Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Planning Algorithms: Introductory Material

A centroidal Voronoi tessellation is a Voronoi tessellation whose generating points are the centroids (centers of mass) of the corresponding Voronoi regions. We give some applications of such tessellations to problems in image compression, quadrature, finite difference methods, distribution of resources, cellular biology, statistics, and the territorial behavior of animals. We discuss methods for computing these tessellations, provide some analyses concerning both the tessellations and the methods for their determination, and, finally, present the results of some numerical experiments.

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Centroidal Voronoi Tessellations: Applications and Algorithms

A comprehensive overview of Monte Carlo simulation that explores the latest topics, techniques, and real-world applications
	
	More and more of today’s numerical problems found in engineering and finance are solved through Monte Carlo methods. The heightened popularity of these methods and their continuing development makes it important for researchers to have a comprehensive understanding of the Monte Carlo approach. Handbook of Monte Carlo Methods provides the theory, algorithms, and applications that helps provide a thorough understanding of the emerging dynamics of this rapidly-growing field.
	
	The authors begin with a discussion of fundamentals such as how to generate random numbers on a computer. Subsequent chapters discuss key Monte Carlo topics and methods, including:
	- Random variable and stochastic process generation
	- Markov chain Monte Carlo, featuring key algorithms such as the Metropolis-Hastings method, the Gibbs sampler, and hit-and-run
	- Discrete-event simulation
	- Techniques for the statistical analysis of simulation data including the delta method, steady-state estimation, and kernel density estimation
	- Variance reduction, including importance sampling, latin hypercube sampling, and conditional Monte Carlo
	- Estimation of derivatives and sensitivity analysis
	- Advanced topics including cross-entropy, rare events, kernel density estimation, quasi Monte Carlo, particle systems, and randomized optimization
	
	The presented theoretical concepts are illustrated with worked examples that use MATLAB, a related Web site houses the MATLAB code, allowing readers to work hands-on with the material and also features the author's own lecture notes on Monte Carlo methods. Detailed appendices provide background material on probability theory, stochastic processes, and mathematical statistics as well as the key optimization concepts and techniques that are relevant to Monte Carlo simulation.
	
	Handbook of Monte Carlo Methods is an excellent reference for applied statisticians and practitioners working in the fields of engineering and finance who use or would like to learn how to use Monte Carlo in their research. It is also a suitable supplement for courses on Monte Carlo methods and computational statistics at the upper-undergraduate and graduate levels.

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Handbook of Monte Carlo Methods

EPW: Electron-phonon coupling, transport and superconducting properties using maximally localized Wannier functions

The Monte Carlo method.- Sampling from known distributions.- Pseudorandom number generators.- Variance reduction techniques.- Quasi-Monte Carlo constructions.- Using quasi-Monte Carlo constructions.- Using quasi-Monte Carlo in practice.- Financial applications.- Beyond numerical integration.- Review of algebra.- Error and variance analysis for Halton sequences.- References.- Index.

Monte Carlo and Quasi-Monte Carlo Sampling

Integration lattices are one of the main types of low discrepancy sets used in quasi-Monte Carlo methods. However, they have the disadvantage of being of fixed size. This article describes the construction of an infinite sequence of points, the first bm of which forms a lattice for any nonnegative integer m. Thus, if the quadrature error using an initial lattice is too large, the lattice can be extended without discarding the original points. Generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice. The spectral test used for finding pseudorandom number generators is one important example of such a discrepancy. The performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems.

https://epubs.siam.org/doi/pdf/10.1137/S1064827599356638

Extensible Lattice Sequences for Quasi-Monte Carlo Quadrature

Random scrambling of deterministic (t, m, s)-nets and (t, s)-sequences eliminates their inherent bias while retaining their low-discrepancy properties. This article describes an implementation of two types of random scrambling, one proposed by Owen and another proposed by Faure and Tezuka. The four different constructions of digital sequences implemented are those proposed by Sobol', Faure, Niederreiter, and Niederreiter and Xing. Because the random scrambling involves manipulating all digits of each point, the code must be written carefully to minimize the execution time. Computed root mean square discrepancies of the scrambled sequences are compared to known theoretical results. Furthermore, the performances of these sequences on various test problems are discussed.

Algorithm 823: Implementing scrambled digital sequences

An L 2 -type discrepancy arises in the average- and worst-case error analyses for multidimensional quadrature rules. This discrepancy is uniquely defined by K(x,y), which serves as the covariance kernel for the space of random functions in the average-case analysis and a reproducing kernel for the space of functions in the worst-case analysis. This article investigates the asymptotic order of the root mean square discrepancy for randomized (0,m,s)-nets in base b. For moderately smooth K(x,y) the discrepancy is O(N -1 [log(N)] (s-1)/2 ), and for K(x,y) with greater smoothness the discrepancy is O(N-3/2[log(N)] (s-1)/2 ), where N = b m is the number of points in the net. Numerical experiments indicate that the (t,m,s)-nets of Faure, Niederreiter and Sobol' do not necessarily attain the higher order of decay for sufficiently smooth kernels. However, Niederreiter nets may attain the higher order for kernels corresponding to spaces of periodic functions.

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The asymptotic efficiency of randomized nets for quadrature

The distribution of the discrepancy of scrambled digital ( t, m, s )-nets

Quasi-Monte Carlo methods are a way of improving the efficiency of Monte Carlo methods. Digital nets and sequences are one of the low discrepancy point sets used in quasi-Monte Carlo methods. This thesis presents the three new results pertaining to digital nets and sequences: implementing randomized digital nets, finding the distribution of the discrepancy of scrambled digital nets, and obtaining better quality of digital nets through evolutionary computation. Finally, applications of scrambled and non-scrambled digital nets are provided.

Hee Sun Hong

Papers

Extensible Lattice Sequences for Quasi-Monte Carlo Quadrature

Algorithm 823: Implementing scrambled digital sequences

The asymptotic efficiency of randomized nets for quadrature

The distribution of the discrepancy of scrambled digital ( t, m, s )-nets

Digital Nets and Sequences for Quasi-Monte Carlo Methods