Numerical Biases Research Articles

Solving deep-learning density functional theory via variational autoencoders

In recent years, machine learning models, chiefly deep neural networks, have revealed suited to learn accurate energy-density functionals from data. However, problematic instabilities have been shown to occur in the search of ground-state density profiles via energy minimization. Indeed, any small noise can lead astray from realistic profiles, causing the failure of the learned functional and, hence, strong violations of the variational property. In this article, we employ variational autoencoders (VAEs) to build a compressed, flexible, and regular representation of the ground-state density profiles of various quantum models. Performing energy minimization in this compressed space allows us to avoid both numerical instabilities and variational biases due to excessive constraints. Our tests are performed on one-dimensional single-particle models from the literature in the field and, notably, on a three-dimensional disordered potential. In all cases, the ground-state energies are estimated with errors below the chemical accuracy and the density profiles are accurately reproduced without numerical artifacts. Furthermore, we show that it is possible to perform transfer learning, applying pre-trained VAEs to different potentials.

Predicting Model Biases of the APOLLO3 Neutronics Code Using Machine Learning: Toward a Multiscale and Multifidelity Approach

As part of the verification, validation, and uncertainty quantification process applied to neutronics deterministic codes, there is a requirement to expand the validation domain, especially to accommodate new third-generation reactors. The objective of the present work is to estimate the numerical biases arising from the several approximations used in deterministic codes and across different points in the phase space. Typically, this is accomplished by comparing the deterministic code to be validated with a Monte Carlo or stochastic reference code (without significant approximations). Since these reference calculations are computationally expensive, this paper proposes an alternative approach for predicting model biases of the APOLLO3® deterministic code for third-generation pressurized water reactors using machine learning algorithms. Three types of metamodels are employed (polynomial regression, kriging, and neural networks). Two scales are investigated, from a single assembly to a cluster of 3 × 3 assemblies [small two-dimensional (2-D) core], with model biases evaluated for APOLLO3 schemes with various levels of accuracy (lattice and core solvers, with high- to low-fidelity approaches). For the small 2-D core, numerical biases are observed for reactivity and power peak, representing both global and local quantities of interest. Throughout the study, the best results are achieved using kriging or neural networks, even if polynomial regression provides satisfactory predictions in some cases. The possibility of predicting biases for different quantities is also introduced. In conclusion, this paper discusses the prospects of extending the applicability of these metamodels to small and large third-generation pressurized water reactor cores, the idea being to potentially use these metamodels to support safety demonstrations for the new reactors in the long term.

Uncertainty quantification analysis of Reynolds-averaged Navier–Stokes simulation of spray swirling jets undergoing vortex breakdown

The computational fluid dynamics-based design of next-generation aeronautical combustion chambers is challenging due to many geometrical and operational parameters to be optimized and several sources of uncertainty that arise from numerical modeling. The present work highlights the potential benefits of exploiting Bayesian uncertainty quantification at the preliminary design stage. A prototypical configuration of an acetone/air spray swirling jet is investigated through an Eulerian–Lagrangian method under non-reactive conditions. Two direct numerical simulations (DNSs) provide reference data, coping with different vortex breakdown states. Consequently, a set of Reynolds-averaged Navier–Stokes simulations is conducted. Polynomial chaos expansion (PCE) is adopted to propagate the uncertainty associated with the spray dispersion model and the turbulent Schmidt number, delivering confidence intervals and the sensitivity of the output variance to each uncertain input. Consequently, the most significant sources of modeling uncertainty may be identified and eventually removed via a calibration procedure, thus making it possible to carry out a combustion chamber optimization process that is no longer affected by numerical biases. The uncertainty quantification analysis in the current study demonstrates that the spray dispersion model slightly affects the fuel vapor spatial distribution under vortex breakdown flow conditions, compared with the output variance induced by the selection of the turbulent Schmidt number. As a result, additional high-fidelity experimental and numerical campaigns should exclusively address the development of an ad hoc model characterizing the spatial distribution of the latter in the presence of vortex breakdown phenomenology, discarding any effort to improve the spray dispersion formulation.

