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Results for “approximate Bayesian inference”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

A critical reassessment of particle Dark Matter limits from dwarf satellites

Dwarf satellite galaxies are ideal laboratories for identifying particle Dark Matter signals. When setting limits on particle Dark Matter properties from null searches, it becomes however crucial the level at which the Dark Matter density profile within these systems is constrained by observations. In the limit in which the spherical Jeans equation is assumed to be valid for a given tracer stellar population, we study the solution of this equation having the Dark Matter mass profile as an output rather than as a trial parametric input. Within our new formulation, we address to what level dwarf spheroidal galaxies feature a reliable mass estimator. We assess then possible extrapolation of the density profiles in the inner regions and—keeping explicit the dependence on the orbital anisotropy profile of the tracer population—we derive general trends on the line-of-sight integral of the density profile squared, a quantity commonly dubbed J -factor and crucial to estimate fluxes from prompt Dark Matter pair annihilations. Taking Ursa Minor as a study case among Milky Way satellites, we perform Bayesian inference using the available kinematical data for this galaxy. Contrary to all previous studies, we avoid marginalization over quantities poorly constrained by observations or by theoretical arguments. We find minimal J -factors to be about 2 to 4 times smaller than commonly quoted estimates, approximately relaxing by the same amount the limit on Dark Matter pair annihilation cross section from gamma-ray surveys of Ursa Minor. At the same time, if one goes back to a fixed trial parametric form for the density, e.g. using a NFW or Burkert profile, we show that the minimal J can hardly be reduced by more than a factor of 1.5.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2017

A Framework for Inferring Taxonomic Class of Asteroids.

Introduction: Taxonomic classification of asteroids based on their visible / near-infrared spectra or multi band photometry has proven to be a useful tool to infer other properties about asteroids. Meteorite analogs have been identified for several taxonomic classes, permitting detailed inference about asteroid composition. Trends have been identified between taxonomy and measured asteroid density. Thanks to NEOWise (Near-Earth-Object Wide-field Infrared Survey Explorer) and Spitzer (Spitzer Space Telescope), approximately twice as many asteroids have measured albedos than the number with taxonomic classifications. (If one only considers spectroscopically determined classifications, the ratio is greater than 40.) We present a Bayesian framework that provides probabilistic estimates of the taxonomic class of an asteroid based on its albedo. Although probabilistic estimates of taxonomic classes are not a replacement for spectroscopic or photometric determinations, they can be a useful tool for identifying objects for further study or for asteroid threat assessment models. Inputs and Framework: The framework relies upon two inputs: the expected fraction of each taxonomic class in the population and the albedo distribution of each class. Luckily, numerous authors have addressed both of these questions. For example, the taxonomic distribution by number, surface area and mass of the main belt has been estimated and a diameter limited estimate of fractional abundances of the near earth asteroid population was made. Similarly, the albedo distributions for taxonomic classes have been estimated for the combined main belt and NEA (Near Earth Asteroid) populations in different taxonomic systems and for the NEA population specifically. The framework utilizes a Bayesian inference appropriate for categorical data. The population fractions provide the prior while the albedo distributions allow calculation of the likelihood an albedo measurement is consistent with a given taxonomic class. These inputs allows calculation of the probability an asteroid with a specified albedo belongs to any given taxonomic class.

Asteroids↗
NASA NTRS2004

Filtering in Hybrid Dynamic Bayesian Networks

We demonstrate experimentally that inference in a complex hybrid Dynamic Bayesian Network (DBN) is possible using the 2 - T i e Slice DBN (2T-DBN) from [Koller & Lerner, 20001 to model fault detection in a watertank system. In [Koller & Lerner, 20001 a generic Particle Filter (PF) is used for inference. We extend the experiment and perform approximate inference using The Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF). Furthermore, we combine these techniques in a 'non-strict' Rao-Blackwellisation framework and apply it to the watertank system. We show that UKF and UKF in a PF framework outperfom the generic PF, EKF and EKF in a PF framework with respect to accuracy and robustness in terms of estimation RMSE. Especially we demonstrate the superiority of UKF in a PF framework when our beliefs of how data was generated are wrong. We also show that the choice of network structure is very important for the performance of the generic PF and the EKF algorithms, but not for the UKF algorithms. Furthermore, we investigate the influence of data noise in the water[ank simulation. Theory and implementation is based on the theory presented.

