Integral theorem on large deviations of integer-valued additive functions

The method of analytical preconditioning combines the discretization and the analytical regularization of a singular integral equation in a single step. In a recent paper by the author, such a method has been applied to a spectral domain integral equation formulation devised to analyze the propagation in polygonal cross-section microstrip lines, which are widely used as high-speed interconnects in monolithic microwave and millimeter waves integrated circuits. By choosing analytically Fourier transformable expansion functions reconstructing the behavior of the fields on the wedges, fast convergence is achieved, and the convolution integrals are expressed in closed form. However, the coefficient matrix elements are one-dimensional improper integrals of oscillating and, in the worst cases, slowly decaying functions. In this paper, a novel technique for the efficient evaluation of such kind of integrals is proposed. By means of a procedure based on Cauchy integral theorem, the general coefficient matrix element is written as a linear combination of fast converging integrals. As shown in the numerical results section, the proposed technique always outperforms the analytical asymptotic acceleration technique, especially when highly accurate solutions are required.

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Run-and-Tumble Motion: The Role of Reversibility

Journal of Statistical Physics ◽

10.1007/s10955-021-02787-1 ◽

2021 ◽

Vol 183 (3) ◽

Author(s):

Bart van Ginkel ◽

Bart van Gisbergen ◽

Frank Redig

Keyword(s):

Free Energy ◽

Diffusion Coefficient ◽

Large Deviations ◽

Energy Function ◽

Internal State ◽

Rate Function ◽

Free Energy Function ◽

Large Deviations Principle ◽

Preferred Direction ◽

State Process

AbstractWe study a model of active particles that perform a simple random walk and on top of that have a preferred direction determined by an internal state which is modelled by a stationary Markov process. First we calculate the limiting diffusion coefficient. Then we show that the ‘active part’ of the diffusion coefficient is in some sense maximal for reversible state processes. Further, we obtain a large deviations principle for the active particle in terms of the large deviations rate function of the empirical process corresponding to the state process. Again we show that the rate function and free energy function are (pointwise) optimal for reversible state processes. Finally, we show that in the case with two states, the Fourier–Laplace transform of the distribution, the moment generating function and the free energy function can be computed explicitly. Along the way we provide several examples.

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Weighted approximate Bayesian computation via Sanov’s theorem

Computational Statistics ◽

10.1007/s00180-021-01093-4 ◽

2021 ◽

Author(s):

Cecilia Viscardi ◽

Michele Boreale ◽

Fabio Corradi

Keyword(s):

Large Deviations ◽

Posterior Distribution ◽

Approximate Bayesian Computation ◽

Bayesian Computation ◽

Information Theoretic ◽

Discrete Random Variables ◽

Positive Weights ◽

Approximate Bayesian ◽

Information Theoretic Method ◽

Computational Resources

AbstractWe consider the problem of sample degeneracy in Approximate Bayesian Computation. It arises when proposed values of the parameters, once given as input to the generative model, rarely lead to simulations resembling the observed data and are hence discarded. Such “poor” parameter proposals do not contribute at all to the representation of the parameter’s posterior distribution. This leads to a very large number of required simulations and/or a waste of computational resources, as well as to distortions in the computed posterior distribution. To mitigate this problem, we propose an algorithm, referred to as the Large Deviations Weighted Approximate Bayesian Computation algorithm, where, via Sanov’s Theorem, strictly positive weights are computed for all proposed parameters, thus avoiding the rejection step altogether. In order to derive a computable asymptotic approximation from Sanov’s result, we adopt the information theoretic “method of types” formulation of the method of Large Deviations, thus restricting our attention to models for i.i.d. discrete random variables. Finally, we experimentally evaluate our method through a proof-of-concept implementation.

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