John Burkardt

Algorithm 596: a program for a locally parameterized

is an open, o n e d i m e n s i o n a l C L m a n i f o l d in R n. W e are i n t e r e s t ed in c o m p u t i n g the c o n n e c t e d c o m p o n e n t 8R (F, x °) of £R (F) wh ich c o n t a i n s x °. B y a f u n d a m e n t a l r esu l t of d i f ferent ia l geome t ry (see, e.g., [5]), 8R(F, x °) is d i f feomorphic e i t he r to a circle or to some in t e rva l ( connec ted subse t ) of R ~. F o r s impl i c i ty we call (~R (F, x 0) the so lu t ion curve of (1.2) t h r o u g h x °.

Comparison of pure and Latinized centroidal Voronoi tessellation against various other statistical sampling methods

Abstract A recently developed centroidal Voronoi tessellation (CVT) sampling method is investigated here to assess its suitability for use in statistical sampling applications. CVT efficiently generates a highly uniform distribution of sample points over arbitrarily shaped M -dimensional parameter spaces. On several 2-D test problems CVT has recently been found to provide exceedingly effective and efficient point distributions for response surface generation. Additionally, for statistical function integration and estimation of response statistics associated with uniformly distributed random-variable inputs (uncorrelated), CVT has been found in initial investigations to provide superior points sets when compared against latin-hypercube and simple-random Monte Carlo methods and Halton and Hammersley quasi-random sequence methods. In this paper, the performance of all these sampling methods and a new variant (“Latinized” CVT) are further compared for non -uniform input distributions. Specifically, given uncorrelated normal inputs in a 2-D test problem, statistical sampling efficiencies are compared for resolving various statistics of response: mean, variance, and exceedence probabilities.

Centroidal Voronoi Tessellation-Based Reduced-Order Modeling of Complex Systems

\noindent A reduced-order modeling methodology based on centroidal Voronoi tessellations (CVTs) is introduced. CVTs are special Voronoi tessellations for which the generators of the Voronoi diagram are also the centers of mass (means) of the corresponding Voronoi cells. For discrete data sets, CVTs are closely related to the h-means and k-means clustering techniques. A discussion of reduced-order modeling for complex systems such as fluid flows is given to provide a context for the application of reduced-order bases. Then, detailed descriptions of CVT-based reduced-order bases and how they can be constructed from snapshot sets and how they can be applied to the low-cost simulation of complex systems are given. Subsequently, some concrete incompressible flow examples are used to illustrate the construction and use of CVT-based reduced-order bases. The CVT-based reduced-order modeling methodology is shown to be effective for these examples.

Insensitive Functionals, Inconsistent Gradients, Spurious Minima, and Regularized Functionals in Flow Optimization Problems

We use the simple context of Navier-Stokes flow in a channel with a bump to examine problems caused by the insensitivity of functionals with respect to design parameters, the inconsistency of functional gradient approximations, and the appearance of spurious minima in discretized functionals. We discuss how regularization can help overcome these problems. Along the way, we compare the discretize-then-differentiate and differentiate-then-discretize approaches to optimization, especially as they relate to the issue of inconsistent functional gradients. We close with a discussion of the implications that our observations have on more practical flow control and optimization problems.

Constrained CVT meshes and a comparison of triangular mesh generators

Mesh generation in regions in Euclidean space is a central task in computational science, and especially for commonly used numerical methods for the solution of partial differential equations, e.g., finite element and finite volume methods. We focus on the uniform Delaunay triangulation of planar regions and, in particular, on how one selects the positions of the vertices of the triangulation. We discuss a recently developed method, based on the centroidal Voronoi tessellation (CVT) concept, for effecting such triangulations and present two algorithms, including one new one, for CVT-based grid generation. We also compare several methods, including CVT-based methods, for triangulating planar domains. To this end, we define several quantitative measures of the quality of uniform grids. We then generate triangulations of several planar regions, including some having complexities that are representative of what one may encounter in practice. We subject the resulting grids to visual and quantitative comparisons and conclude that all the methods considered produce high-quality uniform grids and that the CVT-based grids are at least as good as any of the others.

