A computable characterisation of model uncertainty

Liam Blake

with Dr. John Maclean and A/Prof. Sanjeeva Balasuriya

We are interested in a state variable \(w_t \in \mathbb{R}^n\)

Our best available model is

\[ \frac{\mathrm{d}w_t}{\mathrm{d}t} = u\left(w_t, t\right), \quad w_0 = x, \] for \(t \in [0,T]\).

Solutions are summarised by the flow map \(F_0^t(x) \equiv w_t\).

An example: Rossby wave with oscillatory perturbation in 2D (Samelson and Wiggins 2006).

• Idealised model of Rossby waves observed in the atmosphere and ocean.

• The velocity field is given analytically.

• Here we have one such determinstic trajectory - we fix an initial condition x, and solve the ODE numerically to get the corresponding prediction at time t, F_0^t(x)

But there is unavoidable uncertainty…

• Observation error in \(u\)

• Discretisation and interpolation error

• Other unexplainable phenomena

How can we model the uncertainty in \(F_0^t\)?

\[ \frac{dy_t}{dt} = u\left(y_t, t\right) + \text{``noise''} \]

Multiplicative noise is needed in practice (e.g. Sura et al. 2005)

\[ \frac{dy_t}{dt} = u\left(y_t, t\right) + \varepsilon\sigma\left(y_t, t\right) \cdot \text{``noise''} \]

where \(0 < \varepsilon \ll 1\).

Examples for multiplicative noise

• Cloud cover in satellite imagery

• Uncertainty as a function of velocity, say

• Uncertainty in parameters

Use the Wiener process \(W_t\); \[ W_s - W_t \sim \mathcal{N}\left(0, (s - t)I_n\right). \]

Mention \(I_n\).

Formalise as an Itô stochastic differential equation (SDE) \[ \mathrm{d}y_t = u\left(y_t, t\right)\mathrm{d}t + \varepsilon\sigma\left(y_t, t\right)\mathrm{d}W_t, \quad y_0 = x. \]

This is the true model.

Some key properties (e.g. Kallianpur and Sundar (2014)):

• Unique solutions exist.

• The solution \(y_t\) is now a stochastic process.

\[ \mathrm{d}y_t = u\left(y_t, t\right)\mathrm{d}t + \varepsilon\sigma\left(y_t, t\right)\mathrm{d}W_t, \quad y_0 = x. \]

Solving SDEs is hard.

We can only solve analytically in a few very specific cases.

We can solve numerically to obtain samples (e.g. Kloeden and Platen 1992), but this is computationally expensive.

Any theoretical insight into the behaviour of \(y_t\) is valuable.

\[ \set{y_\tau}_{\tau \in [0,t]} \]

\[ y_t \]

\[ y_t \sim \,??? \]

Extending the approach of Balasuriya (2020)

\[ z_t^{(\varepsilon)}(x) \coloneqq \frac{y_t - F_0^t(x)}{\varepsilon} \]

What happens as \(\varepsilon \to 0\)?

Define \(z_t(x)\) as the solution to the linearised SDE. \[ \mathrm{d}z_t(x) = \nabla u\left(F_0^t(x), t\right)z_t(x) \mathrm{d}t + \sigma\left(F_0^t(x), t\right)\mathrm{d}W_t. \]

We can solve this explicitly

\[ z_t(x) \sim \mathcal{N}\left(0, \Sigma(x,t)\right). \]

We have rigorously shown that \[ z_t^{(\varepsilon)}(x) \xrightarrow{\text{mean}} z_t(x), \quad\text{as } \varepsilon\downarrow 0 \]

so for small \(\varepsilon\), \[ z_t^{(\varepsilon)}(x) \,\dot\sim\, z_t(x). \]

or equivalently \[ y_t^{(\varepsilon)}(x) \,\dot\sim\, F_0^t(x) + \varepsilon z_t(x). \]

\[ y_t \,\dot\sim\, \mathcal{N}\left(F_0^t(x), \varepsilon^2\Sigma(x,t)\right) \\ \]

\[ \Sigma\left(x,t\right) = \int_0^t{L\left(x, t, \tau\right)L\left(x, t, \tau\right)^{T}\mathrm{d}\tau} \] \[ L(x, t, \tau) = {\color{white}\underbrace{\color{black}\left[\nabla F_0^t(x)\right]^{-1} \nabla F_0^\tau(x)}_{\text{model dynamics}}}{\color{white}\underbrace{\color{black}\sigma\left(F_0^\tau(x), \tau\right)}_{\text{multiplicative noise}}} \]

