Showing 1 - 4 of 4 Items

Date: 2022-01-01

Creator: Sonia K. Shah

Access: Open access

The Nelder-Mead optimization method is a numerical method used to find the minimum of an objective function in a multidimensional space. In this paper, we use this method to study functions - specifically functions with three-dimensional graphs - and create images of the basin of attraction of the function. Three different methods are used to create these images named the systematic point method, randomized centroid method, and systemized centroid method. This paper applies these methods to different functions. The first function has two minima with an equivalent function value. The second function has one global minimum and one local minimum. The last function studied has several minima of different function values. The systematic point method is a reliable method in particular scenarios but is extremely sensitive to changes in the initial simplex. The randomized centroid method was not found to be useful as the basin of attraction images are difficult to understand. This made it particularly troublesome to know when the method was working effectively and when it was not. The systemized centroid method appears to be the most precise and effective method at creating the basin of attraction in most cases. This method rarely fails to find a minimum and is particularly adept at finding global minima more effectively compared to local minima. It is important to remember that these conclusions are simply based off the results of the methods and functions studied and that more effective methods may exist.

Date: 2022-01-01

Creator: Gillian King

Access: Open access

This project is an analysis of the effectiveness of five distinct optimization methods in their ability in producing clear images of the basins of attraction, which is the set of initial points that approach the same minimum for a given function. Basin images are similar to contour plots, except that they depict the distinct regions of points--in unique colors--that approach the same minimum. Though distinct in goal, contour plots are useful to basin research in that idealized basin images can be inferred from the steepness levels and location of extrema they depict. Effectiveness of the method changes slightly depending on the function, but is generally defined as how closely the basin image models contour information on where the true minima are located, and by the clarity of the resulting image in depicting well-defined regions. The methods are tested on four distinct functions which were chosen to assess how each method performs in the presence of various challenges. This project ranks the five methods for their overall effectiveness and consistency across the four functions, and also analyzes the sensitivity of the methods when small changes are made to the function. In general, less sensitive and consistently effective methods are more applicable and reliable in applied optimization research.

Date: 2024-01-01

Creator: David Ma

Access: Open access

Particle swarm optimization (PSO) is a metaheuristic optimization method that finds near- optima by spawning particles which explore within a given search space while exploiting the best candidate solutions of the swarm. PSO algorithms emulate the behavior of, say, a flock of birds or a school of fish, and encapsulate the randomness that is present in natural processes. In this paper, we discuss different initialization schemes and meta-optimizations for PSO, its performances on various multi-minima functions, and the unique intricacies and obstacles that the method faces when attempting to produce images for basins of attraction, which are the sets of initial points that are mapped to the same minima by the method. This project compares the relative strengths and weaknesses of the Particle Swarm with other optimization methods, namely gradient-descent, in the context of basin mapping and other metrics. It was found that with proper parameterization, PSO can amply explore the search space regardless of initialization. For all functions, the swarm was capable of finding, within some tolerance, the global minimum or minima in fewer than 60 iterations by having sufficiently well chosen parameters and parameterization schemes. The shortcomings of the Particle Swarm method, however, are that its parameters often require fine-tuning for different search spaces to most efficiently optimize and that the swarm cannot produce the analytical minimum. Overall, the PSO is a highly adaptive and computationally efficient method with few initial restraints that can be readily used as the first step of any optimization task.

Date: 2014-05-01

Creator: Lauren A Skerritt

Access: Open access

In the American lobster (Homarus americanus), neurogenic stimulation of the heart drives fluxes of calcium (Ca2+) into the cytoplasm of a muscle cell resulting in heart muscle contraction. The heartbeat is completed by the active transport of calcium out of the cytoplasm into extracellular and intracellular spaces. An increase in the frequency of calcium release is expected to increase amplitude and duration of muscle contraction. This makes sense because an increase in cytoplasmic calcium should increase the activation of the muscle contractile elements (actin and myosin). Since calcium cycling is a reaction-diffusion process, the extent to which calcium mediates contraction amplitude and frequency will depend on the specific diffusion relationships of calcium in this system. Despite the importance of understanding this relationship, it is difficult to obtain experimental information on the dynamics of cytoplasmic calcium. Thus, we developed a mathematical diffusion model of the myofibril (muscle cell) to simulate calcium cycling in the lobster cardiac muscle cell. The amplitude and duration of the force curves produced by the model empirically mirrored that of the experimental data over a range of calcium diffusion coefficients (1-16), nerve stimulation durations (1/6-1/3 of a contraction period), and frequencies (40-80 Hz). The characteristics that alter the response of the lobster cardiac muscle system are stimulation duration (i.e., burst duration), burst frequency, and the rate of calcium diffusion into the cell’s cytoplasm. For this reason, we developed protocols that allow parameters representing these characteristics in the calcium-force model to be determined from isolated whole muscle experiments on lobster hearts (Phillips et al., 2004). These parameters are used to predict variability in lobster heart muscle function consistent with data recorded in experiments. Within the physiological range of nerve stimulation parameters (burst duration and cycle period), calcium increased the cell’s force output for increased burst duration. For example, increased duration of stimulation increased the muscle contraction period and vice versa. In terms of diffusion, a slower rate of calcium diffusion out of the sarcoplasmic reticulum decreased both the calcium level and the contraction duration of the cell. Finally, changes in stimulation frequency did not produce changes in contraction amplitude and duration. When considered in conjunction with experimental stimulations using lobster heart muscle cells, these data illustrate the prominent role for calcium diffusion in governing contraction-relaxation cycles in lobster hearts.