Optimal Design and Control

This volume is a collection of papers written by engineers and mathematicians actively involved in innovative research in control and optimization, with emphasis placed on problems governed by partial differential equations. The papers arose from a workshop on control and optimization sponsored by SIAM. The volume presents research conducted at laboratory, industrial and academic institutions so that analyses, algorithms, implementations and applications are all well-presented in the papers. An overriding impression that ccan be gleaned from this volume is the complexity of problems addressed by not only those authors engaged in applications, but also by those engaged in algorithmic development and even mathematical analyses. Thus, in many instances, systematic approaches using fully nonlinear constraint equations are routinely used to solve control and optimization problems, in some cases replacing ad hoc or empirically based procedures. Many algorithmic issues are addressed, including sensitivity analyses, novel optimization methods especially tailored for specific applications, multilevel methods and programming techniques. Although many different applications (such as metal forging, heat transfer, contact problems, structures and acoustics) are considered in the volume, flow control plays a central role. Another recurring theme is shape control - that is, the use of shape of the boundary to effect control or to achieve optimal configuration. The book is intended for researchers and graduate students working in structural, fluid, thermal and electromagnetic design and control, or in the design and implementation of optimization algorithms.

Control of Steady Incompressible 2D Channel Flow

We consider steady incompressible flows in a 2D channel with flow quantities measured along some fixed, transverse sampling line. From a set of allowable flows it is desired to produce a flow that matches a given set of measurements as closely as possible. Allowable flows are completely specified by a set of control parameters which determine the shape of the inflow at the boundary and the shape of an internal bump which partially obstructs the flow. Difficulties concerning the transformation of this problem into a standard optimization problem are discussed, including the correct choice of functional and algorithm, and the existence of local minima.

Hyperspherical Sparse Approximation Techniques for High-Dimensional Discontinuity Detection

This work proposes a hyperspherical sparse approximation framework for detecting jump discontinuities in functions in high-dimensional spaces. The need for a novel approach results from the theoretical and computational inefficiencies of well-known approaches, such as adaptive sparse grids, for discontinuity detection. Our approach constructs the hyperspherical coordinate representation of the discontinuity surface of a function. Then sparse approximations of the transformed function are built in the hyperspherical coordinate system, with values at each point estimated by solving a one-dimensional discontinuity detection problem. Due to the smoothness of the hypersurface, the new technique can identify jump discontinuities with significantly reduced computational cost, compared to existing methods. Several approaches are used to approximate the transformed discontinuity surface in the hyperspherical system, including adaptive sparse grid and radial basis function interpolation, discrete least squares proj...

Initial Application and Evaluation of a Promising New Sampling Method for Response Surface Generation: Centroidal Voronoi Tessellation *

A recently developed Centroidal Voronoi Tessellation (CVT) sampling method is investigated here to assess its suitability for use in response surface generation. CVT is an unstructured sampling method that can generate nearly uniform point spacing over arbitrarily shaped Mdimensional parameter spaces. For rectangular parameter spaces (hypercubes), CVT appears to extend to higher dimensions more effectively and inexpensively than “Distributed” and “Improved Distributed” Latin Hypercube Monte Carlo methods, and CVT does not appear to suffer from spurious correlation effects in higher dimensions and at high sampling densities as quasi-Monte-Carlo methods such as Halton and Sobol sequences typically do. CVT is described briefly in this paper and its impact on response surface accuracy in a 2D test problem is compared to the accuracy yielded by Latin Hypercube Sampling (LHS) and a deterministic structured-uniform sampling method. To accommodate the different point patterns over the parameter space given by the different sampling methods, Moving Least Squares (MLS) for interpolation of arbitrarily located data points is used. It is found that CVT performs better than LHS in 11 of 12 test cases investigated here, and as often as not performs better than the structured sampling method with its deterministically uniform point placement over the 2-D parameter space.

The dual variable method for the solution of compressible fluid flow problems

Discretizations of the Navier–Stokes equations describing a compressible flow problem can be viewed as systems defining flows on an associated network. This observation provides a means of economizing on their numerical solution.