Lognormal Case Research Articles

Exploiting locality in sparse polynomial approximation of parametric elliptic PDEs and application to parameterized domains

This work studies how the choice of the representation for parametric, spatially distributed inputs to elliptic partial differential equations (PDEs) affects the efficiency of a polynomial surrogate, based on Taylor expansion, for the parameter-to-solution map. In particular, we show potential advantages of representations using functions with localized supports. As model problem, we consider the steady-state diffusion equation, where the diffusion coefficient and right-hand side depend smoothly but potentially in a highly nonlinear way on a parameter y ∈ [−1, 1]N. Following previous work for affine parameter dependence and for the lognormal case, we use pointwise instead of norm-wise bounds to prove ℓp-summability of the Taylor coefficients of the solution. As application, we consider surrogates for solutions to elliptic PDEs on parametric domains. Using a mapping to a nominal configuration, this case fits in the general framework, and higher convergence rates can be attained when modeling the parametric boundary via spatially localized functions. The theoretical results are supported by numerical experiments for the parametric domain problem, illustrating the efficiency of the proposed approach and providing further insight on numerical aspects. Although the methods and ideas are carried out for the steady-state diffusion equation, they extend easily to other elliptic and parabolic PDEs.

Induced gravitational waves from statistically anisotropic scalar perturbations

Scalar-induced gravitational waves (SIGWs) are attracting growing attention for probing extremely short-scale scalar perturbations via gravitational wave measurements. In this paper, we investigate the SIGWs from statistically anisotropic scalar perturbations, which are motivated in inflationary scenarios in the presence of, e.g., a vector field. While the ensemble average of the SIGW energy spectrum is isotropic for the standard statistically isotropic scalar perturbations, the statistical anisotropy in the source introduces the multipole moments of the differential SIGW energy spectrum. We consider quadrupole anisotropy in the scalar power spectrum and show that the SIGW spectrum has anisotropies up to $\ensuremath{\ell}=4$. We present generic formulas of the multipole moments and then apply them to the delta-function-like and log-normal source spectra. We find analytic expressions for the former case and show that the infrared scalings of the multipole moments are the same as the isotropic SIGWs. Interestingly, the monopole has an additional local minimum in the high-$k$ tail, a key feature to distinguish from the isotropic SIGWs. The latter log-normal case is analytic for the narrow-peak source, and we perform the numerical calculation for the broad peak. As one expects, the multipole moments become broader with increasing source width. Our results are helpful to test the isotropy of primordial density perturbations at extremely small scales through SIGWs.

Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates

In this study, we present an adaptive multilevel Monte Carlo (AMLMC) algorithm for approximating deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient and geometric singularities in bounded domains of Rd. Our AMLMC algorithm is built on the results of the weak convergence rates in the work (Moon et al., 2006) for an adaptive algorithm using isoparametric d-linear quadrilateral finite element approximations and the dual weighted residual error representation in a deterministic setting. Designed to suit the geometric nature of the singularities in the solution, our AMLMC algorithm uses a sequence of deterministic, non-uniform auxiliary meshes as a building block. The above-mentioned deterministic adaptive algorithm generates these meshes, corresponding to a geometrically decreasing sequence of tolerances. In particular, for a given realization of the diffusivity coefficient and accuracy level, AMLMC constructs its approximate sample using the first mesh in the hierarchy that satisfies the corresponding bias accuracy constraint. This adaptive approach is particularly useful for the lognormal case treated here, which lacks uniform coercivity and thus produces functional outputs that vary over orders of magnitude when sampled. Furthermore, we discuss iterative solvers and compare their efficiency with direct ones. To reduce computational work, we propose a stopping criterion for the iterative solver with respect to the quantity of interest, the realization of the diffusivity coefficient, and the desired level of AMLMC approximation.From the numerical experiments, based on a Fourier expansion of the diffusivity coefficient field, we observe improvements in efficiency compared with both standard Monte Carlo (MC) and standard MLMC (SMLMC) for a problem with a singularity similar to that at the tip of a slit modeling a crack.

