260 IGPM260.pdf        July 2006
TITLE Compressed Sensing and Best k-Term Approximation
AUTHORS Albert Cohen, Wolfgang Dahmen, Ronald DeVore
ABSTRACT We consider a relatively simple model for pool-boiling processes. This model involves only the temperature distribution within the heater and describes the heat exchange with the boiling medium via a nonlinear boundary condition imposed on the fluid-heater interface. This results in a standard heat-transfer problem with a nonlinear Neumann boundary condition on part of the boundary. In a recent paper [18] we analysed this nonlinear heat-transfer problem for the case of two space dimensions and in particular studied the qualitative structure of steady-state solutions. The study revealed that, depending on system parameters, the model allows both multiple homogeneous and multiple heterogeneous temperature distributions on the fluid-heater interface. In the present paper we show that the analysis from Speetjens et al. [18] can be generalised to the physically more realistic case of three space dimensions. A fundamental shift-invariance property is derived that implies multiplicity of heterogeneous solutions. We present a numerical bifurcation analysis that demonstrates the multiple solution structure in this mathematical model by way of a representative case study.
KEYWORDS Compressed sensing, best k-term approximation, instance optimality, instance optimality in probability, restricted isometry property, null space property, mixed norm estimates
DOI 10.1090/S0894-0347-08-00610-3
PUBLICATION Journal of the American Mathematical Society
22(1), 211-231 (2009)