We propose and analyze deterministic multilevel approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive gaussian measurement data. The algorithms use a multilevel (ML) approach based on deterministic, higher order quasi-Monte Carlo (HoQMC) quadrature for approximating the high-dimensional expectations, which arise in the Bayesian estimators, and a Petrov-Galerkin (PG) method for approximating the solution to the underlying partial differential equation (PDE). This extends the previous single-level approach from [J. Dick, R. N. Gantner, Q. T. Le Gia and Ch. Schwab, Higher order Quasi-Monte Carlo integration for Bayesian Estimation. Report 2016-13, Seminar for Applied Mathematics, ETH Zürich (in review)].Compared to the single-level approach, the present convergence analysis of the multilevel method requires stronger assumptions on holomorphy and regularity of the countably-parametric uncertaintyto-observation maps of the forward problem. As in the single-level case and in the affine-parametric case analyzed in [ J. Dick, F.Y. Kuo, Q. T. Le Gia and Ch. Schwab, Multi-level higher order QMC Galerkin discretization for affine parametric operator equations. Accepted for publication in SIAM J. Numer. Anal., 2016], we obtain sufficient conditions which allow us to achieve arbitrarily high, algebraic convergence rates in terms of work, which are independent of the dimension of the parameter space. The convergence rates are limited only by the spatial regularity of the forward problem, the discretization order achieved by the Petrov Galerkin discretization, and by the sparsity of the uncertainty parametrization.We provide detailed numerical experiments for linear elliptic problems in two space dimensions, with s = 1024 parameters characterizing the uncertain input, confirming the theory and showing that the ML HoQMC algorithms outperform, in terms of error vs. computational work, both multilevel Monte Carlo (MLMC) methods and single-level (SL) HoQMC methods.
We develop a strong diagnostic for bubbles and crashes in bitcoin, by analyzing the coincidence (and its absence) of fundamental and technical indicators. Using a generalized Metcalfe's law based on network properties, a fundamental value is quantified and shown to be heavily exceeded, on at least four occasions, by bubbles that grow and burst. In these bubbles, we detect a universal super-exponential unsustainable growth. We model this universal pattern with the Log-Periodic Power Law Singularity (LPPLS) model, which parsimoniously captures diverse positive feedback phenomena, such as herding and imitation. The LPPLS model is shown to provide an ex-ante warning of market instabilities, quantifying a high crash hazard and probabilistic bracket of the crash time consistent with the actual corrections; although, as always, the precise time and trigger (which straw breaks the camel's back) being exogenous and unpredictable. Looking forward, our analysis identifies a substantial but not unprecedented overvaluation in the price of bitcoin, suggesting many months of volatile sideways bitcoin prices ahead (from the time of writing, March 2018).
The efficient construction of higher-order interlaced polynomial lattice rules introduced recently in [4] is considered. After briefly reviewing the principles of their construction by the "fast component-by-component" (CBC) algorithm due to [1,10] as well as recent theoretical results on their convergence rates, we indicate algorithmic details of their construction. Instances of such rules are applied to highdimensional test integrands which belong to weighted function spaces with weights of product and of SPOD type. Practical considerations that lead to improved quantitative convergence behavior for various classes of test integrands are reported. The use of (analytic or numerical) bounds on the Walsh coefficients of the integrand are found to improve the convergence behavior. The sharpness of theoretical bounds on memory usage and operation counts, with respect to the number of points N and dimension s of the integration domain is verified experimentally. The efficiency of the proposed algorithms for computation of the generating vectors is confirmed for the considered classes of functions in dimensions s = 10, ..., 1000.
We develop a strong diagnostic for bubbles and crashes in Bitcoin, by analysing the coincidence (and its absence) of fundamental and technical indicators. Using a generalized Metcalfe’s Law based on network properties, a fundamental value is quantified and shown to be heavily exceeded, on at least four occasions, by bubbles that grow and burst. In these bubbles, we detect a universal super-exponential unsustainable growth. We model this universal pattern with the Log-Periodic Power Law Singularity (LPPLS) model, which parsimoniously captures diverse positive feedback phenomena, such as herding and imitation. The LPPLS model is shown to provide an ex ante warning of market instabilities, quantifying a high crash hazard and probabilistic bracket of the crash time consistent with the actual corrections; although, as always, the precise time and trigger (which straw breaks the camel’s back) is exogenous and unpredictable. Looking forward, our analysis identifies a substantial but not unprecedented overvaluation in the price of Bitcoin, suggesting many months of volatile sideways Bitcoin prices ahead (from the time of writing, March 2018).
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