Andersen, Morten Nonboe↗
DOE OSTI2017

Quantification of uncertainty in photon source spot size inference during laser-driven radiography experiments at TRIDENT

Images of the R2DTO resolution target were obtained during laser-driven-radiography experiments performed at the TRIDENT laser facility, and analysis of these images using the Bayesian Inference Engine (BIE) determines a most probable full-width half maximum (FWHM) spot size of 78 μm. However, significant uncertainty prevails due to variation in the measured detector blur. Propagating this uncertainty in detector blur through the forward model results in an interval of probabilistic ambiguity spanning approximately 35-195 μm when the laser energy impinges on a thick (1 mm) tantalum target. In other phases of the experiment, laser energy is deposited on a thin (~100 nm) aluminum target placed 250 μm ahead of the tantalum converter. When the energetic electron beam is generated in this manner, upstream from the bremsstrahlung converter, the inferred spot size shifts to a range of much larger values, approximately 270-600 μm FWHM. This report discusses methods applied to obtain these intervals as well as concepts necessary for interpreting the result within a context of probabilistic quantitative inference.

47 OTHER INSTRUMENTATION↗
DOE OSTI2018

Soil moisture estimation using tomographic ground penetrating radar in a MCMC–Bayesian framework

In this paper, we focus on a hydrogeological inverse problem specifically targeting monitoring soil moisture variations using tomographic ground penetrating radar (GPR) travel time data. Technical challenges exist in the inversion of GPR tomographic data for handling non-uniqueness, nonlinearity and high-dimensionality of unknowns. We have developed a new method for estimating soil moisture fields from crosshole GPR data. It uses a pilot-point method to provide a low-dimensional representation of the relative dielectric permittivity field of the soil, which is the primary object of inference: the field can be converted to soil moisture using a petrophysical model. We integrate a multi-chain Markov chain Monte Carlo (MCMC)–Bayesian inversion framework with the pilot point concept, a curved-ray GPR travel time model, and a sequential Gaussian simulation algorithm, for estimating the dielectric permittivity at pilot point locations distributed within the tomogram, as well as the corresponding geostatistical parameters (i.e., spatial correlation range). We infer the dielectric permittivity as a probability density function, thus capturing the uncertainty in the inference. The multi-chain MCMC enables addressing high-dimensional inverse problems as required in the inversion setup. The method is scalable in terms of number of chains and processors, and is useful for computationally demanding Bayesian model calibration in scientific and engineering problems. The proposed inversion approach can successfully approximate the posterior density distributions of the pilot points, and capture the true values. The computational efficiency, accuracy, and convergence behaviors of the inversion approach were also systematically evaluated, by comparing the inversion results obtained with different levels of noises in the observations, increased observational data, as well as increased number of pilot points.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Cosmic Inference: Constraining Parameters with Observations and a Highly Limited Number of Simulations

We look at cosmological probes that pose an inverse problem where the measurement result is obtained through observations, and the objective is to infer values of model parameters that characterize the underlying physical system-our universe, from these observations and theoretical forward-modeling. The only way to accurately forward-model physical behavior on small scales is via expensive numerical simulations, which are further “emulated” due to their high cost. Emulators are commonly built with a set of simulations covering the parameter space with Latin hypercube sampling and an interpolation procedure; the aim is to establish an approximately constant prediction error across the hypercube. In this paper, we provide a description of a novel statistical framework for obtaining accurate parameter constraints. The proposed framework uses multi-output Gaussian process emulators that are adaptively constructed using Bayesian optimization methods with the goal of maintaining a low emulation error in the region of the hypercube preferred by the observational data. In this paper, we compare several approaches for constructing multi-output emulators that enable us to take possible inter-output correlations into account while maintaining the efficiency needed for inference. Using a Lyα forest flux power spectrum, we demonstrate that our adaptive approach requires considerably fewer-by a factor of a few in the Lyα P(k) case considered here-simulations compared to the emulation based on Latin hypercube sampling, and that the method is more robust in reconstructing parameters and their Bayesian credible intervals.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Applying Bayesian inference and deterministic anisotropy to retrieve the molecular structure ∣Ψ(R)∣2 distribution from gas-phase diffraction experiments

