We present a fundamental procedure for instant rendering from the radiance equation. Operating directly on the textured scene description, the very efficient and simple algorithm produces photorealistic images without any finite element kernel or solution discretization of the underlying integral equation. Rendering rates of a few seconds are obtained by exploiting graphics hardware, the deterministic technique of the quasi-random walk for the solution of the global illumination problem, and the new method of jittered low discrepancy sampling.

Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed (IID) U(0, 1) random variables (i.e., continuous random variables distributed uniformly over the interval

Rasterization hardware provides interactive frame rates for rendering dynamic scenes, but lacks the ability of ray tracing required for efficient global illumination simulation. Existing ray tracing based methods yield high quality renderings but are far too slow for interactive use. We present a new parallel global illumination algorithm that perfectly scales, has minimal preprocessing and communication overhead, applies highly efficient sampling techniques based on randomized quasi-Monte Carlo integration, and benefits from a fast parallel ray tracing implementation by shooting coherent groups of rays. Thus a performance is achieved that allows for applying arbitrary changes to the scene, while simulating global illumination including shadows from area light sources, indirect illumination, specular effects, and caustics at interactive frame rates. Ceasing interaction rapidly provides high quality renderings.

Recently quasi-Monte Carlo algorithms have been successfully used for multivariate integration of high dimension d, and were significantly more efficient than Monte Carlo algorithms. The existing theory of the worst case error bounds of quasi-Monte Carlo algorithms does not explain this phenomenon. This paper presents a partial answer to why quasi-Monte Carlo algorithms can work well for arbitrarily large d. It is done by identifying classes of functions for which the effect of the dimension d is negligible. These are weighted classes in which the behavior in the successive dimensions is moderated by a sequence of weights. We prove that the minimal worst case error of quasi-Monte Carlo algorithms does not depend on the dimension d iff the sum of the weights is finite. We also prove that under this assumption the minimal number of function values in the worst case setting needed to reduce the initial error by " is bounded by C " \Gammap , where the exponent p 2 [1; 2], and C depends ...

Abstract. An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the L p-star discrepancy and Pα that arises in the study of lattice rules.

this paper were obtained using FINDER. One of the improvements was developing the table of primitive polynomials and initial direction numbers for dimensions up to 360. This paper is based on two years of software construction and testing. Preliminary results were presented to a number of New York City financial houses in the Fall of 1993 and the Spring of 1994. A January, 1994 article in Scientific American [5] discussed the theoretical issues and reported that "Preliminary results obtained by testing certain finance problems suggest the superiority of the deterministic methods in practice." Further results were reported at a number of conferences in the summer and fall of 1994. A June, 1994 article in Business Week [1] indicates the possible superiority of low discrepancy sequences. Details on the CMO, the numerical methods, and the test results are presented in [3]. Here we limit ourselves to stating our main conclusions and indicating typical results. For brevity, we shall refer to the method which uses Sobol points as the Sobol method. We summarize our main conclusions regarding the evaluation of this CMO. The conclusions may be divided into three groups.

this paper is to provide good CMRGs of different sizes, selected via the spectral test up to 32 (or 24) dimensions, and a faster implementation than in L'Ecuyer (1996) using floating-point arithmetic. Why do we need different parameter sets? Firstly, different types of implementations require different constraints on the modulus and multipliers. For example, a floating-point implementation with 53 bits of precision allows moduli of more than 31 bits and this can be exploited to increase the period length for free. Secondly, as 64-bit computers get more widespread, there is demand for generators implemented in 64-bit integer arithmetic. Tables of good parameters for such generators must be made available. Thirdly, RNGs are somewhat like cars: a single model and single size for the entire world is not the most satisfactory solution. Some people want a fast and relatively small RNG, while others prefer a bigger and more robust one, with longer period and good equidistribution properties in larger dimensions. Naively, one could think that an RNG with period length near 2

This paper introduces Latin supercube sampling (LSS) for very high dimensional simulations, such as arise in particle transport, finance and queuing. LSS is developed as a combination of two widely used methods: Latin hypercube sampling (LHS), and Quasi-Monte Carlo (QMC). In LSS, the input variables are grouped into subsets, and a lower dimensional QMC method is used within each subset. The QMC points are presented in random order within subsets. QMC methods have been observed to lose effectiveness in high dimensional problems. This paper shows that LSS can extend the benefits of QMC to much higher dimensions, when one can make a good grouping of input variables. Some suggestions for grouping variables are given for the motivating examples. Even a poor grouping can still be expected to do as well as LHS. The paper also extends LHS and LSS to infinite dimensional problems. The paper includes a survey of QMC methods, randomized versions of them (RQMC) and previous methods for extending Q...

Abstract: This paper introduces random versions of successive approximations and multigrid algorithms for computing approximate solutions to a class of finite and infinite horizon Markovian decision problems (MDPs). We prove that these algorithms succeed in breaking the “curse of dimensionality ” for a subclass of MDPs known as discrete decision processes (DDPs). 1

Tausworthe random number generators based on a primitive trinomial allow an easy and fast implementation when their parameters obey certain restrictions. However, such generators, with those restrictions, have bad statistical properties unless we combine them. A generator is called maximally equidistributed if its vectors of successive values have the best possible equidistribution in all dimensions. This paper shows how to find maximally equidistributed combinations in an efficient manner, and gives a list of generators with that property. Such generators have a strong theoretical support and lend themselves to very fast software implementations.