Numerical assessment of morphological and hydraulic properties of moss, lichen and peat from a permafrost peatland

Abstract. Due to its insulating and draining role, assessing ground vegetation cover properties is important for high-resolution hydrological modeling of permafrost regions. In this study, morphological and effective hydraulic properties of Western Siberian Lowland ground vegetation samples (lichens, Sphagnum mosses, peat) are numerically studied based on tomography scans. Porosity is estimated through a void voxels counting algorithm, showing the existence of representative elementary volumes (REVs) of porosity for most samples. Then, two methods are used to estimate hydraulic conductivity depending on the sample's homogeneity. For homogeneous samples, direct numerical simulations of a single-phase flow are performed, leading to a definition of hydraulic conductivity related to a REV, which is larger than those obtained for porosity. For heterogeneous samples, no adequate REV may be defined. To bypass this issue, a pore network representation is created from computerized scans. Morphological and hydraulic properties are then estimated through this simplified representation. Both methods converged on similar results for porosity. Some discrepancies are observed for a specific surface area. Hydraulic conductivity fluctuates by 2 orders of magnitude, depending on the method used. Porosity values are in line with previous values found in the literature, showing that arctic cryptogamic cover can be considered an open and well-connected porous medium (over 99 % of overall porosity is open porosity). Meanwhile, digitally estimated hydraulic conductivity is higher compared to previously obtained results based on field and laboratory experiments. However, the uncertainty is less than in experimental studies available in the literature. Therefore, biological and sampling artifacts are predominant over numerical biases. This could be related to compressibility effects occurring during field or laboratory measurements. These numerical methods lay a solid foundation for interpreting the homogeneity of any type of sample and processing some quantitative properties' assessment, either with image processing or with a pore network model. The main observed limitation is the input data quality (e.g., the tomographic scans' resolution) and its pre-processing scheme. Thus, some supplementary studies are compulsory for assessing syn-sampling and syn-measurement perturbations in experimentally estimated, effective hydraulic properties of such a biological porous medium.

The influence of visual illusion perception on numerosity estimation could be evolutionarily conserved: exploring the numerical Delboeuf illusion in humans (Homo sapiens) and fish (Poecilia reticulata).

Discriminating between different quantities is an essential ability in daily life that has been demonstrated in a variety of non-human vertebrates. Nonetheless, what drives the estimation of numerosity is not fully understood, as numerosity intrinsically covaries with several other physical characteristics. There is wide debate as to whether the numerical and spatial abilities of vertebrates are processed by a single magnitude system or two different cognitive systems. Adopting a novel approach, we aimed to investigate this issue by assessing the interaction between area size and numerosity, which has never been conceptualized with consideration for subjective experience in non-human animals. We examined whether the same perceptual biases underlying one of the best-known size illusions, the Delboeuf illusion, can be also identified in numerical estimation tasks. We instructed or trained human participants and guppies, small teleost fish, to select a target numerosity (larger or smaller) of squares between two sets that actually differed in their numerosity. Subjects were also presented with illusory trials in which the same numerosity was presented in two different contexts, against a large and a small background, resembling the Delboeuf illusion. In these trials, both humans and fish demonstrated numerical biases in agreement with the perception of the classical version of the Delboeuf illusion, with the array perceived as larger appearing more numerous. Thus, our results support the hypothesis of a single magnitude system, as perceptual biases that influence spatial decisions seem to affect numerosity judgements in the same way.

Connecting the Dots: Numerical Randomized Hamiltonian Monte Carlo with State-Dependent Event Rates

Numerical generalized randomized Hamiltonian Monte Carlo is introduced, as a robust, easy to use and computationally fast alternative to conventional Markov chain Monte Carlo methods for continuous target distributions. A wide class of piecewise deterministic Markov processes generalizing Randomized HMC (Bou-Rabee and Sanz-Serna) by allowing for state-dependent event rates is defined. Under very mild restrictions, such processes will have the desired target distribution as an invariant distribution. Second, the numerical implementation of such processes, based on adaptive numerical integration of second order ordinary differential equations (ODEs) is considered. The numerical implementation yields an approximate, yet highly robust algorithm that, unlike conventional Hamiltonian Monte Carlo, enables the exploitation of the complete Hamiltonian trajectories (hence, the title). The proposed algorithm may yield large speedups and improvements in stability relative to relevant benchmarks, while incurring numerical biases that are negligible relative to the overall Monte Carlo errors. Granted access to a high-quality ODE code, the proposed methodology is both easy to implement and use, even for highly challenging and high-dimensional target distributions. Supplementary materials for this article are available online.

