arXiv subjects

Chandrajit Bajaj

Publications and source records attributed to Chandrajit Bajaj.

48 records · Page 3Linked to original sources

arXiv2016

Quantifying and Visualizing Uncertainties in Molecular Models

Computational molecular modeling and visualization has seen significant progress in recent years with sev- eral molecular modeling and visualization software systems in use today. Nevertheless the molecular biology community lacks techniques and tools for the rigorous analysis, quantification and visualization of the associated errors in molecular structure and its associated properties. This paper attempts at filling this vacuum with the introduction of a systematic statistical framework where each source of structural uncertainty is modeled as a ran- dom variable (RV) with a known distribution, and properties of the molecules are defined as dependent RVs. The framework consists of a theoretical basis, and an empirical implementation where the uncertainty quantification (UQ) analysis is achieved by using Chernoff-like bounds. The framework enables additionally the propagation of input structural data uncertainties, which in the molecular protein world are described as B-factors, saved with almost all X-ray models deposited in the Protein Data Bank (PDB). Our statistical framework is also able and has been applied to quantify and visualize the uncertainties in molecular properties, namely solvation interfaces and solvation free energy estimates. For each of these quantities of interest (QOI) of the molecular models we provide several novel and intuitive visualizations of the input, intermediate, and final propagated uncertainties. These methods should enable the end user achieve a more quantitative and visual evaluation of various molecular PDB models for structural and property correctness, or the lack thereof.

cs.CE↗

arXiv2016

Construction of scalar and vector finite element families on polygonal and polyhedral meshes

We combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart-Thomas, and Brezzi-Douglas-Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space with regards to global continuity and that they reproduce the requisite polynomial differential forms described by finite element exterior calculus. We present a method to count the number of basis functions required to ensure these two key properties.

math.NA↗

arXiv2015

Characterization and Construction of a Family of Highly Symmetric Spherical Polyhedra with Application in Modeling Self-Assembling Structures

The regular polyhedra have the highest order of 3D symmetries and are exceptionally at- tractive templates for (self)-assembly using minimal types of building blocks, from nano-cages and virus capsids to large scale constructions like glass domes. However, they only represent a small number of possible spherical layouts which can serve as templates for symmetric assembly. In this paper, we formalize the necessary and sufficient conditions for symmetric assembly using exactly one type of building block. All such assemblies correspond to spherical polyhedra which are edge-transitive and face-transitive, but not necessarily vertex-transitive. This describes a new class of polyhedra outside of the well-studied Platonic, Archimedean, Catalan and and Johnson solids. We show that this new family, dubbed almost-regular polyhedra, can be pa- rameterized using only two variables and provide an efficient algorithm to generate an infinite series of such polyhedra. Additionally, considering the almost-regular polyhedra as templates for the assembly of 3D spherical shell structures, we developed an efficient polynomial time shell assembly approximation algorithm for an otherwise NP-hard geometric optimization problem.

cs.CG↗

arXiv2015

On the construction of general cubature formula by flat extensions

We describe a new method to compute general cubature formulae. The problem is initially transformed into the computation of truncated Hankel operators with flat extensions. We then analyse the algebraic properties associated to flat extensions and show how to recover the cubature points and weights from the truncated Hankel operator. We next present an algorithm to test the flat extension property and to additionally compute the decomposition. To generate cubature formulae with a minimal number of points, we propose a new relaxation hierarchy of convex optimization problems minimizing the nuclear norm of the Hankel operators. For a suitably high order of convex relaxation, the minimizer of the optimization problem corresponds to a cubature formula. Furthermore cubature formulae with a minimal number of points are associated to faces of the convex sets. We illustrate our method on some examples, and for each we obtain a new minimal cubature formula.

