RESEARCH WORKS OF ALBERT COHEN
My early research works (from my PhD of 1990, supervised by Yves Meyer, until 1998)
were concerned with the development of the theory of wavelet bases in relation with
algorithms used in signal and image processing, or in computer aided geometric design.
One significant achievement was the derivation, together with Ingrid Daubechies and
Jean-Christophe Feauveau, of biorthogonal wavelet bases which are used in the state
of the art image compression standard JPEG 2000.
Since 1998, my research is oriented in various applicative directions, with as a
common denominator its theoretical foundations in nonlinear approximation theory and
harmonic analysis. In particular, it has led to the development and analysis of adaptive
and sparsity-based numerical methods in various application contexts such as (i) data
compression, (ii) statistical estimation and learning theory or (iii) discretizations
of partial differential equations.
I am often joining forces with my colleagues Wolfgang Dahmen and Ronald DeVore.
Some of our most significant results are concerned with the analysis of adaptive methods
for PDE's, the space BV, greedy algorithms, and statistical learning theory. A topic
of particular interest the present time is high-dimensional approximation problems arising
in learning theory and in the numerical treatment of parametric and stochastic PDE's.
My current research is concerned with problems that involve a very large
number of variables, and whose efficient numerical treatment is therefore challenged
by the so-called curse of dimensionality, meaning that computational complexity increases
exponentially in the variable dimension. Such problems are ubiquitous in an increasing
number of applicative areas, among which statistical or active learning theory,
parametric and stochastic partial differential equations, parameter optimization in
numerical codes, with a high demand from the industrial world of efficient numerical
methods. Central scientific objectives in this context are (i) to identify
fundamental mathematical principles behind overcoming the curse of dimensionality,
(ii) to understand how these principles enter in relevant instances of the applications
described above, and, (iii) based on these principles to develop broadly applicable
concrete adaptive numerical strategies that benefit from such mechanisms.
This research is supported by the Advanced ERC grant BREAD (Breaking the Curse of
Dimensionality in Analysis and Simulation) awarded in 2013.