ABSTRACT
Many successful variational regularization methods employed to solve linear inverse problems in imaging applications (such as image deblurring, image inpainting, and computed tomography) aim at enhancing edges in the solution, and often involve non-smooth regularization terms (e.g., total variation). Such regularization methods can be treated as iteratively reweighted least squares problems (IRLS), which are usually solved by the repeated application of a Krylov projection method. This approach gives rise to an inner-outer iterative scheme where the outer iterations update the weights and the inner iterations solve a least squares problem with fixed weights. Recently, flexible or generalized Krylov solvers, which avoid inner-outer iterations by incorporating iteration-dependent weights within a single approximation subspace for the solution, have been devised to efficiently handle IRLS problems. Indeed, substantial computational savings are generally possible by avoiding the repeated application of a traditional Krylov solver. This paper aims to extend the available flexible Krylov algorithms in order to handle a variety of edge-enhancing regularization terms, with computationally convenient adaptive regularization parameter choice. In order to tackle both square and rectangular linear systems, flexible Krylov methods based on the so-called flexible Golub-Kahan decomposition are considered. Some theoretical results are presented (including a convergence proof) and numerical comparisons with other edge-enhancing solvers show that the new methods compute solutions of similar or better quality, with increased speedup.
ABSTRACT
Two established thermal properties of enzymes are the Arrhenius activation energy and thermal stability. Arising from anomalies found in the variation of enzyme activity with temperature, a comparison has been made of experimental data for the activity and stability properties of five different enzymes with theoretical models. The results provide evidence for a new and fundamental third thermal parameter of enzymes, T(eq), arising from a subsecond timescale-reversible temperature-dependent equilibrium between the active enzyme and an inactive (or less active) form. Thus, at temperatures above its optimum, the decrease in enzyme activity arising from the temperature-dependent shift in this equilibrium is up to two orders of magnitude greater than what occurs through thermal denaturation. This parameter has important implications for our understanding of the connection between catalytic activity and thermostability and of the effect of temperature on enzyme reactions within the cell. Unlike the Arrhenius activation energy, which is unaffected by the source ("evolved") temperature of the enzyme, and enzyme stability, which is not necessarily related to activity, T(eq) is central to the physiological adaptation of an enzyme to its environmental temperature and links the molecular, physiological, and environmental aspects of the adaptation of life to temperature in a way that has not been described previously. We may therefore expect the effect of evolution on T(eq) with respect to enzyme temperature/activity effects to be more important than on thermal stability. T(eq) is also an important parameter to consider when engineering enzymes to modify their thermal properties by both rational design and by directed enzyme evolution.