Emphasise computable

\[ y_t \,\dot\sim\, \mathcal{N}\left(F_0^t(x), \varepsilon^2\Sigma(x,t)\right) \\ \] \[ \Sigma\left(x,t\right) = \int_0^t{L\left(x, t, \tau\right)L\left(x, t, \tau\right)^{T}\mathrm{d}\tau} \] \[ L(x, t, \tau) = {\color{blue}\underbrace{\color{blue}\left[\nabla F_0^t(x)\right]^{-1} \nabla F_0^\tau(x)}_{\text{model dynamics}}}{\color{white}\underbrace{\color{black}\sigma\left(F_0^\tau(x), \tau\right)}_{\text{multiplicative noise}}} \]

\[ y_t \,\dot\sim\, \mathcal{N}\left(F_0^t(x), \varepsilon^2\Sigma(x,t)\right) \\ \] \[ \Sigma\left(x,t\right) = \int_0^t{L\left(x, t, \tau\right)L\left(x, t, \tau\right)^{T}\mathrm{d}\tau} \] \[ L(x, t, \tau) = {\color{white}\underbrace{\color{black}\left[\nabla F_0^t(x)\right]^{-1} \nabla F_0^\tau(x)}_{\text{model dynamics}}}{\color{blue}\underbrace{\color{blue}\sigma\left(F_0^\tau(x), \tau\right)}_{\text{multiplicative noise}}} \]

References

Balasuriya, Sanjeeva. 2020. “Stochastic Sensitivity: A Computable Lagrangian Uncertainty Measure for Unsteady Flows.” SIAM Review 62 (November): 781–816.

Kallianpur, G., and P. Sundar. 2014. Stochastic Analysis and Diffusion Processes. First edition. Oxford Graduate Texts in Mathematics 24. Oxford, United Kingdom: Oxford University Press.

Kloeden, Peter E., and Eckhard Platen. 1992. Numerical Solution of Stochastic Differential Equations. 1st ed. Applications of Mathematics. Berlin, Heidelberg: Springer.

Leutbecher, Martin, Sarah-Jane Lock, Pirkka Ollinaho, et al. 2017. “Stochastic Representations of Model Uncertainties at ECMWF: State of the Art and Future Vision.” Quarterly Journal of the Royal Meteorological Society 143 (707): 2315–39.

Samelson, Roger M., and Stephen Wiggins. 2006. Lagrangian Transport in Geophysical Jets and Waves: The Dynamical Systems Approach. Vol. 31. Interdisciplinary Applied Mathematics. New York, NY: Springer.

Sura, Philip, Matthew Newman, Cécile Penland, and Prashant Sardeshmukh. 2005. “Multiplicative Noise and Non-Gaussianity: A Paradigm for Atmospheric Regimes?” Journal of the Atmospheric Sciences 62 (5): 1391–1409.

For any \(0 < \varepsilon \ll 1\), \(x \in \mathbb{R}^n\), \(t \in [0,T]\) and \(r \geq 1\), there is a constant \(D_r(t)\) such that \[ \mathbb{E}\left[\left|\!\left|z_t^{(\varepsilon)}(x) - z_t(x)\right|\!\right|^r\right] \leq D_r(t)\varepsilon^r, \] so \[ \lim_{\varepsilon \to 0} \mathbb{E}\left[\left|\!\left|z_t^{(\varepsilon)}(x) - z_t(x)\right|\!\right|^r\right] = 0. \]

Just a few future extensions/applications:

• Uncertain initial conditions

• Imperfect knowlege of \(F_0^t(\cdot)\)

• Constructing efficient stochastic parameterisation schemes (Leutbecher et al. 2017)

• Measuring linearisation error of extended Kalman Filter

• Extracting Lagrangian coherent structures

• Bayesian inference of diffusivity \(\sigma\)

\[ S^2(x,t) = \left|\!\left|\Sigma(x,t)\right|\!\right| \]