Establishing a non-hydrostatic global atmospheric modeling system at 3-km horizontal resolution with aerosol feedbacks on the Sunway supercomputer of China

During the era of global warming and highly urbanized development, extreme and high impact weather as well as air pollution incidents influence everyday life and might even cause the incalculable loss of life and property. Despite the vast development of atmospheric models, there still exist substantial numerical forecast biases objectively. To accurately predict extreme weather, severe air pollution, and abrupt climate change, numerical atmospheric model requires not only to simulate meteorology and atmospheric compositions simultaneously involving many sophisticated physical and chemical processes but also at high spatiotemporal resolution. Global integrated atmospheric simulation at spatial resolutions of a few kilometers remains challenging due to its intensive computational and input/output (I/O) requirement. Through multi-dimension-parallelism structuring, aggressive and finer-grained optimizing, manual vectorizing, and parallelized I/O fragmenting, an integrated Atmospheric Model Across Scales (iAMAS) was established on the new Sunway supercomputer platform to significantly increase the computational efficiency and reduce the I/O cost. The global 3-km atmospheric simulation for meteorology with online integrated aerosol feedbacks with iAMAS was scaled to 39,000,000 processor cores and achieved the speed of 0.82 simulation day per hour (SDPH) with routine I/O, which enabled us to perform 5-day global weather forecast at 3-km horizontal resolution with online natural aerosol impacts. The results demonstrate the promising future that the increasing of spatial resolution to a few kilometers with online integrated aerosol feedbacks may significantly improve the global weather forecast.

Determining soil particle-size distribution from infrared spectra using machine learning predictions: Methodology and modeling.

Accuracy of infrared (IR) models to measure soil particle-size distribution (PSD) depends on soil preparation, methodology (sedimentation, laser), settling times and relevant soil features. Compositional soil data may require log ratio (ilr) transformation to avoid numerical biases. Machine learning can relate numerous independent variables that may impact on NIR spectra to assess particle-size distribution. Our objective was to reach high IRS prediction accuracy across a large range of PSD methods and soil properties. A total of 1298 soil samples from eastern Canada were IR-scanned. Spectra were processed by Stochastic Gradient Boosting (SGB) to predict sand, silt, clay and carbon. Slope and intercept of the log-log relationships between settling time and suspension density function (SDF) (R2 = 0.84–0.92) performed similarly to NIR spectra using either ilr-transformed (R2 = 0.81–0.93) or raw percentages (R2 = 0.76–0.94). Settling times of 0.67-min and 2-h were the most accurate for NIR predictions (R2 = 0.49–0.79). The NIR prediction of sand sieving method (R2 = 0.66) was more accurate than sedimentation method(R2 = 0.53). The NIR 2X gain was less accurate (R2 = 0.69–0.92) than 4X (R2 = 0.87–0.95). The MIR (R2 = 0.45–0.80) performed better than NIR (R2 = 0.40–0.71) spectra. Adding soil carbon, reconstituted bulk density, pH, red-green-blue color, oxalate and Mehlich3 extracts returned R2 value of 0.86–0.91 for texture prediction. In addition to slope and intercept of the SDF, 4X gain, method and pre-treatment classes, soil carbon and color appeared to be promising features for routine SGB-processed NIR particle-size analysis. Machine learning methods support cost-effective soil texture NIR analysis.

Neural Network Method for Solving Time-Fractional Telegraph Equation

Recently, the development of neural network method for solving differential equations has made a remarkable progress for solving fractional differential equations. In this paper, a neural network method is employed to solve time-fractional telegraph equation. The loss function containing initial/boundary conditions with adjustable parameters (weights and biases) is constructed. Also, in this paper, a time-fractional telegraph equation was formulated as an optimization problem. Numerical examples with known analytic solutions including numerical results, their graphs, weights, and biases were also discussed to confirm the accuracy of the method used. Also, the graphical and tabular results were analyzed thoroughly. The mean square errors for different choices of neurons and epochs have been presented in tables along with graphical presentations.

The influence of locative expressions on context-dependency of endpoint control in aiming.

