcal{I}}_{h}(u)$ and $\tilde{I}_{h}(u)$ be as in (43) and (44), respectively. Then there exists a constant c>0 such that

• (i) $\|u-\tilde{\mathcal{I}}_{h}(u)\|_{L^{2}(\varOmega)} \le c (h \|\nabla u\|_{L^{2}(\varOmega)} + h^{2} \| {\partial^{2} u\over\partial x_{1}\partial x_{2}} \|_{L^{2}(\varOmega)} )$.

• (ii) $\| (u-\langle u \rangle )-\tilde{I}_{h}(u)\| _{L^{2}(\varOmega)} \le c (h \|\nabla u\|_{L^{2}(\varOmega)} + h^{2} \| {\partial^{2} u\over\partial x_{1}\partial x_{2}} \|_{L^{2}(\varOmega)} )$.

• (iii) $\| \nabla\tilde{I}_{h}(u)\|_{L^{2}(\varOmega)}=\| \nabla \tilde {\mathcal{I}}_{h}(u)\|_{L^{2}(\varOmega)} \le c ( \|\nabla u\|_{L^{2}(\varOmega)} + h \| {\partial^{2} u\over\partial x_{1}\partial x_{2}} \|_{L^{2}(\varOmega)} )$.

• (iv) $\| \nabla(u - \tilde{I}_{h}(u))\|_{L^{2}(\varOmega)} =\| \nabla(u - \tilde{\mathcal{I}}_{h}(u))\|_{L^{2}(\varOmega)}\le c ( \|\nabla u\|_{L^{2}(\varOmega)} + h \| {\partial^{2} u\over\partial x_{1}\partial x_{2}} \|_{L^{2}(\varOmega)} )$.

Moreover, if 〈u〉=0, then there exists a constant c>0 such that we can replace the term $\| {\partial^{2} u\over\partial x_{1}\partial x_{2}} \|_{L^{2}(\varOmega)}$, in the above estimates, by $c \| \Delta u \|_{L^{2}(\varOmega)}$.

Proof

In what follows we will use some of the properties stated in Propositions 6 and 7.

$$\begin{aligned} \big\| u-\tilde{\mathcal{I}}_h(u)\big\| _{L^2(\varOmega)}^2 =&\int _{\varOmega} \bigg|u(x) -\sum_{\alpha\in \mathcal{J}} u(x_\alpha)\tilde{\psi}_\alpha(x) \bigg|^2 \,\mathrm{d}x \\ =&\int_{\varOmega} \bigg| \sum_{\alpha\in \mathcal{J}} \bigl(u(x)-u(x_\alpha) \bigr) \tilde{\psi}_\alpha(x) \bigg|^2 \, \mathrm{d}x \\ =&\int_{\varOmega} \sum_{\alpha,\beta\in \mathcal{J}} \bigl(u(x)-u(x_\alpha) \bigr) \cdot \bigl(u(x)-u(x_\beta) \bigr) \tilde{\psi}_\alpha(x) \tilde{\psi}_\beta(x)\,\mathrm{d}x. \end{aligned}$$

Since $\tilde{\psi}_{\alpha}(x)\tilde{\psi}_{\beta}(x)=0$ for $\alpha-\beta\notin\mathcal{K}$ (see Proposition 6) we have

$$\begin{aligned} &\big\| u-\tilde{\mathcal{I}}_h(u)\big\| _{L^2(\varOmega)}^2 \\ &\quad\le\int_\varOmega\sum_{\gamma\in\mathcal{K}} \sum_{\alpha\in \mathcal{J}} \big|u(x)-u(x_\alpha)\big| \big|u(x)-u(x_{\alpha+\gamma})\big| \tilde{\psi}_\alpha(x)\tilde{\psi}_{\alpha+\gamma}(x)\,\mathrm{d}x \\ &\quad\le\sum_{\gamma\in\mathcal{K}} \biggl(\sum _{\alpha\in \mathcal{J}}\int_\varOmega \big|u(x)-u(x_\alpha)\big|^2 \bigl(\tilde{\psi}_\alpha(x) \bigr)^2\,\mathrm{d}x \biggr)^{1/2} \\ &\quad\quad{}\times \biggl(\sum_{\alpha\in \mathcal{J}}\int _\varOmega \big|u(x)-u(x_{\alpha+\gamma})\big|^2 \bigl( \tilde{\psi}_{\alpha+\gamma}(x) \bigr)^2\,\mathrm{d}x \biggr)^{1/2} \\ &\quad=9\sum_{\alpha\in \mathcal{J}}\int_\varOmega \big|u(x)-u(x_\alpha)\big|^2 \bigl(\tilde{\psi}_\alpha(x) \bigr)^2\,\mathrm{d}x \\ &\quad \le9\sum_{\alpha\in \mathcal{J}}\int _{\mathcal{U}_\alpha} \big|u(x)-u(x_\alpha)\big|^2\,\mathrm{d}x \\ &\quad\le9\sum_{\alpha\in \mathcal{J}}\int_{\tilde{Q}_\alpha} \big|u(x)-u(x_\alpha)\big|^2\,\mathrm{d}x. \end{aligned}$$

