Solution of the Fokker-Planck Equation by Sequentially Optimized Meshfree Approximation
This thesis presents a technique to solve the Fokker-Planck equation by applica- tion of the Sequentially Optimized Meshfree Approximation (SOMA) method. It is well known that numerical solution of the Fokker-Planck equation is made di cult by the challenges of positivity enforcement, in nite domain, and high dimensionality. Through the use of optimization, radial basis functions, and its mesh-free architecture, respectively, the SOMA method attempts to address these challenges and sidestep the exponential growth of dimensionality, which hinders traditional numerical methods. Results are presented for one, two, and four-dimensional Fokker-Planck equations. This work will show that SOMA allows for enforcement of a positivity condition that removes the need for log-transforms, produces a solution domain that does not re- quire arti cial boundary condition enforcement, and provides the capability of solving higher dimensional forms of the Fokker-Planck equation.
Fokker-Planck equation, FPE, Sequentially optimized meshfree approximation, SOMA, Oscillator, Curse of dimensionality, Positivity, Radial basis functions, Series expansion, Optimization
Networked Digital Library of Theses and Dissertations