Abstract Currently, our general approach to retrieving molecular structures from ultrafast gas-phase diffraction heavily relies on complex ab initio electronic or vibrational excited state simulations to make conclusive interpretations. Without such simulations, inverting this measurement for the structural probability distribution is typically intractable. This creates a so-called inverse problem. Here we address this inverse problem by developing a broadly applicable method that approximates the molecular frame structure ∣Ψ( R , t )∣ 2 distribution independent of these complex simulations. We retrieve the vibronic ground state ∣Ψ( R )∣ 2 for both simulated stretched NO 2 and measured N 2 O. From measured N 2 O, we observe 40 mÅ coordinate-space resolution from 3.75 Å −1 reciprocal space range and poor signal-to-noise, a 50X improvement over traditional Fourier transform methods. In simulated NO 2 diffraction experiments, typical to high signal-to-noise levels predict 100–1000X resolution improvements, down to 0.1 mÅ. By directly measuring the width of ∣Ψ( R )∣ 2 , we open ultrafast gas-phase diffraction capabilities to measurements beyond current analysis approaches. This method has the potential to effectively turn gas-phase ultrafast diffraction into a discovery-oriented technique to probe systems that are prohibitively difficult to simulate.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2017

Multilevel Sequential Monte Carlo Samplers for Normalizing Constants

This article considers the sequential Monte Carlo (SMC) approximation of ratios of normalizing constants associated to posterior distributions which in principle rely on continuum models. Therefore, the Monte Carlo estimation error and the discrete approximation error must be balanced. A multilevel strategy is utilized to substantially reduce the cost to obtain a given error level in the approximation as compared to standard estimators. Two estimators are considered and relative variance bounds are given. The theoretical results are numerically illustrated for two Bayesian inverse problems arising from elliptic partial differential equations (PDEs). The examples involve the inversion of observations of the solution of (i) a 1-dimensional Poisson equation to infer the diffusion coefficient, and (ii) a 2-dimensional Poisson equation to infer the external forcing.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Peering down the barrel with DESI DR2: 10 000+ inflows at $z$ < 0.6 reveal how galaxies accrete cold gas

Direct observational constraints on how galaxies acquire their gas remain remarkably limited, hindering our understanding of the baryon cycle. We present a search for down-the-barrel NaI D absorption towards 15.6 million galaxies at $z < 0.6$ in DESI Data Release 2. We use Bayesian evidence ratios to assess whether the absorption requires additional components tracing interstellar gas distinct from the systemic component of the galaxy. We construct a catalogue of 50 088 (27 420) galaxies with moderate (strong) evidence for down-the-barrel absorption. The inferred absorption components are broadly distributed in velocity, with approximately 50% at $v_{\rm flow} < -50$ km/s, 30% within 50 km/s of the systemic velocity and the remaining 20% at $v_{\rm flow} > 50$ km/s. We find strong evidence for a large population of low-velocity, infalling absorbers with velocities $\sim$20 km/s in edge-on galaxies, consistent with radial inflows predicted in simulations. The stronger correlation in early-type galaxies between inflow velocity and stellar velocity dispersion, compared to that with stellar mass, suggests that a portion of these inflows may be associated with accreting satellites. These results reveal the multiple pathways in which galaxies accrete gas at redshift $z < 0.6$ for the first time in a statistically significant sample.

