Maximum Entropy Reconstruction Research Articles

Precision MARS Mass Reconstruction of A2744: Synergizing the Largest Strong-lensing and Densest Weak-lensing Data Sets from JWST

We present a new high-resolution free-form mass model of A2744 that combines both weak-lensing (WL) and strong-lensing (SL) data sets from JWST. The SL data set comprises 286 multiple images, presenting the most extensive SL constraint to date for a single cluster. The WL data set, employing photo-z selection, yields a source density of ∼350arcmin−2 , marking the densest WL constraint ever. The combined mass reconstruction enables the highest-resolution mass map of A2744 within the ∼1.8 Mpc × 1.8 Mpc reconstruction region to date, revealing an isosceles triangular structure with two legs of ∼1 Mpc and a base of ∼0.6 Mpc. Although our algorithm, which is called MAximum-entropy ReconStruction (MARS), is entirely blind to the cluster galaxy distribution, the resulting mass reconstruction traces the brightest cluster galaxies remarkably well. The five strongest mass peaks coincide with the five most luminous cluster galaxies within ≲2″. We do not detect any unusual mass peaks that are not traced by the cluster galaxies, unlike the findings in previous studies. Our mass model shows the smallest scatter of SL multiple images in both source (∼0.″05) and image (∼0.″1) planes, which is lower than in previous studies by a factor of ∼4. Although MARS represents the mass field with an extremely large number of free parameters (∼300,000), it converges to a solution within a few hours because we use a deep-learning technique. We make our mass and magnification maps publicly available.

Model-independent Mass Reconstruction of the Hubble Frontier Field Clusters with MARS Based on Self-consistent Strong-lensing Data

We present a new strong-lensing (SL) mass reconstruction of the six Hubble Frontier Fields (HFF) clusters with the MAximum-entropy ReconStruction (MARS) algorithm. MARS is a new free-form inversion method, which suppresses spurious small-scale fluctuations while achieving excellent convergence in positions of multiple images. For each HFF cluster, we obtain a model-independent mass distribution from the compilation of the self-consistent SL data in the literature. With 100–200 multiple images per cluster, we reconstruct solutions with small scatters of multiple images in both source (∼0.″02) and image planes (0.″05–0.″1), which are lower than the previous results by a factor of 5–10. An outstanding case is the MACS J0416.1-2403 mass reconstruction, which is based on the largest high-quality SL data set where all 236 multiple images/knots have spectroscopic redshifts. Although our solution is smooth on a large scale, it reveals group/galaxy-scale peaks where the substructures are required by the data. We find that in general, these mass peaks are in excellent spatial agreement with the member galaxies, although MARS never uses the galaxy distributions as priors. Our study corroborates the flexibility and accuracy of the MARS algorithm and demonstrates that MARS is a powerful tool in the JWST era, when a 2–3 times larger number of multiple image candidates become available for SL mass reconstruction, and self-consistency within the data set becomes a critical issue.

MAximum-entropy ReconStruction (MARS): A New Strong-lensing Reconstruction Algorithm for the JWST Era

AbstractThe MAximum-entropy ReconStruction (MARS) method is a free-form strong-lensing (SL) reconstruction algorithm, which adopts the maximum cross-entropy as a regularization. MARS shows remarkable convergence of multiple images in both source (∼0.”02) and image planes (∼0.”05 – 0.”1) while suppressing spurious fluctuations. Although the reconstruction requires a large number of free parameters exceeding ∼19,000, our implementation through PyTorch can obtain the reconstruction within hours. From our test using the publicly available synthetic clusters, we have verified that the reconstructed radial mass profiles are consistent with the truth within 1 percent. This makes MARS one of the best-performing SL reconstruction methods. We apply MARS to the six Hubble Frontier Fields clusters and present new mass reconstruction results. We also reconstruct a mass model of Abell 2744 using both weak-lensing (WL) and SL data from the JWST observations, with the largest dataset of Abell 2744, including 286 SL multiple images and ∼350 arcmin−2 WL constraints.

