Section: New Results

Reduction and emulation

The use of environmental models raise a number of problems due to:

• the dimension of their inputs, which can easily be ${10}^5--{10}^8$ at every time step;

• the dimension of their state vector, which is usually ${10}^5--{10}^7$;

• their high computational cost.

In particular, the application of data assimilation methods and uncertainty quantification techniques may require dimension reduction and cost reduction. The dimension reduction consists in projecting the inputs and the state vector to low-dimensional subspaces. The cost reduction can be carried out by emulation, i.e., the replacement of costly components with fast surrogates.

Reduction and emulation of a chemistry-transport model

Participants : Vivien Mallet, Serge Guillas [University College London] .

Both reduction and emulation were applied to the dynamic air quality model Polair3D from Polyphemus. The reduction relied on proper orthogonal decomposition on the input data and on the state vector. The dimension of the reduced subspace for the input data is about 80, while the dimension of the reduced state vector is less than 10. The projection of the state vector on its reduced subspace can be carried out before every integration time step, so that one can reproduce a full state trajectory (in time) using the reduced model.

Significant advances were made to emulate the reduced model, which requires about 90 inputs (reduced input data and reduced state vector) and computes about 10 outputs (reduced state vector). 90 inputs is however a large number to build an emulator using a classical approach like krigging. Promising results were however obtained with an interpolation method based on inverse distance weighting.

Reduction and emulation of a static air quality model

Participants : Vivien Mallet, Anne Tilloy, Fabien Brocheton [Numtech] , David Poulet [Numtech] .

The dimension reduction was applied to the outputs of the urban air quality model ADMS Urban, which is a static model with low-dimensional inputs and high-dimensional outputs. A proper orthogonal decomposition on the outputs allowed us to drastically reduce their dimension, from ${10}^4$ to just a few scalars. First attempts of emulation of the reduced model rely on Gaussian process emulation.

Motion estimation from images with a wavelets reduced model

Participants : Giuseppe Papari, Isabelle Herlin, Etienne Huot, Karim Drifi.

The dimension reduction was applied to an image model, composed of Lagrangian constancy of velocity and transport of image brightness. Wavelets basis have been computed on the image domain for subspaces of images, motion fields and divergence-free motion fields. Image assimilation with this reduced model allows to estimate smooth velocity fields with properties defined by user. This also solves the issue of complex geographical domains and the difficulty of applying boundary conditions on these domains. First results are obtained with a reduced dimension of motion to a few scalars, to be compared with the original problem that has the size of image domain.