Uncertainty quantification of time-average quantities of chaotic systems using sensitivity-enhanced polynomial chaos expansion

We consider the effect of multiple stochastic parameters on the time-average quantities of chaotic systems. We employ the recently proposed sensitivity-enhanced generalized polynomial chaos expansion, se-gPC, to quantify efficiently this effect. se-gPC is an extension of gPC expansion, enriched with the sensitivity of the time-averaged quantities with respect to the stochastic variables. To compute these sensitivities, the adjoint of the shadowing operator is derived in the frequency domain. Coupling the adjoint operator with gPC provides an efficient uncertainty quantification algorithm, which, in its simplest form, has computational cost that is independent of the number of random variables. The method is applied to the Kuramoto-Sivashinsky equation and is found to produce results that match very well with Monte Carlo simulations. The efficiency of the proposed method significantly outperforms sparse-grid approaches, such as Smolyak quadrature. These properties make the method suitable for application to other dynamical systems with many stochastic parameters.

Figure 1

Block diagram of the proposed UQ method. Arrows denote flow of information between blocks.

Figure 2

Bell-shaped profile $\varphi \left(x\right)$ with amplitude $\mathrm{\Phi }$.

Figure 3

Variation of (a) $\overline{J^{\left(1\right)}}$ with the forcing amplitude $\mathrm{\Phi }$ and (b) sensitivity $\frac{d\overline{J^{\left(1\right)}}}{d\mathrm{\Phi }}$ computed using the adjoint operator and finite differences. Results are averaged over 200 random initial conditions.

Figure 4

Variation of (a) $\overline{J^{\left(2\right)}}$ with the forcing amplitude $\mathrm{\Phi }$ and (b) sensitivity $\frac{d\overline{J^{\left(2\right)}}}{d\mathrm{\Phi }}$ computed using the adjoint operator and finite differences. Results are averaged over 200 random initial conditions.

Figure 5

Contour plots of the (a) state $\mathit{u}(x,t)$ and (b) adjoint $\lambda (x,t)$ variables for the unforced system, $\varphi =0$. Results from a single realization with a random initial condition.

Figure 6

$\mathcal{E}\left[\overline{J^{\left(1\right)}\left(x\right)}\right]$ for the unforced and forced KS system with $\mathrm{\Phi }\sim \mathcal{N}(0,0.01)$ for $T=100$. Results are averaged over 1000 random initial conditions.

Figure 7

Mean forcing $\varphi \left(x\right)$ with $m=10$ stochastic parameters, ${\mathrm{\Phi }}_i$ $(i=1,\cdots ,m$). The forcing amplitudes are ${\mathrm{\Phi }}_1=0.001$, ${\mathrm{\Phi }}_2=-0.001$, ${\mathrm{\Phi }}_3=0.005$, ${\mathrm{\Phi }}_4=0.002$, ${\mathrm{\Phi }}_5=0.007$, ${\mathrm{\Phi }}_6=-0.003$, ${\mathrm{\Phi }}_7=-0.001$, ${\mathrm{\Phi }}_8=-0.002$, ${\mathrm{\Phi }}_9=0.0005$, ${\mathrm{\Phi }}_{10}=-0.002$.

Figure 8

Contour plots of (a) the state variable and (b) the adjoint variable for the KS system forced with the profile shown in Fig. 7. Results from a random initial condition.

Figure 9

Spectra of $\lambda (x,t)$ at $x=\frac{L}{4}$, $\frac{L}{2}$ and $\frac{3L}{4}$ for (a) the unforced and (b) the forced KS system.

Figure 10

Comparison of the sensitivities of $\overline{J^{\left(1\right)}}$ and $\overline{J^{\left(2\right)}}$ with respect to the amplitudes ${\mathrm{\Phi }}_i$ computed using finite differences and the adjoint shadowing operator. Results are averaged over 100 initial conditions for the adjoint shadowing operator.

Figure 11

Maximum Lyapunov exponent ${\lambda }_{max}$ for 2000 realizations of $m=10$ stochastic variables.

Figure 12

Convergence rates of mean and standard deviation of $\overline{J^{\left(1\right)}}$ against number of evaluations computed with WLS, Smolyak quadrature and se-gPC for $p=2$. The stochastic input is shown in Fig. 7.

Figure 13

Convergence rates of mean and standard deviation of $\overline{J^{\left(2\right)}}$ against number of evaluations computed with WLS, Smolyak quadrature and se-gPC for $p=2$. The stochastic input is shown in Fig. 7.