Section: New Results

Reduction and emulation

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

  • the dimension of the inputs, which can easily be 105--108 at every time step;

  • the dimension of the state vector, which is usually 105--107;

  • the 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 (POD) 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 approaches. Promising results were however obtained with radial basis functions and an adapted kriging-based method.

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 (POD) on the outputs allowed us to drastically reduce their dimension, from 104 to just a few scalars. The emulation of the reduced model itself was successfully carried out with radial basis functions or an adapted kriging-based method. The resulting reduced/emulated model exhibited meaningful response to all variables. Its performance compared to observations was the same as the original model. The computational cost of the full model is about 8 minutes on 16 cores (for a single time step), while the reduced/emulated model requires only 50 ms on one core [29] .

Motion estimation from images with a waveforms reduced model

Participants : Etienne Huot, Isabelle Herlin, Giuseppe Papari [Lithicon, Norway] , Karim Drifi.

Dimension reduction is applied to an image model, composed of Lagrangian constancy of velocity and transport of image brightness. Waveforms basis are obtained on the image domain for subspaces of images, motion fields and divergence-free motion fields, as eigenvectors of quadratic functions. Image assimilation with th reduced model allows to estimate velocity fields satisfying space-time properties defined by user and traduced as a quadratic function. This approach also solves the issue of complex geographical domains and the difficulty of applying boundary conditions on these domains. 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 [31][26][25] .