Reconstruction of neutron multiplicity distributions from low-order statistical information

A point-kinetic methodology for rapid and accurate computation of neutron multiplicity distributions in fixed time intervals is demonstrated. The approach uses an optimal combination of low-order count probabilities and low-order statistical moments, both of which are inexpensively computed from a backward Master equation, to reconstruct distributions valid for all count numbers. For high count numbers a generalized Laguerre polynomial representation is shown to accurately reconstruct the multiplicity distribution from five or fewer statistical moments when the zero-count probability is negligible. For intermediate count numbers and when the zero-count probability is significant, maximum information entropy reconstruction constrained by low-order moments and low-order count probabilities reproduces the overall count distribution very accurately. Numerical results for count number distributions and reconstruction accuracy as a function of number of constraints, relative to a system state-updating stochastic simulation MC algorithm (SSA), are presented to demonstrate the efficacy of this approach.

Reconstructing firm-level interactions in the Dutch input–output network from production constraints

Recent crises have shown that the knowledge of the structure of input–output networks, at the firm level, is crucial when studying economic resilience from the microscopic point of view of firms that try to rewire their connections under supply and demand constraints. Unfortunately, empirical inter-firm network data are protected by confidentiality, hence rarely accessible. The available methods for network reconstruction from partial information treat all pairs of nodes as potentially interacting, thereby overestimating the rewiring capabilities of the system and the implied resilience. Here, we use two big data sets of transactions in the Netherlands to represent a large portion of the Dutch inter-firm network and document its properties. We, then, introduce a generalized maximum-entropy reconstruction method that preserves the production function of each firm in the data, i.e. the input and output flows of each node for each product type. We confirm that the new method becomes increasingly more reliable in reconstructing the empirical network as a finer product resolution is considered and can, therefore, be used as a realistic generative model of inter-firm networks with fine production constraints. Moreover, the likelihood of the model directly enumerates the number of alternative network configurations that leave each firm in its current production state, thereby estimating the reduction in the rewiring capability of the system implied by the observed input–output constraints.

MARS: A New Maximum-entropy-regularized Strong Lensing Mass Reconstruction Method

Free-form strong-lensing (SL) mass reconstructions typically suffer from overfitting, which manifests itself as false-positive small-scale fluctuations. We present a new free-form MAximum-entropy ReconStruction (MARS) method without the assumption that light traces mass (LTM). The MARS algorithm enables us to achieve excellent convergence in source positions (∼0.″001), minimize spurious small-scale fluctuations, and provide a quasi-unique solution independently of initial conditions. Our method is tested with the publicly available synthetic SL data FF-SIMS. The comparison with the truth shows that the mass reconstruction quality is on par with those of the best-performing LTM methods published in the literature, which have been demonstrated to outperform existing free-form methods. In terms of the radial mass profile reconstruction, we achieve <1% agreement with the truth for the regions constrained by multiple images. Finally, we apply MARS to A1689 and find that the cluster mass in the SL regime is dominated by the primary halo centered on the brightest cluster galaxy and the weaker secondary halo is also coincident with the bright cluster member ∼160 kpc northeast. Within the SL field, the A1689 radial profile is well described by a Navarro–Frenk–White profile with c 200 = 5.53 ± 0.77 and kpc, and we find no evidence that A1689 is overconcentrated.

On the dependence of orientation averaging mean field homogenization on planar fourth-order fiber orientation tensors

A comprehensive study on the influence of planar fourth-order fiber orientation tensors on effective linear elastic stiffnesses predicted by orientation averaging mean field homogenization is given. Fiber orientation states of sheet molding compound (SMC) are identified to be in most cases approximately planar. In the planar case, all possible fourth-order fiber orientation tensors are given by a minimal invariant set of structurally differing planar fourth-order fiber orientation tensors. This set defines a three-dimensional body and forms the basis for a comprehensive study on the influence of a fiber orientation distribution in terms of a fourth-order tensor on homogenized stiffnesses. The methodology of this study is the main contribution of this work and can be adopted to analyze the orientation dependence of any quantity which is a function of a planar fourth-order fiber orientation tensor. At specific points inside the set of planar fiber orientation tensors, effective stiffnesses are calculated with selected mean field homogenization schemes. These schemes are based on orientation averaging of transversely isotropic elasticity tensors following Advani and Tucker (1987), which is explicitly recast as linear invariant composition in the fiber orientation tensors of second and fourth order of Kanatani third kind. A maximum entropy reconstruction of a fiber orientation distribution function based on leading fiber orientation tensors, enables a new numerical formulation of the Advani and Tucker average for the special planar case. Polar plots of Young’s modulus and generalized bulk modulus obtained by selected homogenization schemes are arranged on two-dimensional slices within the body of admissible fiber orientation tensors, visualizing the influence of the orientation tensor on the stiffness tensor. The orientation-dependence of the generalized bulk modulus differs significantly between selected homogenizations. Restrictions on the effective anisotropic material response caused by orthotropy of closure approximations are discussed.

