Application of projection algorithms to differential equations: boundary value problems

The Douglas-Rachford method has been employed successfully to solve many kinds of non-convex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary valued problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well-suited to parallelization. We explore the stability of the method by applying it to several examples of BVPs, including cases where the traditional Newton's method fails

Computable Centering Methods for Spiraling Algorithms and their Duals, with Motivations from the theory of Lyapunov Functions

Splitting methods like Douglas--Rachford (DR), ADMM, and FISTA solve problems whose objectives are sums of functions that may be evaluated separately, and all frequently show signs of spiraling. Circumcentering reflection methods (CRMs) have been shown to obviate spiraling for DR for certain feasibility problems. Under conditions thought to typify local convergence for splitting methods, we first show that Lyapunov functions generically exist. We then show for prototypical feasibility problems that CRMs, subgradient projections, and Newton--Raphson are all describable as gradient-based methods for minimizing Lyapunov functions constructed for DR operators, with the former returning the minimizers of quadratic surrogates for the Lyapunov function. Motivated thereby, we introduce a centering method that shares these properties but with the added advantages that it: 1) does not rely on subproblems (e.g. reflections) and so may be applied for any operator whose iterates spiral; 2) provably has the aforementioned Lyapunov properties with few structural assumptions and so is generically suitable for primal/dual implementation; and 3) maps spaces of reduced dimension into themselves whenever the original operator does. We then introduce a general approach to primal/dual implementation of a centering method and provide a computed example (basis pursuit), the first such application of centering. The new centering operator we introduce works well, while a similar primal/dual adaptation of CRM fails to solve the problem, for reasons we explain

Optimization of resource allocation can explain the temporal dynamics and honesty of sexual signals

In species in which males are free to dynamically alter their allocation to sexual signaling over the breeding season, the optimal investment in signaling should depend on both a male’s state and the level of competition he faces at any given time. We developed a dynamic optimization model within a game‐theoretical framework to explore the resulting signaling dynamics at both individual and population levels and tested two key model predictions with empirical data on three‐spined stickleback (Gasterosteus aculeatus) males subjected to dietary manipulation (carotenoid availability): (1) fish in better nutritional condition should be able to maintain their signal for longer over the breeding season, resulting in an increasingly positive correlation between nutritional status and signal (i.e., increasing signal honesty), and (2) female preference for more ornamented males should thus increase over the breeding season. Both predictions were supported by the experimental data. Our model shows how such patterns can emerge from the optimization of resource allocation to signaling in a competitive situation. The key determinants of the honesty and dynamics of sexual signaling are the condition dependency of male survival, the initial frequency distribution of nutritional condition in the male population, and the cost of signaling