It has been claimed that increased reliance on context, or allocentric information, develops when aiming movements are more consciously monitored and/or controlled. Since verbalizing target features requires strong conscious monitoring, we expected an increased reliance on allocentric information when verbalizing a target label (i.e. target number) during movement execution. We examined swiping actions towards a global array of targets embedded in different local array configurations on a tablet under no-verbalization and verbalization conditions. The global and local array configurations allowed separation of contextual-effects from any possible numerical magnitude biases triggered from calling out specific target numbers.The patterns of constant errors in the target directionwere used to assess differences between conditions. Variation in the target context configuration systematically biased movement endpoints in both the no-verbalization and verbalization conditions. Ultimately, our results do not support the assertion that calling out target numbers during movement execution increases the context-dependency of targeted actions.

How biased are halo properties in cosmological simulations?

ABSTRACT Cosmological N-body simulations have been a major tool of theorists for decades, yet many of the numerical issues that these simulations face are still unexplored. This paper measures numerical biases in these large, dark matter-only simulations that affect the properties of their dark matter haloes. We compare many simulation suites in order to provide several tools for simulators and analysts which help mitigate these biases. We summarize our comparisons with practical ‘convergence limits’ that can be applied to a wide range of halo properties, including halo properties which are traditionally overlooked by the testing literature. We also find that the halo properties predicted by different simulations can diverge from one another at unexpectedly high resolutions. We demonstrate that many halo properties depend strongly on force softening scale and that this dependence leads to much of the measured divergence between simulations. We offer an empirical model to estimate the impact of such effects on the rotation curves of a halo population. This model can serve as a template for future empirical models of the biases in other halo properties.

Speed and Biases of Fourier-Based Pricing Choices: Analysis of the Bastes and Asymmetric Variance Gamma Models

This paper compares the CPU effort and numerical biases of six Fourier-based implementations. Our analyses focus on two jump-related models that can consistently price options with different strikes and maturities: (i) the Bates jump-diffusion model, which combines jumps with stochastic volatility and (ii) the Asymmetric Variance Gamma (AVG) model, a pure-jump process where an infinite number of jumps can occur in any interval of time. We show that both truncation and discretization errors significantly increase as we move away from the diffusive Black-Scholes-Merton dynamics. While most pricing choices converge to the Bates reference values, Attari’s formula is the only Fourier-based method that does not completely blow up in any AVG problematic region. In terms of CPU speed, the strike vector computations proposed by Zhu (2010) significantly improve the computational burden, rendering the use of fast Fourier transforms and plain delta-probability decompositions inefficient.

Cognitive Load Affects Numerical and Temporal Judgments in Distinct Ways.

Prominent theories suggest that time and number are processed by a single neural locus or a common magnitude system (e.g., Meck and Church, 1983; Walsh, 2003). However, a growing body of literature has identified numerous inconsistencies between temporal and numerical processing, casting doubt on the presence of such a singular system. Findings of distinct temporal and numerical biases in the presence of emotional content (Baker et al., 2013; Young and Cordes, 2013) are particularly relevant to this debate. Specifically, emotional stimuli lead to temporal overestimation, yet identical stimuli result in numerical underestimation. In the current study, we tested adults’ temporal and numerical processing under cognitive load, a task that compromises attention. Under the premise of a common magnitude system, one would predict cognitive load to have an identical impact on temporal and numerical judgments. Inconsistent with the common magnitude account, results revealed baseline performance on the temporal and numerical task was not correlated and importantly, cognitive load resulted in distinct and opposing quantity biases: numerical underestimation and marginal temporal overestimation. Together, our data call into question the common magnitude account, while also providing support for the role of attentional processes involved in numerical underestimation.