On the multiscale characterization of effective hydraulic conductivity in random heterogeneous media: A historical survey and some new perspectives

In large scale heterogeneous aquifer simulations, determining the appropriate coarsening scale λ to define an effective hydraulic conductivity Keff is a challenging task, that involves a trade-off between accuracy and cost. Efficiently adjusting the scale λ is then key, in particular for uncertainty quantification. In this paper, we obtain improved analytical results for the variance of Keff, valid at any scale, in the context of energy dissipation formulation. Using this formulation, we then derive an efficient Keff numerical estimator, and compare it with those of the potential-flow average and permeameter formulations in 2D, for lognormal and binary media, over a wide range of λ and of heterogeneity. We analyze the probability density function (pdf), mean, and variance, of these estimators, comparing them with the analytical results. In the lognormal case, the pdf’s are rather similar for the three estimators, and remain lognormal at all scales. In the binary case, slow convergence to an asymptotic regime is observed close to the percolation threshold.

High-precision Monte Carlo modelling of galaxy distribution

We revisit the case of fast Monte Carlo simulations of galaxy positions for a non-Gaussian field. More precisely, we address the question of generating a 3D field with a given one-point function (e.g. log-normal) and some power spectrum fixed by cosmology. We highlight and investigate a problem that occurs in the log-normal case when the field is filtered, and we identify a regime where this approximation still holds. However, we show that the filtering is unnecessary if aliasing effects are taken into account and the discrete sampling step is carefully controlled. In this way we demonstrate a sub-percent precision of all our spectra up to the Nyquist frequency. We extend the method to generate a full light cone evolution, comparing two methods for this process, and validate our method with a tomographic analysis. We analytically and numerically investigate the structure of the covariance matrices obtained with such simulations which may be useful for future large and deep surveys.

PSO-KS Algorithm for Fitting Lognormal Distribution: Simulation and Empirical Implementation to Women’s Age at First Marriage Data

Lognormal distribution plays an essential role in the distribution modeling of right-skewed data in many areas. In social sciences, for instance, it can be used to model women’s age at first marriage pattern, a key indicator in studying fertility patterns. Distribution fitting is a fundamental and essential stage of data modeling before doing advancing the analysis. Kolmogorov-Smirnov (KS) distance is applicable as maximum goodness-of-fit (GOF) estimators for distribution parameters. Minimizing KS distance is optimization problem. Particle swarm optimization (PSO) algorithm is a general optimizer that can handle various optimization problems. This study assesses the characteristics of minimum KS distance estimator for lognormal distribution parameters. KS distance estimators were obtained via optimization using the PSO algorithm, so the combination of these is called the PSO-KS algorithm. We conducted a simulation to assess the performance of PSO-KS, Maximum Likelihood (MLE), Method of Moment (MME). The bias and mean square error (MSE) of point estimators were used in simulation to assess the characteristics of estimators. Meanwhile, MSE of distribution fitting, KS distance, and log-likelihood value were used to evaluate the GOF characteristics. Moreover, we demonstrated the performance of the algorithm by implementing it to women’s age at first marriage data in Indonesia. The results show that based on the bias and MSE properties, the PSO-KS point estimators yield similar characteristics with MLE, but better than MME. From the GOF perspective, PSO-KS outperforms in MSE of distribution fitting and KS distance, but not in log-likelihood value. We also observed these patterns in the women’s age at first marriage data. The contributions of this study are two-fold, first to assess the PSO-KS algorithm in the lognormal distribution case. Second, it implements the algorithm on women’s age at first marriage data, which has broad social, economic, and public health implications.

On the Sum of Order Statistics and Applications to Wireless Communication Systems Performances

We consider the problem of evaluating the cumulative distribution function (CDF) of the sum of order statistics, which serves to compute outage probability (OP) values at the output of generalized selection combining receivers. Generally, closed-form expressions of the CDF of the sum of order statistics are unavailable for many practical distributions. Moreover, the naive Monte Carlo (MC) method requires a substantial computational effort when the probability of interest is sufficiently small. In the region of small OP values, we instead propose two effective variance reduction techniques that yield a reliable estimate of the CDF with small computing cost. The first estimator, which can be viewed as an importance sampling estimator, has bounded relative error under a certain assumption that is shown to hold for most of the challenging distributions. A possible improvement of this estimator is then proposed for the Pareto and the Weibull cases. The second is a conditional MC estimator that achieves the bounded relative error property for the generalized Gamma case and the logarithmic efficiency for the Log-normal case. Finally, the efficiency of these estimators is compared via various numerical simulations.

