The Expected Fitness Cost of a Mutation Fixation under the One-dimensional Fisher Model

This paper employs Fisher’s model of adaptation to understand the expected fitness effect of fixing a mutation in a natural population. Fisher’s model in one dimension admits a closed form solution for this expected fitness effect. A combination of different parameters, including the distribution of mutation lengths, population sizes, and the initial state that the population is in, are examined to see how they affect the expected fitness effect of state transitions. The results show that the expected fitness change due to the fixation of a mutation is always positive, regardless of the distributional shapes of mutation lengths, effective population sizes, and the initial state that the population is in. The further away the initial state of a population is from the optimal state, the slower the population returns to the optimal state. Effective population size (except when very small) has little effect on the expected fitness change due to mutation fixation. The always positive expected fitness change suggests that small populations may not necessarily be doomed due to the runaway process of fixation of deleterious mutations

Probability-one Homotopies in Computational Science

Probability-one homotopy algorithms are a class of methods for solving nonlinear systems of equations that,under mild assumptions,are globally convergent for a wide range of problems in science and engineering.Convergence theory, robust numerical algorithms,and production quality mathematical software exist for general nonlinear systems of equations, and special cases suc as Brouwer fixed point problems,polynomial systems,and nonlinear constrained optimization.Using a sample of challenging scientific problems as motivation,some pertinent homotopy theory and algorithms are presented. The problems considered are analog circuit simulation (for nonlinear systems),reconfigurable space trusses (for polynomial systems),and fuel-optimal orbital rendezvous (for nonlinear constrained optimization).The mathematical software packages HOMPACK90 and POLSYS_PLP are also briefly described

Modern Homotopy Methods in Optimization

Probability-one homotopy methods are a class of algorithms for solving nonlinear systems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely. These new techniques have been successfully applied to solve Brouwer faced point problems, polynomial systems of equations, and discretizations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements. This paper summarizes the theory of globally convergent homotopy algorithms for unconstrained and constrained optimization, and gives some examples of actual application of homotopy techniques to engineering optimization problems

Message Length Effects for Solving Polynomial Systems on a Hypercube

Comparisons between problems solved on uniprocessor systems and those solved on distributed computing systems generally ignore the overhead associated with information transfer from one process to another. This paper considers the solution of polynomial systems of equations via a globally convergent homotopy algorithm on a hypercube and some timing results for different situations

Multidisciplinary Design Optimization with Mixed Integer Quasiseparable Subsystems

Numerous hierarchical and nonhierarchical decomposition strategies for the optimization of large scale systems, comprised of interacting subsystems, have been proposed. With a few exceptions, all of these strategies have proven theoretically unsound. Recent work considered a class of optimization problems, called quasiseparable, narrow enough for a rigorous decomposition theory, yet general enough to encompass many large scale engineering design problems. The subsystems for these problems involve local design variables and global system variables, but no variables from other subsystems. The objective function is a sum of a global system criterion and the subsystems' criteria. The essential idea is to give each subsystem a budget and global system variable values, and then ask the subsystems to independently maximize their constraint margins. Using these constraint margins, a system optimization then adjusts the values of the system variables and subsystem budgets. The subsystem margin problems are totally independent, always feasible, and could even be done asynchronously in a parallel computing context. An important detail is that the subsystem tasks, in practice, would be to construct response surface approximations to the constraint margin functions, and the system level optimization would use these margin surrogate functions. The present paper extends the quasiseparable necessary conditions for continuous variables to include discrete subsystem variables, although the continuous necessary and sufficient conditions do not extend to include integer variables

Performance Analysis of a Novel GPU Computation-to-core Mapping Scheme for Robust Facet Image Modeling

Though the GPGPU concept is well-known in image processing, much more work remains to be done to fully exploit GPUs as an alternative computation engine. This paper investigates the computation-to-core mapping strategies to probe the efficiency and scalability of the robust facet image modeling algorithm on GPUs. Our fine-grained computation-to-core mapping scheme shows a significant performance gain over the standard pixel-wise mapping scheme. With in-depth performance comparisons across the two different mapping schemes, we analyze the impact of the level of parallelism on the GPU computation and suggest two principles for optimizing future image processing applications on the GPU platform

A fully Distributed Parallel Global Search Algorithm

The n-dimensional direct search algorithm DIRECT of Jones,Perttunen, and Stuckman has attracted recent attention from the multidisciplinary design optimization community. Since DIRECT only requires function values (or ranking)and balances global exploration with local refinement better than n-dimensional bisection, it is well suited to the noisy function values typical of realistic simulations. While not efficient for high accuracy optimization, DIRECT is appropriate for the sort of global design space exploration done in large scale engineering design. Direct and pattern search schemes have the potential to exploit massive parallelism, but efficient use of massively parallel machines is nontrivial to achieve. This paper presents a fully distribute control version of DIRECT that is designed for massively parallel (distribute memory architectures. Parallel results are presented for a multidisciplinary design optimization problem — configuration design of a high speed civil transport

Optimization by nonhierarchical asynchronous decomposition

Large scale optimization problems are tractable only if they are somehow decomposed. Hierarchical decompositions are inappropriate for some types of problems and do not parallelize well. Sobieszczanski-Sobieski has proposed a nonhierarchical decomposition strategy for nonlinear constrained optimization that is naturally parallel. Despite some successes on engineering problems, the algorithm as originally proposed fails on simple two dimensional quadratic programs. The algorithm is carefully analyzed for quadratic programs, and a number of modifications are suggested to improve its robustness

A Probability-one Homotopy Algoithm for Non-Smooth Equations and Mixed Complementarity Problems

A probability-one homotopy algorithm for solving nonsmooth equations is described. This algorithm is able to solve problems involving highly nonlinear equations,where the norm of the residual has non-global local minima.The algorithm is based on constructing homotopy mappings that are smooth in the interior of their domains.The algorithm is specialized to solve mixed complementarity problems through the use of MCP functions and associated smoothers.This specialized algorithm includes an option to ensure that all iterates remain feasible.Easily satisfiable sufficient conditions are given to ensure that the homotopy zero curve remains feasible,and global convergence properties for the MCP algorithm are developed.Computational results on the MCPLIB test library demonstrate the effectiveness of the algorithm

Large Deformations of a Whirling Elastic Cable

The large deformations of a whirling elastic cable is studied. The ends of the cable are hinged but otherwise free to translate along the rotational axis. The nonlinear governing equations depend on a rotation-elasticity parameter J. Bifurcation about the straight, axially rotating case occurs when J is greater than or equal to n(pi). Perturbation solutions about the bifurcation points and matched asymptotic solutions for large J are found to second order. Exact numerical solutions are obtained using quasi-Newton and homotopy methods