Direct radiative effects during intense Mediterranean desert dust outbreaks

Abstract. The direct radiative effect (DRE) during 20 intense and widespread dust outbreaks, which affected the broader Mediterranean basin over the period March 2000–February 2013, has been calculated with the NMMB-MONARCH model at regional (Sahara and European continent) and short-term temporal (84 h) scales. According to model simulations, the maximum dust aerosol optical depths (AODs) range from ∼ 2.5 to ∼ 5.5 among the identified cases. At midday, dust outbreaks locally induce a NET (shortwave plus longwave) strong atmospheric warming (DREATM values up to 285 W m−2; Niger–Chad; dust AODs up to ∼ 5.5) and a strong surface cooling (DRENETSURF values down to −337 W m−2), whereas they strongly reduce the downward radiation at the ground level (DRESURF values down to −589 W m−2 over the Eastern Mediterranean, for extremely high dust AODs, 4.5–5). During night-time, reverse effects of smaller magnitude are found. At the top of the atmosphere (TOA), positive (planetary warming) DREs up to 85 W m−2 are found over highly reflective surfaces (Niger–Chad; dust AODs up to ∼ 5.5) while negative (planetary cooling) DREs down to −184 W m−2 (Eastern Mediterranean; dust AODs 4.5–5) are computed over dark surfaces at noon. Dust outbreaks significantly affect the mean regional radiation budget, with NET DREs ranging from −8.5 to 0.5 W m−2, from −31.6 to 2.1 W m−2, from −22.2 to 2.2 W m−2 and from −1.7 to 20.4 W m−2 for TOA, SURF, NETSURF and ATM, respectively. Although the shortwave DREs are larger than the longwave ones, the latter are comparable or even larger at TOA, particularly over the Sahara at midday. As a response to the strong surface day-time cooling, dust outbreaks cause a reduction in the regional sensible and latent heat fluxes by up to 45 and 4 W m−2, respectively, averaged over land areas of the simulation domain. Dust outbreaks reduce the temperature at 2 m by up to 4 K during day-time, whereas a reverse tendency of similar magnitude is found during night-time. Depending on the vertical distribution of dust loads and time, mineral particles heat (cool) the atmosphere by up to 0.9 K (0.8 K) during day-time (night-time) within atmospheric dust layers. Beneath and above the dust clouds, mineral particles cool (warm) the atmosphere by up to 1.3 K (1.2 K) at noon (night-time). On a regional mean basis, negative feedbacks on the total emitted dust (reduced by 19.5 %) and dust AOD (reduced by 6.9 %) are found when dust interacts with the radiation. Through the consideration of dust radiative effects in numerical simulations, the model positive and negative biases for the downward surface SW or LW radiation, respectively, with respect to Baseline Surface Radiation Network (BSRN) measurements, are reduced. In addition, they also reduce the model near-surface (at 2 m) nocturnal cold biases by up to 0.5 K (regional averages), as well as the model warm biases at 950 and 700 hPa, where the dust concentration is maximized, by up to 0.4 K. However, improvements are relatively small and do not happen in all episodes because other model first-order errors may dominate over the expected improvements, and the misrepresentation of the dust plumes' spatiotemporal features and optical properties may even produce a double penalty effect. The enhancement of dust forecasts via data assimilation techniques may significantly improve the results.

Evaluation of Neutron-induced Cross Sections and their Related Covariances with Physical Constraints

Nuclear data, along with numerical methods and the associated calculation schemes, continue to play a key role in reactor design, reactor core operating parameters calculations, fuel cycle management and criticality safety calculations. Due to the intensive use of Monte-Carlo calculations reducing numerical biases, the final accuracy of neutronic calculations increasingly depends on the quality of nuclear data used. This paper gives a broad picture of all ingredients treated by nuclear data evaluators during their analyses. After giving an introduction to nuclear data evaluation, we present implications of using the Bayesian inference to obtain evaluated cross sections and related uncertainties. In particular, a focus is made on systematic uncertainties appearing in the analysis of differential measurements as well as advantages and drawbacks one may encounter by analyzing integral experiments.The evaluation work is in general done independently in the resonance and in the continuum energy ranges giving rise to inconsistencies in evaluated files. For future evaluations on the whole energy range, we call attention to two innovative methods used to analyze several nuclear reaction models and impose constraints. Finally, we discuss suggestions for possible improvements in the evaluation process to master the quantification of uncertainties. These are associated with experiments (microscopic and integral), nuclear reaction theories and the Bayesian inference.

The impact of emotion on numerical estimation: A developmental perspective.

Recent literature has revealed underestimation effects in numerical judgments when adult participants are presented with emotional stimuli (as opposed to neutral). Whether these numerical biases emerge early in development however, or instead reflect overt, learned responses to emotional stimuli across development are unclear. Moreover, reported links between numerical acuity and mathematics achievement point to the importance of exploring how numerical approximation abilities in childhood may be influenced in real-world affective contexts. In this study, children (aged 6-10 years) and adults were presented with happy and neutral facial stimuli in the context of a numerical bisection task. Results reveal that children, like adults, underestimate number following emotional (i.e., happy) faces (relative to neutral). However, children's, but not adult's, responses were also significantly more precise following emotional stimuli. In a second experiment, adult judgments revealed a similar increase in precision following emotional stimuli when numerical discriminations were more challenging (involving larger sets). Together, results are the first to reveal children, like adults, underestimate number in the context of emotional stimuli and this underestimation bias is accompanied with enhanced response precision.