Vanna-Volga Method for Normal Volatilities

Vanna-Volga is a popular method for the interpolation/extrapolation of volatility smiles. The technique is widely used in the FX markets context, due to its ability to consistently construct the entire Lognormal smile using only three Lognormal market quotes. However, the derivation of the Vanna-Volga method itself is free of distributional assumptions. With this is mind, it is surprising there have been no attempts to apply the method to Normal volatilities (the current standard for interest rate markets). We show how the method can be modified to build Normal volatility smiles. As it turns out, only minor modifications are required compared to the Lognormal case. Moreover, as the inversion of Normal volatilities from option prices is easier in the Normal case, the smile construction can occur at a machine-precision level using analytical formulae, making the approximations via Taylor-series unnecessary. Apart from being based on practical and intuitive hedging arguments, the Vanna-Volga has further important advantages. In comparison to the Normal SABR model, the Vanna-Volga can easily fit both classical convex and atypical concave smiles (‘frowns’). Concave smile patterns are sometimes observed around ATM strikes in the interest rate markets, particularly in the situations of anticipated jumps (with an unclear outcome) in interest rates. Besides, concavity is often observed towards the lower/left end of the Normal volatility smiles of interest rates. At least in these situations, the Vanna-Volga can be expected to interpolate/extrapolate better than SABR.

Robust allocation of operating rooms: A cutting plane approach to handle lognormal case durations

The problem of allocating operating rooms (OR) to surgical cases is a challenging task, involving both combinatorial aspects and uncertainty handling. We formulate this problem as a parallel machines scheduling problem, in which job durations follow a lognormal distribution, and a fixed assignment of jobs to machines must be computed. We propose a cutting-plane approach to solve the robust counterpart of this optimization problem. To this end, we develop an algorithm based on fixed-point iterations that identifies worst-case scenarios and generates cut inequalities. The main result of this article uses Hilbert’s projective geometry to prove the convergence of this procedure under mild conditions. We also propose two exact solution methods for a similar problem, but with a polyhedral uncertainty set, for which only approximation approaches were known. Our model can be extended to balance the load over several planning periods in a rolling horizon. We present extensive numerical experiments for instances based on real data from a major hospital in Berlin. In particular, we find that: (i) our approach performs well compared to a previous model that ignored the distribution of case durations; (ii) compared to an alternative stochastic programming approach, robust optimization yields solutions that are more robust against uncertainty, at a small price in terms of average cost; (iii) the longest expected processing time first (LEPT) heuristic performs well and efficiently protects against extreme scenarios, but only if a good prediction model for the durations is available. Finally, we draw a number of managerial implications from these observations.

Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation

This article provides a survey of recent research efforts on the application of quasi-Monte Carlo (QMC) methods to elliptic partial differential equations (PDEs) with random diffusion coefficients. It considers and contrasts the uniform case versus the lognormal case, single-level algorithms versus multi-level algorithms, first-order QMC rules versus higher-order QMC rules, and deterministic QMC methods versus randomized QMC methods. It gives a summary of the error analysis and proof techniques in a unified view, and provides a practical guide to the software for constructing and generating QMC points tailored to the PDE problems. The analysis for the uniform case can be generalized to cover a range of affine parametric operator equations.

Performance of single and dual-polarized optically preamplified M-ary PPM systems with finite extinction ratios over FSO fading channels

M-ary pulse position modulation (PPM) systems have been considered in free-space optical (FSO) communications, optical fiber links, and passive optical networks. In this paper, we study the error performance of direct-detection optically preamplified M-ary PPM systems over slowly fading FSO channels. The study considers the combined effects on the probability of bit error, Pb, of channel fading with a given scintillation index, σp2, the transmitter finite extinction ratio, r, and the preamplifier spontaneous emission (ASE) noise. We provide results for both single and dual-polarized systems with symbol sizes M∈{2,4,…,1024} at Pb=10−4 and Pb=10−9. The fading models considered in this study are the exponential, log-normal, and gamma–gamma channels. For single-polarized systems with infinite extinction ratios, we provide closed-form expressions for the bit error probabilities for the three channel models. For the dual-polarized systems we compute them numerically. The results indicate that gamma–gamma fading imposes a more severe penalty than the log-normal case. In this study, the power penalty at Pb=10−9 ranges between 1.8 and 14dB for the log-normal channel, whereas it ranges between 2.2 and 30.7dB for the gamma–gamma channel. The study also demonstrates that the power penalty due to the combined effects of transmitter finite r and channel fading is the sum of the penalty due to fading alone and the penalty due to a finite r alone, and that the power penalty for dual-polarized systems is about 0.4dB larger than single-polarized ones.

