Geometric Computational Electrodynamics with Variational Integrators and Discrete Differential Forms

In this paper, we develop a structure-preserving discretization of the Lagrangian framework for electrodynamics, combining the techniques of variational integrators and discrete differential forms. This leads to a general family of variational, multisymplectic numerical methods for solving Maxwell’s equations that automatically preserve key symmetries and invariants. In doing so, we show that Yee’s finite-difference time-domain (FDTD) scheme and its variants are multisymplectic and derive from a discrete Lagrangian variational principle. We also generalize the Yee scheme to unstructured meshes, not just in space but in 4-dimensional spacetime, which relaxes the need to take uniform time steps or even to have a preferred time coordinate. Finally, as an example of the type of methods that can be developed within this general framework, we introduce a new asynchronous variational integrator (AVI) for solving Maxwell’s equations. These results are illustrated with some prototype simulations that show excellent numerical behavior and absence of spurious modes, even for an irregular mesh with asynchronous time stepping

Non-convex clustering using expectation maximization algorithm with rough set initialization

An integration of a minimal spanning tree (MST) based graph-theoretic technique and expectation maximization (EM) algorithm with rough set initialization is described for non-convex clustering. EM provides the statistical model of the data and handles the associated uncertainties. Rough set theory helps in faster convergence and avoidance of the local minima problem, thereby enhancing the performance of EM. MST helps in determining non-convex clusters. Since it is applied on Gaussians rather than the original data points, time required is very low. These features are demonstrated on real life datasets. Comparison with related methods is made in terms of a cluster quality measure and computation time

Adjoint-Based Aerodynamic Design of Complex Aerospace Configurations

An overview of twenty years of adjoint-based aerodynamic design research at NASA Langley Research Center is presented. Adjoint-based algorithms provide a powerful tool for efficient sensitivity analysis of complex large-scale computational fluid dynamics (CFD) simulations. Unlike alternative approaches for which computational expense generally scales with the number of design parameters, adjoint techniques yield sensitivity derivatives of a simulation output with respect to all input parameters at the cost of a single additional simulation. With modern large-scale CFD applications often requiring millions of compute hours for a single analysis, the efficiency afforded by adjoint methods is critical in realizing a computationally tractable design optimization capability for such applications

Non-parametric Methods for Correlation Analysis in Multivariate Data with Applications in Data Mining

In this thesis, we develop novel methods for correlation analysis in multivariate data, with a special focus on mining correlated subspaces. Our methods handle major open challenges arisen when combining correlation analysis with subspace mining. Besides traditional correlation analysis, we explore interaction-preserving discretization of multivariate data and causality analysis. We conduct experiments on a variety of real-world data sets. The results validate the benefits of our methods

Automatic Linear and Curvilinear Mesh Generation Driven by Validity Fidelity and Topological Guarantees

Image-based geometric modeling and mesh generation play a critical role in computational biology and medicine. In this dissertation, a comprehensive computational framework for both guaranteed quality linear and high-order automatic mesh generation is presented. Starting from segmented images, a quality 2D/3D linear mesh is constructed. The boundary of the constructed mesh is proved to be homeomorphic to the object surface. In addition, a guaranteed dihedral angle bound of up to 19:47o for the output tetrahedra is provided. Moreover, user-specified guaranteed bounds on the distance between the boundaries of the mesh and the boundaries of the materials are allowed. The mesh contains a small number of mesh elements that comply with these guarantees, and the runtime is compatible in performance with other software. Then the curvilinear mesh generator allows for a transformation of straight-sided meshes to curvilinear meshes with C1 or C2 smooth boundaries while keeping all elements valid and with good quality as measured by their Jacobians. The mathematical proof shows that the meshes generated by our algorithm are guaranteed to be homeomorphic to the input images, and all the elements inside the meshes are guaranteed to be with good quality. Experimental results show that the mesh boundaries represent the objects\u27 shapes faithfully, and the accuracy of the representation is improved compared to the corresponding linear mesh

Continuous and Orientation-preserving Correspondences via Functional Maps

We propose a method for efficiently computing orientation-preserving and approximately continuous correspondences between non-rigid shapes, using the functional maps framework. We first show how orientation preservation can be formulated directly in the functional (spectral) domain without using landmark or region correspondences and without relying on external symmetry information. This allows us to obtain functional maps that promote orientation preservation, even when using descriptors, that are invariant to orientation changes. We then show how higher quality, approximately continuous and bijective pointwise correspondences can be obtained from initial functional maps by introducing a novel refinement technique that aims to simultaneously improve the maps both in the spectral and spatial domains. This leads to a general pipeline for computing correspondences between shapes that results in high-quality maps, while admitting an efficient optimization scheme. We show through extensive evaluation that our approach improves upon state-of-the-art results on challenging isometric and non-isometric correspondence benchmarks according to both measures of continuity and coverage as well as producing semantically meaningful correspondences as measured by the distance to ground truth maps.Comment: 16 pages, 22 figure