Parameter optimization for mathematical modeling

Abstract

Mathematical modeling is used to explain and forecast complex systems, and parameter optimization methods have a crucial role to find the optimal set of parameters obtained by minimizing an objective function. Also, the management of computational resources is essential for handling big models in real-time scenarios. A. Duran and G. Caginalp (2008) propose a hybrid parameter optimization forecast algorithm for asset prices via asset flow differential equations. In this thesis, we propose a new mathematical method for an inverse problem of parameter vector optimization in asset flow theory. For this purpose, we use quasi-Newton (QN) and Monte Carlo simulations to optimize the function F[K] for each selected event and initial parameter vector. We present grid and random methods and conclude that the grid approach is better than the random approach in the unconstrained optimization problem. This study also presents a parallel numerical parameter optimization algorithm for dynamical systems used in financial applications. It achieves speed-up for up to 512 cores and considers more extensive financial market situations. Moreover, it also evaluates the convergence of the model parameter vector via nonlinear least squares error, and maximum improvement factor. In this thesis, we also examine the performance, scalability, and robustness of OpenFOAM on the GPGPU cluster for bio-medical fluid flow simulations. It compared the CPU performance of iterative solver icoFoam with direct solver SuperLU_DIST 4.0 and hybrid parallel codes of MPI+OpenMP+CUDA versus MPI+OpenMP implementation of SuperLU_DIST 4.0. Results showed speed-up for large matrices up to 20 million x 20 million. Besides that, we investigate the usage of eigenvalues to examine the spectral effects of large matrices on the performance of scalable direct solvers. Gerschgorin's theorem can be used to bound the spectrum of square matrices, and behaviors such as disjoint, overlapped, or clustered Gerschgorin circles can give clues. We define the minimum number of cores and show that it depends on the sparsity level and size of the matrix, increasing slightly as the sparsity level decreases and the order increases. In sum, this thesis presents new methods for initial parameter selection and a new algorithm for parallel numerical parameter optimization. Also, we define new metrics and show that the importance of right matching for computational systems and the optimal minimum number of cores are important in mathematical modeling and simulation.