Preface: Multivariate algorithms and information-based complexity

Abstract

The authors of this book include several of the invited speakers in the workshopMultivariate Algorithms and Information-Based Complexity, which was part of the RICAM Special Semester onMultivariate Algorithms and their Foundations in Number Theory in the fall of 2018. The special semester consisted of four larger and two smaller workshops on various topics ranging fromPseudo-Randomness andDiscrepancy Theory to Information-Based Complexity and Uncertainty Quantification. This book arises from the second workshop, which took place at the Johann Radon Institute for Computational andAppliedMathematics (RICAM) of the Austrian Academy of Sciences in Linz, Austria, on November 5–9, 2018. Multivariate continuous problems occur in a multitude of practical applications, such as physics, finance, computer graphics, and chemistry. The number of variables involved, d, can be in the hundreds or thousands. The information complexity of a given problem is the minimal number of information operations required by the best algorithm to solve the problem for a prescribed set of inputs within a certain error threshold, ε. Typical examples of information operations are function values and linear functionals. The field of information-based complexity (IBC), founded by Traub andWozniakowski in the 1980s, analyzes the information complexity for multivariate problemsanddetermineshow it depends ond and ε. A crucial question is underwhich circumstances one can avoid a curse of dimensionality, namely, exponential growth of the information complexity with d. This book addresses the analysis of multivariate (continuous) problems, especially from the IBC viewpoint. The problems discussed by the authors reflect the breadth of current inquiry under the umbrella of multivariate algorithms and IBC. The chapter entitled“The control variate integration algorithm for multivariate functions defined at scattered data points” studies a method of approximating the integral of a multivariate function, in which one uses the exact integral of a control variate based on a least-squares multivariate quasiinterpolant. Numerical examples demonstrate that such an algorithm can overcome the curse of dimensionality formultivariate least-squares fits. The second chapter, titled “An adaptive random bit multilevel algorithm for SDEs”, considers the approximations of expectations for functionals applied to the solutions of stochastic differential equations by employing Monte Carlo methods based on random bits instead of random numbers. An adaptive random bit multilevel algorithm is provided and compared numerically to other methods. The chapter “RBF-based penalized least-squares approximation of noisy scattered data on the sphere” deals with the approximation of noisy scattered data on the 2-dimensional unit sphere. In particular, global and local penalized least-squares approximation based on radial basis functions (RBFs) are explored. The authors of the fourth chapter in this book, “On the power of random information”, consider a problem from the core of IBC theory,