Weng, S. [Marseille, Lab. Astrophys.]↗
DOE OSTI2017

Stan : A Probabilistic Programming Language

Stan is a probabilistic programming language for specifying statistical models. A Stan program imperatively defines a log probability function over parameters conditioned on specified data and constants. As of version 2.14.0, Stan provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the No-U-Turn sampler, an adaptive form of Hamiltonian Monte Carlo sampling. Penalized maximum likelihood estimates are calculated using optimization methods such as the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm. Stan is also a platform for computing log densities and their gradients and Hessians, which can be used in alternative algorithms such as variational Bayes, expectation propagation, and marginal inference using approximate integration. To this end, Stan is set up so that the densities, gradients, and Hessians, along with intermediate quantities of the algorithm such as acceptance probabilities, are easily accessible. Stan can also be called from the command line using the cmdstan package, through R using the rstan package, and through Python using the pystan package. All three interfaces support sampling and optimization-based inference with diagnostics and posterior analysis. rstan and pystan also provide access to log probabilities, gradients, Hessians, parameter transforms, and specialized plotting.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Scalable Bayesian Physics-Informed Kolmogorov-Arnold Networks

Uncertainty quantification (UQ) plays a pivotal role in scientific machine learning, especially when surrogate models are used to approximate complex systems. Although multilayer perceptions (MLPs) are commonly employed as surrogates, they often suffer from overfitting due to their large number of parameters. Kolmogorov-Arnold networks (KANs) offer an alternative solution with fewer parameters. However, gradient-based inference methods, such as Hamiltonian Monte Carlo (HMC), may result in computational inefficiency when applied to KANs, especially for large-scale datasets, due to the high cost of back-propagation. To address these challenges, we propose a novel approach, combining the dropout Tikhonov ensemble Kalman inversion (DTEKI) with Chebyshev KANs. This gradient-free method effectively mitigates overfitting and enhances numerical stability. In addition, we incorporate the active subspace method to reduce the parameter-space dimensionality, allowing us to improve the accuracy of predictions and obtain more reliable uncertainty estimates. Extensive experiments demonstrate the efficacy of our approach in various test cases, including scenarios with large datasets and high noise levels. Our results show that the new method achieves comparable or better accuracy, much higher efficiency as well as stability compared to HMC, in addition to scalability. Moreover, by leveraging the low-dimensional parameter subspace, our method preserves prediction accuracy while substantially reducing further the computational cost.

97 MATHEMATICS AND COMPUTING↗
NASA NTRSDate not supplied

Probabilistic Calibration of Expensive Models using Efficiently Trained Surrogates

Calibration of computational models in the presence of uncertainty is often cast as a Bayesian inference problem and solved via sampling methods, e.g., Markov chain Monte Carlo. When the computational model is expensive, this task becomes intractable due to the large number of samples required to accurately estimate the posterior distribution of the calibration parameters. A popular solution to this problem is to use machine learning to develop a faster-to-evaluate, lower-fidelity substitute for the original model to serve as a surrogate while solving the inference problem. Although considered an offline cost, generating training data to construct this surrogate model can still be an expensive task in practice. An active learning algorithm is presented that focuses training on improving surrogate accuracy specifically in and around the bulk of the posterior distribution, as this is where the model is exercised during calibration. Candidate samples are drawn from families of distributions related to an approximation of the posterior. The sample maximizing predictive variance is then selected for evaluation by the original computational model, yielding a label for the training point. Iterating this approach increases efficiency relative to space filling designs (e.g., Latin hypercube sampling) by avoiding low probability points. Practical considerations are discussed, including the benefits of using a sequential Monte Carlo sampling approach, convergence heuristics, and the importance of both exploration and exploitation given that the true posterior is unknown a priori.

uncertainty quantification↗
NASA NTRS2010

Understanding the Scalability of Bayesian Network Inference Using Clique Tree Growth Curves