math.AG↗

arXiv2014

On Low Discrepancy Samplings in Product Spaces of Motion Groups

Deterministically generating near-uniform point samplings of the motion groups like SO(3), SE(3) and their n-wise products SO(3)^n, SE(3)^n is fundamental to numerous applications in computational and data sciences. The natural measure of sampling quality is discrepancy. In this work, our main goal is construct low discrepancy deterministic samplings in product spaces of the motion groups. To this end, we develop a novel strategy (using a two-step discrepancy construction) that leads to an almost exponential improvement in size (from the trivial direct product). To the best of our knowledge, this is the first nontrivial construction for SO(3)^n, SE(3)^n and the hypertorus T^n. We also construct new low discrepancy samplings of S^2 and SO(3). The central component in our construction for SO(3) is an explicit construction of N points in S^2 with discrepancy \tildeØ(1/\sqrt{N}) with respect to convex sets, matching the bound achieved for the special case of spherical caps in \cite{ABD_12}. We also generalize the discrepancy of Cartesian product sets \cite{Chazelle04thediscrepancy} to the discrepancy of local Cartesian product sets. The tools we develop should be useful in generating low discrepancy samplings of other complicated geometric spaces.

cs.CG↗

arXiv2012

Interpolation Error Estimates for Mean Value Coordinates over Convex Polygons

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in [Gillette et al., AiCM, doi:10.1007/s10444-011-9218-z], we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach pi.

math.NA↗

arXiv2012

Quadratic Serendipity Finite Elements on Polygons Using Generalized Barycentric Coordinates

We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex n-gon satisfying simple geometric criteria, our construction produces 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n+1)/2 basis functions known to obtain quadratic convergence. The technique broadens the scope of the so-called `serendipity' elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Uniform `a priori' error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of generic convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.

math.NA↗

arXiv2011

On a Nanoscopically-Informed Shell Theory of Single-Wall Carbon Nanotubes

This paper proposes a bottom-up sequence of modeling steps leading to a nanoscopically informed continuum, and as such macroscopic, theory of single-walled carbon nanotubes (SWCNTs). We provide a description of the geometry of the two most representative types of SWCNTs, armchair (A-) and zigzag (Z-), of their modules and of their elementary bond units. We believe ours to be the simplest shell theory that accounts accurately for the linearly elastic response of both A- and Z- CNTs. In fact, our theory can be shown to fit SWCNTs of whatever chirality; its main novel feature is perhaps the proposition of chirality-dependent concepts of effective thickness and effective radius.

cond-mat.mes-hall↗

arXiv2011

Dual Formulations of Mixed Finite Element Methods with Applications

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the numerical stability of the method. Additionally, we define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods. Examples from magnetostatics and Darcy flow are examined in detail.

math.DG↗

arXiv2011

Error Estimates for Generalized Barycentric Interpolation

We prove the optimal convergence estimate for first order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.

math.NA↗

arXiv2009

Complementary Space for Enhanced Uncertainty and Dynamics Visualization

Given a computer model of a physical object, it is often quite difficult to visualize and quantify any global effects on the shape representation caused by local uncertainty and local errors in the data. This problem is further amplified when dealing with hierarchical representations containing varying levels of detail and / or shapes undergoing dynamic deformations. In this paper, we compute, quantify and visualize the complementary topological and geometrical features of 3D shape models, namely, the tunnels, pockets and internal voids of the object. We find that this approach sheds a unique light on how a model is affected by local uncertainty, errors or modifications and show how the presence or absence of complementary shape features can be essential to an object's structural form and function.

cs.CG↗

arXiv2009

Stable Mesh Decimation

Current mesh reduction techniques, while numerous, all primarily reduce mesh size by successive element deletion (e.g. edge collapses) with the goal of geometric and topological feature preservation. The choice of geometric error used to guide the reduction process is chosen independent of the function the end user aims to calculate, analyze, or adaptively refine. In this paper, we argue that such a decoupling of structure from function modeling is often unwise as small changes in geometry may cause large changes in the associated function. A stable approach to mesh decimation, therefore, ought to be guided primarily by an analysis of functional sensitivity, a property dependent on both the particular application and the equations used for computation (e.g. integrals, derivatives, or integral/partial differential equations). We present a methodology to elucidate the geometric sensitivity of functionals via two major functional discretization techniques: Galerkin finite element and discrete exterior calculus. A number of examples are given to illustrate the methodology and provide numerical examples to further substantiate our choices.

math.AP↗