Bayesian accelerated life testing under competing log-location-scale family of causes of failure

This article provides Bayesian analyses of data arising from multi-stress accelerated life testing of series systems. The component log-lifetimes are assumed to independently belong to some log-concave location-scale family of distributions. The location parameters are assumed to depend on the stress variables through a linear stress translation function. Bayesian analyses and associated predictive inference of reliability characteristics at usage stresses are performed using Gibbs sampling from the joint posterior. The developed methodology is numerically illustrated by analyzing a real data set through Bayesian model averaging of the two popular cases of Weibull and log-normal, with the later getting a special focus in this article as a slightly easier example of the log-location-scale family. A detailed simulation study is also carried out to compare the performance of various Bayesian point estimators for the log-normal case.

BER of annular and flat-topped beams in strong turbulence

The average bit error rate (<BER>) of annular and flat-topped beams are evaluated in strong turbulence. In this respect, our earlier results on the scintillation indices obtained by the unified Rytov method are employed and the intensity is taken to be gamma–gamma distributed. For comparison purposes, <BER> for the log-normal intensity distribution is also evaluated. It is found that for the annular beams, the ones that are thinner, possessing smaller ratio of primary to secondary beam size, and smaller focal lengths will have smaller average BER in strong turbulence. For the flat-topped beams, the ones that are flatter and possessing large source sizes have smaller average BER in strong turbulence. Large average SNR substantially reduces the average BER in weak and moderate turbulence, whereas in strong turbulence, the average BER stays at the same value no matter what the average SNR is. Comparison of the log-normal and the gamma–gamma statistics for the intensity shows that the average BER will be higher for the log-normal case when the average SNR is small and the reverse relationship holds at large average SNR. For both the gamma–gamma and the log-normal intensity distributions, <BER> obtained for the annular and the flat-topped beams in strong turbulence is advantageous over the Gaussian beam <BER> values.

KPZ formula for log-infinitely divisible multifractal random measures

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in [E. Bacry et al. Comm. Math. Phys. 236 (2003) 449–475]. If M is a non degenerate multifractal measure with associated metric ρ(x,y) = M([x,y]) and structure function ζ, we show that we have the following relation between the (Euclidian) Hausdorff dimension dimH of a measurable set K and the Hausdorff dimension dimHρ with respect to ρ of the same set: ζ(dimHρ(K)) = dimH(K). Our results can be extended to all dimensions: inspired by quantum gravity in dimension 2, we focus on the log normal case in dimension 2.

Pricing arithmetic asian options under the cev process

This paper discusses the pricing of arithmetic Asian options when the underlying stock follows the constant elasticity of variance (CEV) process. We build a binomial tree method to estimate the CEV process and use it to price arithmetic Asian options. We find that the binomial tree method for the lognormal case can effectively solve the computational problems arising from the inherent complexities of arithmetic Asian options when the stock price follows CEV process. We present numerical results to demonstrate the validity and the convergence of the approach for the different parameter values set in CEV process.