One of the main approaches to performing computation in Bayesian networks (BNs) is clique tree clustering and propagation. The clique tree approach consists of propagation in a clique tree compiled from a Bayesian network, and while it was introduced in the 1980s, there is still a lack of understanding of how clique tree computation time depends on variations in BN size and structure. In this article, we improve this understanding by developing an approach to characterizing clique tree growth as a function of parameters that can be computed in polynomial time from BNs, specifically: (i) the ratio of the number of a BN s non-root nodes to the number of root nodes, and (ii) the expected number of moral edges in their moral graphs. Analytically, we partition the set of cliques in a clique tree into different sets, and introduce a growth curve for the total size of each set. For the special case of bipartite BNs, there are two sets and two growth curves, a mixed clique growth curve and a root clique growth curve. In experiments, where random bipartite BNs generated using the BPART algorithm are studied, we systematically increase the out-degree of the root nodes in bipartite Bayesian networks, by increasing the number of leaf nodes. Surprisingly, root clique growth is well-approximated by Gompertz growth curves, an S-shaped family of curves that has previously been used to describe growth processes in biology, medicine, and neuroscience. We believe that this research improves the understanding of the scaling behavior of clique tree clustering for a certain class of Bayesian networks; presents an aid for trade-off studies of clique tree clustering using growth curves; and ultimately provides a foundation for benchmarking and developing improved BN inference and machine learning algorithms.

Mengshoel, Ole J.↗
DOE OSTI2021

Bayesian calibration of strength model parameters from Taylor impact data

Materials strength plays a key role in determining the mechanical response of engineered structures. As such, accurate strength models are crucial in simulations involving complex loading conditions, particularly when deformation in the plastic regime is deemed important. In this work, a Gaussian process based surrogate for a finite element simulation of a Taylor impact test is developed and used for Bayesian calibration of the Preston–Tonks–Wallace strength model. Here, the surrogate model is shown to closely approximate the salient features of the Taylor cylinder deformation and is validated against simulation output before being used in the strength model calibration routine. The results show that Taylor impact test data can be used in the calibration of constitutive equations through the use of a combination of data science techniques, namely Gaussian processes and Bayesian inference.

36 MATERIALS SCIENCE↗
DOE OSTI2014

Breaking Computational Barriers: Real-time Analysis and Optimization with Large-scale Nonlinear Models via Model Reduction

Model reduction for dynamical systems is a promising approach for reducing the computational cost of large-scale physics-based simulations to enable high-fidelity models to be used in many- query (e.g., Bayesian inference) and near-real-time (e.g., fast-turnaround simulation) contexts. While model reduction works well for specialized problems such as linear time-invariant systems, it is much more difficult to obtain accurate, stable, and efficient reduced-order models (ROMs) for systems with general nonlinearities. This report describes several advances that enable nonlinear reduced-order models (ROMs) to be deployed in a variety of time-critical settings. First, we present an error bound for the Gauss-Newton with Approximated Tensors (GNAT) nonlinear model reduction technique. This bound allows the state-space error for the GNAT method to be quantified when applied with the backward Euler time-integration scheme. Second, we present a methodology for preserving classical Lagrangian structure in nonlinear model reduction. This technique guarantees that important properties--such as energy conservation and symplectic time-evolution maps--are preserved when performing model reduction for models described by a Lagrangian formalism (e.g., molecular dynamics, structural dynamics). Third, we present a novel technique for decreasing the temporal complexity --defined as the number of Newton-like iterations performed over the course of the simulation--by exploiting time-domain data. Fourth, we describe a novel method for refining projection-based reduced-order models a posteriori using a goal-oriented framework similar to mesh-adaptive h -refinement in finite elements. The technique allows the ROM to generate arbitrarily accurate solutions, thereby providing the ROM with a 'failsafe' mechanism in the event of insufficient training data. Finally, we present the reduced-order model error surrogate (ROMES) method for statistically quantifying reduced- order-model errors. This enables ROMs to be rigorously incorporated in uncertainty-quantification settings, as the error model can be treated as a source of epistemic uncertainty. This work was completed as part of a Truman Fellowship appointment. We note that much additional work was performed as part of the Fellowship. One salient project is the development of the Trilinos-based model-reduction software module Razor , which is currently bundled with the Albany PDE code and currently allows nonlinear reduced-order models to be constructed for any application supported in Albany. Other important projects include the following: 1. ROMES-equipped ROMs for Bayesian inference: K. Carlberg, M. Drohmann, F. Lu (Lawrence Berkeley National Laboratory), M. Morzfeld (Lawrence Berkeley National Laboratory). 2. ROM-enabled Krylov-subspace recycling: K. Carlberg, V. Forstall (University of Maryland), P. Tsuji, R. Tuminaro. 3. A pseudo balanced POD method using only dual snapshots: K. Carlberg, M. Sarovar. 4. An analysis of discrete v. continuous optimality in nonlinear model reduction: K. Carlberg, M. Barone, H. Antil (George Mason University). Journal articles for these projects are in progress at the time of this writing.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

