The distribution of resources between enzymes of photosynthetic carbon metabolism might be assumed to have been optimized by natural selection. However, natural selection for survival and fecundity does not necessarily select for maximal photosynthetic productivity. Further, the concentration of a key substrate, atmospheric CO 2 , has changed more over the past 100 years than the past 25 million years, with the likelihood that natural selection has had inadequate time to reoptimize resource partitioning for this change. Could photosynthetic rate be increased by altered partitioning of resources among the enzymes of carbon metabolism? This question is addressed using an ''evolutionary'' algorithm to progressively search for multiple alterations in partitioning that increase photosynthetic rate. To do this, we extended existing metabolic models of C 3 photosynthesis by including the photorespiratory pathway (PCOP) and metabolism to starch and sucrose to develop a complete dynamic model of photosynthetic carbon metabolism. The model consists of linked differential equations, each representing the change of concentration of one metabolite. Initial concentrations of metabolites and maximal activities of enzymes were extracted from the literature. The dynamics of CO 2 fixation and metabolite concentrations were realistically simulated by numerical integration, such that the model could mimic well-established physiological phenomena. For example, a realistic steady-state rate of CO 2 uptake was attained and then reattained after perturbing O 2 concentration. Using an evolutionary algorithm, partitioning of a fixed total amount of protein-nitrogen between enzymes was allowed to vary. The individual with the higher light-saturated photosynthetic rate was selected and used to seed the next generation. After 1,500 generations, photosynthesis was increased substantially. This suggests that the ''typical'' partitioning in C 3 leaves might be suboptimal for maximizing the light-saturated rate of photosynthesis. An overinvestment in PCOP enzymes and underinvestment in Rubisco, sedoheptulose-1,7-bisphosphatase, and fructose-1,6-bisphosphate aldolase were indicated. Increase in sink capacity, such as increase in ADP-glucose pyrophosphorylase, was also indicated to lead to increased CO 2 uptake rate. These results suggest that manipulation of partitioning could greatly increase carbon gain without any increase in the total protein-nitrogen investment in the apparatus for photosynthetic carbon metabolism.

Many problems in science and engineering require the solution of a long sequence of slowly changing linear systems. We propose and analyze two methods that significantly reduce the total number of matrix-vector products required to solve all systems. We consider the general case where both the matrix and right-hand side change, and we make no assumptions regarding the change in the right-hand sides. Furthermore, we consider general nonsingular matrices, and we do not assume that all matrices are pairwise close or that the sequence of matrices converges to a particular matrix. Our methods work well under these general assumptions, and hence form a significant advancement with respect to related work in this area. We can reduce the cost of solving subsequent systems in the sequence by recycling selected subspaces generated for previous systems. We consider two approaches that allow for the continuous improvement of the recycled subspace at low cost. We consider both Hermitian and non-Hermitian problems, and we analyze our algorithms both theoretically and numerically to illustrate the effects of subspace recycling. We also demonstrate the effectiveness of our algorithms for a range of applications from computational mechanics, materials science, and computational physics.

SUMMARYThe computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis. We propose fast iterative solvers for large threedimensional topology optimization problems to address this problem. Since the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems. In addition, we introduce a MINRES (Minimum Residual method) version with recycling (and a short term recurrence) to make recycling more efficient for symmetric problems. Furthermore, we discuss preconditioning to ensure fast convergence. We show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density. We demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast PC.