School of Science Department of Mathematics 58 Efficient Algorithms for Visualizing Dynamical Systems Supervisor: LEUNG Shing Yu / MATH Student: CHAU Wai Ming / MATH-SFM Course: UROP1100, Summer We present a principal component analysis (PCA) method for computing the finite time variance analysis (FTVA) in uncertain vector fields. The definition of FTVA is developed from the finite-time Lyapunov exponent (FTLE) for deterministic vector fields. We first do the simulations by solving the corresponding stochastic differential equation. The method allows us to get the maximum eigenvalue from the covariance matrix, which measures the arrival location's maximal variance direction. Moreover, we explored the weighted principal component analysis (WPCA) method based on the diagonalization of the weighted covariance matrix. We will apply the methods to the double gyre flow to demonstrate the performance of the two methods. Efficient Algorithms for Visualizing Dynamical Systems Supervisor: LEUNG Shing Yu / MATH Student: TSAI Yun Chen / MATH-IRE Course: UROP1100, Summer Lagrangian coherent structure (LCS) is important for studying dynamical surfaces, and the finite-time Lyapunov exponent (FTLE) is an essential quantity in analyzing the LCS. The current method of visualizing the FTLE field is inefficient in extracting the ridges as a significant fraction of computation resources have been wasted on unrelated regions. We propose a quadtree-based adaptive refinement method to visualize the FTLE field on a 2D plane and test it with the double gyre flow. Also, we develop a rough error estimation to show the effect of the adaptive refinement. Efficient Algorithms for Visualizing Dynamical Systems Supervisor: LEUNG Shing Yu / MATH Student: YAU Wing Yan / SSCI Course: UROP1100, Summer We investigate how infinitesimal portion (area for 2d flow, volume for 3d flow) changes over time in fluid flows. The project tests the performance of two different Lagrangian approaches. The first one approximates the final positions of grid points to obtain the Jacobian matrix, while the other one keeps track of the deformation of the infinitesimal portion. Both methods give results that can show the details of flow structure, yet the former one would have higher accuracy when more refined grids are applied while the latter one has stable performance.