A flexible event reconstruction based on machine learning and likelihood principles

Event reconstruction is a central step in many particle physics experiments, turning detector observables into parameter estimates; for example estimating the energy of an interaction given the sensor readout of a detector. A corresponding likelihood function is often intractable, and approximations need to be constructed. Here, in our work, we first show how the full likelihood for a many-sensor detector can be broken apart into smaller terms, and secondly how we can train neural networks to approximate all terms solely based on forward simulation. Our technique results in a fast, flexible, and close-to-optimal surrogate model proportional to the likelihood and can be used in conjunction with standard inference techniques allowing for a consistent treatment of uncertainties. We illustrate our technique for parameter inference in neutrino telescopes based on maximum likelihood and Bayesian posterior sampling. Given its great flexibility, we also showcase our method for geometry optimization enabling to learn optimal detector designs. Lastly, we apply our method to realistic simulation of a ton-scale water-based liquid scintillator detector.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2026

The Sensitivity of Variational Bayesian Neural Network Performance to Hyperparameters

In scientific applications, predictive modeling is often of limited use without accurate uncertainty quantification (UQ) to indicate when a model may be extrapolating or when more data needs to be collected. Bayesian Neural Networks (BNNs) produce predictive uncertainty by propagating uncertainty in neural network (NN) weights and offer the promise of obtaining not only an accurate predictive model but also accurate UQ. However, in practice, obtaining accurate UQ with BNNs is difficult due in part to the approximations used for model training (such as those made in variational inference) and in part to the need to choose a suitable set of hyperparameters; these hyperparameters outnumber those needed for traditional NNs and often have opaque effects on the results. We aim to shed light on the effects of hyperparameter choices for variational BNNs by performing a global sensitivity analysis of variational BNN performance under varying hyperparameter settings. Our results indicate that many of the hyperparameters interact with each other to affect both predictive accuracy and UQ. For improved usage of variational BNNs in real-world applications, we suggest that thorough hyperparameter tuning, including tuning of prior hyperparameters and loss function parameters, is essential for accurate UQ in variational BNNs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Multi-fidelity Bayesian neural networks: Algorithms and applications

Here we propose a new class of Bayesian neural networks (BNNs) that can be trained using noisy data of variable fidelity, and we apply them to learn function approximations as well as to solve inverse problems based on partial differential equations (PDEs). These multi-fidelity BNNs consist of three neural networks: The first is a fully connected neural network, which is trained following the maximum a posteriori probability (MAP) method to fit the low-fidelity data; the second is a Bayesian neural network employed to capture the cross-correlation with uncertainty quantification between the low- and high-fidelity data; and the last one is the physics-informed neural network, which encodes the physical laws described by PDEs. For the training of the last two neural networks, we first employ the mean-field variational inference (VI) to maximize the evidence lower bound (ELBO) to obtain informative prior distributions for the hyperparameters in the BNNs, and subsequently we use the Hamiltonian Monte Carlo (HMC) method to estimate accurately the posterior distributions for the corresponding hyperparameters. We demonstrate the accuracy of the present method using synthetic data as well as real measurements. Specifically, we first approximate a one- and four-dimensional function, and then infer the reaction rates in one- and two-dimensional diffusion-reaction systems. Moreover, we infer the sea surface temperature (SST) in the Massachusetts and Cape Cod Bays using satellite images and in-situ measurements. Taken together, our results demonstrate that the present method can capture both linear and nonlinear correlation between the low- and high-fidelity data adaptively, identify unknown parameters in PDEs, and quantify uncertainties in predictions, given a few scattered noisy high-fidelity data. Finally, we demonstrate that we can effectively and efficiently reduce the uncertainties and hence enhance the prediction accuracy with an active learning approach, using as examples a specific one-dimensional function approximation and an inverse PDE problem.

97 MATHEMATICS AND COMPUTING↗