Cognitive Mechanisms Underlying Directional and Non-directional Spatial-Numerical Associations across the Lifespan.

There is accumulating evidence suggesting an association of numbers with physical space. However, the origin of such spatial-numerical associations (SNAs) is still debated. In the present study we investigated the development of two SNAs in a cross-sectional study involving children, young and middle-aged adults as well as the elderly: (1) the SNARC (spatial-numerical association of response codes) effect, reflecting a directional SNA; and (2) the numerical bisection bias in a line bisection task with numerical flankers. Results revealed a consistent SNARC effect in all age groups that continuously increased with age. In contrast, a numerical bisection bias was only observed for children and elderly participants, implying an U-shaped distribution of this bias across age groups. Additionally, individual SNARC effects and numerical bisection biases did not correlate significantly. We argue that the SNARC effect seems to be influenced by longer-lasting experiences of cultural constraints such as reading and writing direction and may thus reflect embodied representations. Contrarily, the numerical bisection bias may originate from insufficient inhibition of the semantic influence of irrelevant numerical flankers, which should be more pronounced in children and elderly people due to development and decline of cognitive control, respectively. As there is an ongoing debate on the origins of SNAs in general and the SNARC effect in particular, the present results are discussed in light of these differing accounts in an integrative approach. However, taken together, the present pattern of results suggests that different cognitive mechanisms underlie the SNARC effect and the numerical bisection bias.

MODIS VCF should not be used to detect discontinuities in tree cover due to binning bias. A comment on Hanan et al. (2014) and Staver and Hansen (2015)

AbstractIn their recent paper, Staver and Hansen (Global Ecology and Biogeography, 2015, 24, 985–987) refute the case made by Hanan et al. (Global Ecology and Biogeography, 2014, 23, 259–263) that the use of classification and regression trees (CARTs) to predict tree cover from remotely sensed imagery (MODIS VCF) inherently introduces biases, thus making the resulting tree cover unsuitable for showing alternative stable states through tree cover frequency distribution analyses. Here we provide a new and equally fundamental argument for why the published frequency distributions should not be used for such purposes. We show that the practice of pre‐average binning of tree cover values used to derive cover values to train the CART model will also introduce errors in the frequency distributions of the final product. We demonstrate that the frequency minima found at tree covers of 8–18%, 33–45% and 55–75% can be attributed to numerical biases introduced when training samples are derived from landscapes containing asymmetric tree cover distributions and/or a tree cover gradient. So it is highly likely that the CART, used to produce MODIS VCF, delivers tree cover frequency distributions that do not reflect the real world situation.

Universal and uniquely human factors in spontaneous number perception

A capacity for nonverbal numerical estimation is widespread among humans and animals. However, it is currently unclear whether numerical percepts are spontaneously extracted from the environment and whether nonverbal perception is influenced by human exposure to formal mathematics. We tested US adults and children, non-human primates, and numerate and innumerate Tsimane’ adults on a quantity task in which they could choose to categorize sets of dots on the basis of number alone, surface area alone or a combination of the two. Despite differences in age, species and education, subjects are universally biased to base their judgments on number as opposed to the alternatives. Numerical biases are uniquely enhanced in humans compared to non-human primates, and correlated with degree of mathematics experience in both the US and Tsimane’ groups. We conclude that humans universally and spontaneously extract numerical information, and that human nonverbal numerical perception is enhanced by symbolic numeracy.

Speed and Biases of Fourier-Based Pricing Choices: A Numerical Analysis

This paper compares the CPU effort and numerical biases of six Fourier-based implementations. Our analyses focus on two jump-related models that can consistently price options with different strikes and maturities: (i) the Bates jump-diffusion model, which combines jumps with stochastic volatility and (ii) the Asymmetric Variance Gamma (AVG) model, a pure-jump process where an infinite number of jumps can occur in any interval of time. We show that both truncation and discretization errors significantly increase as we move away from the diffusive Black-Scholes-Merton dynamics. While most pricing choices converge to the Bates reference values, Attari’s formula is the only Fourier-based method that does not completely blow up in any AVG problematic region. In terms of CPU speed, the strike vector computations proposed by Zhu (2010) significantly improve the computational burden, rendering the use of fast Fourier transforms and plain delta-probability decompositions inefficient.