3,536 research outputs found
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 maleās state and the level of competition he faces at any given time. We developed a dynamic optimization model within a gameātheoretical framework to explore the resulting signaling dynamics at both individual and population levels and tested two key model predictions with empirical data on threeāspined 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
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