APDE: Seminars

Seminars in Applied and Computational Analysis

The equations of landscape formation: review and a new model

In this talk, we start by reviewing some models for landscape evolution, many of which are hybrid models, combining fundamental physical laws with empirical modeling. Such models can be valid near equilibrium. Nevertheless, this situation is not satisfactory from the mathematical standpoint, since such models will be valid only for a given landscape or class of landscapes.

We propose a simple landscape model, deduced from mathematical principles, coping with the main features of all models. This model singles out three spatially distributed scalar state variable, namely the landscape elevation, the water elevation, and the sediment concentration in water. These state variables are linked by three partial differential equations. Two of these equations are mere conservation laws. A third equation copes with the three main features identified in the literature as the main phenomena shaping a landscape: erosion, sedimentation and creep.

Numerical results show that a variety of common landscape features can be reproduced. Furthermore changing various parameters in the model can alter the morphology of the landscape and the various features observed, even for the same initial landscape. The conjectured mathematical instability and non-uniqueness of landscape evolution is illustrated numerically. On the other hand numerical stability of real landscape topographies under realistic values for their evolution is also observed. Lastly, the model presented also shows promise in the field of channel network restoration, as river networks tend to become sharper with the proper choice of parameters in the erosion model.

Part of Applied and Computational Analysis
Speaker: Alexander Chen (SAMSI, University of North Carolina)
Thursday 11 April 2013, 15:00-16:00
MR 14, CMS.

What is a flutter shutter good for?

Until recently moving objects could only be photographed with short exposure times, to avoid motion blur. Yet, recently two groundbreaking works in computational photography offer new camera designs allowing arbitrary exposure times. The “flutter shutter” of Agrawal et al. creates an invertible motion blur by using a clever shutter technique to interrupt the photon flux during the exposure time according to a well chosen binary sequence. The “motion-invariant photography” of Levin et al. gets the same result by a uniformly accelerated camera motion. This talk proposes a simple mathematical method for evaluating the image quality of these new cameras. The theory, providing explicit formulas for the SNR obtained after deconvolution, raises a central paradox for these cameras: It shows that even an infinite exposure time cannot bring an SNR increase of more than 17%! Nevertheless, three consequences of the theory permit to mitigate this harsh limitation. First, this SNR gain can be obtained on any video with moderate motion blur by a very simple new video temporal filter. The video improvement by this blind deconvolution is visible. Second, we show that if a probabilistic motion model is available, then one can compute an optimal flutter shutter with SNR exceeding significantly the predicted limit. Third, we show that the “best snapshot’’ for a given exposure time is not obtained with a constant aperture: it is obtained with a flutter shutter.

This is joint work with Yohann Tendero

Part of Applied and Computational Analysis
Speaker: Jean-Michel Morel (ENS Cachan)
Thursday 25 April 2013, 15:00-16:00
MR 14, CMS.

Data assimilation as an inverse problem: mathematical theory and computational challenges

We provide a theoretical framework for data assimilation, a specific type of inverse problem arising in numerical weather prediction, hydrology and geology. We show that data assimilation techniques such as three-dimensional and four-dimensional variational data assimilation (3DVar and 4DVar) as well as the Kalman filter and Bayes data assimilation are, in the linear case, merely a form of cycled Tikhonov regularisation. Furthermore, we provide an error analysis for the data assimilation process in general. We then show that results from regularisation theory can be applied to data assimilation and give numerical examples.