Optimal decision making with biologically realistic neural signals

Action selection in animals requires rapid decision making that can discriminate the most salient requests for behavioral expression. The basal ganglia (BG) are believed to play a critical role in resolving competition between these requests [1] and, recently it has been proposed that the BG and cortex, taken together, implement a decision algorithm known as the multi-hypothesis sequential probability ratio test (MSPRT) [2]. Here, the cortex first integrates noisy ‘evidence’ indicating salience of action requests. The BG then examine this integrated evidence, and report the channel with maximal mean salience. The MSPRT is optimal in the sense that it guarantees the smallest decision time for a given error. The signals used in [2] assume cortical evidence - as neural firing rates - is supplied by Gaussian distributed signals. However, it is known that such rates are not distributed normally. To remedy this we have deduced three new forms for MSPRT that are based on more biologically realistic input signals. They include the Inverse Gaussian, Gamma and Lognormal distributions. All implementations preserve a form which could be mapped, in principle, onto the anatomy of the BG. Figure ​Figure11 shows an example of the performance of the MSPRT models across all distributions, together with the so called race model. All variants used the same input statistics (mean and variance). Counterintuitively, the relative performance of the models did not change with base level of salience or evidence. Figure 1 Performance of original Gaussian MSPRT[2], new implementations of MSPRT, and Race model. Performance is measured as decision time against number of alternatives for selection. The selection criterion was 1% error rate. The results are averaged across ... A critical parameter in the algorithm is the time step Δt used to sample the cortical input. The Gaussian, Inverse Gaussian, and Gamma cases are generated as stable Levy processes and may be scaled indefinitely with decreasing Δt. However, unlike the MSPRT with Gaussian signals, we show that decision time for the inverse Gaussian, and Gamma decreases with Δt. Figure ​Figure11 use a Δt = 1 ms for all models and clearly shows a performance advantage for the cases with more realistic distributions (the lognormal case is shown with similar scaling even though it has not an associated Levy process). Indefinitely small decision times may appear unrealistic, but we present an interpretation of cortical sampling that relates Δt to the inter-spike intervals of an ensemble of neural afferents impinging on the cortical ‘integrators’. This and, other mechanisms, provide a natural lower bound for Δt. We conclude that neurons may take advantage of the properties of natural spike trains to enhance decision making in the cortex-BG complex.

MODERN LIBOR MARKET MODELS: USING DIFFERENT CURVES FOR PROJECTING RATES AND FOR DISCOUNTING

We introduce an extended LIBOR market model that is compatible with the current market practice of building different yield curves for different tenors and for discounting. The new paradigm is based on modeling the joint evolution of FRA rates and forward rates belonging to the discount curve. We will start by analyzing the basic lognormal case, then we will add stochastic volatility. The dynamics of FRA rates under different measures will be obtained and closed form formulas for caplets and swaptions derived in the lognormal and Heston (1993) cases.

Bounds and approximations for sums of dependent log-elliptical random variables

Dhaene, Denuit, Goovaerts, Kaas and Vyncke [Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002a. The concept of comonotonicity in actuarial science and finance: theory. Insurance Math. Econom. 31 (1), 3–33; Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002b. The concept of comonotonicity in actuarial science and finance: Applications. Insurance Math. Econom. 31 (2), 133–161] have studied convex bounds for a sum of dependent random variables and applied these to sums of log-normal random variables. In particular, they have shown how these convex bounds can be used to derive closed-form approximations for several of the risk measures of such a sum. In this paper we investigate to which extent their general results on convex bounds can also be applied to sums of log-elliptical random variables which incorporate sums of log-normals as a special case. Firstly, we show that unlike the log-normal case, for general sums of log-ellipticals the convex lower bound does no longer result in closed-form approximations for the different risk measures. Secondly, we demonstrate how instead the weaker stop-loss order can be used to derive such closed-form approximations. We also present numerical examples to show the accuracy of the proposed approximations.

Extremal behavior of recurrent random sequences

A process Yn, n ≥ 1, satisfying the stochastic recurrent equation Yn = AnYn−1 + Bn, n ≥ 1, Y0 ≥ 0, is studied in the paper; here (An, Bn), n ≥ 1, are independent identically distributed pairs of nonnegative random variables. The cases when the values An have a lognormal and log-Laplace distributions are considered. The tail index κ (for a stationary distribution) and the extremal index ϑ are studied. In the lognormal case, κ is determined and some useful properties of ϑ are established. In the log-Laplace case the both characteristics are obtained in explicit form.

Pricing Geometric Asian Options under the CEV Process

This paper discusses the pricing of geometric Asian options when the underlying stock follows the constant elasticity of variance (CEV) process. We build a binomial tree method to estimate the CEV process and use it to price geometric Asian options. We find that the binomial tree method for the lognormal case can effectively solve the computational problems arising from the inherent complexities of geometric Asian options when the stock price follows the CEV process. We present numerical results to demonstrate the validity and the convergence of the approach for the different parameter values set in the CEV process.