Applying Proposition 4 to each of the squares $\tilde{Q}_{\alpha}$, for $\alpha\in \mathcal{J}$, the above implies

$$\begin{aligned} \big\| u-\tilde{I}_h(u)\big\| _{L^2(\varOmega)}^2 \le&9\sum _{\alpha\in \mathcal{J}} 36h^2 \biggl( 4\|\nabla u \|_{L^2(\tilde{Q}_\alpha)}^2 +h^2 \bigg\| {\partial^2 u\over\partial x_1\partial x_2} \bigg\| _{L^2(\tilde{Q}_\alpha)}^2 \biggr) \\ =& 81\sum_{\alpha\in \mathcal{J}} 36h^2 \biggl( 4\|\nabla u\|_{L^2(Q_\alpha)}^2 +h^2 \bigg\| {\partial^2 u\over\partial x_1\partial x_2} \bigg\| _{L^2(Q_\alpha)}^2 \biggr) \\ =&\gamma_1 h^2 \|\nabla u\|_{L^2(\varOmega)}^2 +\gamma_2 h^4 \bigg\| {\partial u\over\partial x_1\partial x_2} \bigg\| _{L^2(\varOmega)}^2 , \end{aligned}$$

where γ ₁=11664 and γ ₂=2916. By this we conclude (i).

The proof of (ii) follows from (i) by following the same lines as the proof of part (ii) of Proposition 5.

Next, we focus on the proof of (iii). To this end we implement some of the steps used in the proof of part (i), above, and the properties stated in Proposition 6.

$$\begin{aligned} \big\| \nabla\tilde{\mathcal{I}}_h(u)\big\| ^2_{L^2(\varOmega)} =&\bigg\| \sum _{\alpha\in \mathcal{J}} u(x_\alpha)\nabla \tilde{\psi}_\alpha(\cdot)\bigg\| ^2_{L^2(\varOmega)} \\ =& \bigg\| \sum_{\alpha\in \mathcal{J}} u(x_\alpha)\nabla \tilde{\psi}_\alpha(\cdot)- u(\cdot)\nabla \biggl(\sum _{\alpha\in \mathcal{J}} \tilde{\psi}_\alpha(\cdot) \biggr)\bigg\| ^2_{L^2(\varOmega) } \\ =& \bigg\| \sum_{\alpha\in \mathcal{J}} \bigl(u(x_\alpha) - u( \cdot)\bigr)\nabla \tilde{\psi}_\alpha (\cdot)\bigg\| ^2_{L^2(\varOmega)} \\ \le& c \sum_{\alpha\in \mathcal{J}} \big\| \bigl(u(x_\alpha) - u( \cdot)\bigr)\nabla \tilde{\psi}_\alpha(\cdot)\big\| ^2_{L^2(\tilde{Q}_\alpha)} \\ \le&\frac{c}{h^2} \sum_{\alpha\in \mathcal{J}}\big\| \bigl(u(x_\alpha) - u(\cdot)\bigr)\big\| ^2_{L^2(\tilde{Q}_\alpha)}. \end{aligned}$$

Applying Proposition 4 to each of the squares $\tilde{Q}_{\alpha}$, for $\alpha\in \mathcal{J}$, in the above estimate to obtain

$$\begin{aligned} \big\| \nabla\tilde{\mathcal{I}}_h(u)\big\| ^2_{L^2(\varOmega)} \le& c \biggl( \sum_{\alpha\in \mathcal{J}} \|\nabla u\|^2_{L^2(\tilde{Q}_\alpha)} + h^2 \sum_{\alpha\in \mathcal{J}} \bigg\| {\partial^2 u\over\partial x_1\partial x_2} \bigg\| ^2_{L^2(\tilde{Q}_\alpha)} \biggr) \\ \le&9 c \biggl( \sum_{\alpha\in \mathcal{J}} \|\nabla u \|^2_{L^2(Q_\alpha)} + h^2 \sum _{\alpha\in \mathcal{J}} \bigg\| {\partial ^2 u\over\partial x_1\partial x_2} \bigg\| ^2_{L^2( Q_\alpha)} \biggr) \\ =& c \biggl(\|\nabla u\|^2_{L^2(\varOmega)} + h^2 \bigg\| {\partial^2 u\over \partial x_1\partial x_2} \bigg\| ^2_{L^2(\varOmega)} \biggr), \end{aligned}$$

which concludes the proof of point (iii).

Point (iv) is an obvious consequence of (iii). The rest of the proof is similar to Proposition 5. □