The comparative study of three nonparametric methods of SAR tomography for building reconstruction

Over the past years, synthetic aperture radar (SAR) tomography (TomoSAR) has indicated significant potentials for three-dimensional (3D) reconstruction of buildings in urban areas. A large number of SAR images are thus required to perform tomographic inversions in non-parametric spectral estimators. To this end, in the case of a limited choice of SAR acquisitions, the present study aims to evaluate the capabilities of five non-parametric spectral estimation (SE) techniques, including linear prediction (LP), maximum entropy (ME), and minimum norm (MN), Capon, and beamforming (BF) in the tomographic reconstruction of urban environments. The performance analysis is carried out by using both simulated and real SAR datasets. The study results demonstrate that the proposed efficient LP estimator, minimizing the average output signal power over the antenna array elements, can separate the layover scatterers along the height direction. This low-computational SE method is thus able to clean side lobes while using a small number of observations. The proposed algorithms, as applied on TerraSAR-X strip-map images of the city of Tehran, Iran, verify the effectiveness of the non-parametric LP reconstruction technique for urban buildings. The estimated height of the scatterers also indicates that the LP estimator is similar to the field-based measurements once compared with ME, MN, Capon, and BF reconstruction methods.

Maximum Entropy Reconstruction of 4D Transverse Phase Space from 2D Projections: with Application to Laser Wire Measurements in the SNS HEBT

Maximum Entropy Reconstruction of 4D Transverse Phase Space from 2D Projections: with Application to Laser Wire Measurements in the SNS HEBT

SNR enhancement in brillouin microspectroscopy using spectrum reconstruction.

Brillouin spectroscopy can suffer from low signal-to-noise ratios (SNRs). Such low SNRs can render common data analysis protocols unreliable, especially for SNRs below ∼10. In this work we exploit two denoising algorithms, namely maximum entropy reconstruction (MER) and wavelet analysis (WA), to improve the accuracy and precision in determination of Brillouin shifts and linewidth. Algorithm performance is quantified using Monte-Carlo simulations and benchmarked against the Cramér-Rao lower bound. Superior estimation results are demonstrated even at low SNRs (≥ 1). Denoising is furthermore applied to experimental Brillouin spectra of distilled water at room temperature, allowing the speed of sound in water to be extracted. Experimental and theoretical values were found to be consistent to within ±1% at unity SNR.

The influence of the probability density function on spectral quality in nonuniformly sampled multidimensional NMR

The goal of nonuniform sampling (NUS) is to select a subset of free induction decays (FIDs) from the conventional, uniform grid in a manner that sufficiently samples short evolution times needed for improved sensitivity and long evolution times needed for enhanced resolution. In addition to specifying the number of FIDs to be collected from a uniform grid, NUS schemes also specify the distribution of the selected FIDs, which directly impacts sampling-induced artifacts. Sampling schemes typically address these heuristic guidelines by utilizing a probability density function (PDF) to bias the distribution of sampled evolution times. Given this common approach, schemes differentiate themselves by how the evolution times are distributed within the envelope of the PDF. Here, we employ maximum entropy reconstruction and utilize in situ receiver operating characteristic (IROC) to conduct a critical comparison of the sensitivity and resolution that can be achieved by three types of biased sampling schemes: exponential (PDF is exponentially decaying), Poisson-gap (PDF derived from a sine function), and quantile-directed (PDF defined by simple polynomial decay). This methodology reveals practical insights and trends regarding how the sampling schemes and bias can provide the highest sensitivity and resolution for two nonuniformly sampled dimensions in a three-dimensional biomolecular NMR experiment. The IROC analysis circumvents the limitations of common metrics when used with nonlinear spectral estimation (a characteristic of all methods used with NUS) by quantifying the spectral quality via synthetic signals that are added to the empirical dataset. Recovery of these synthetic signals provides a proxy for the quality of the empirical portion of the spectrum. The central finding is that differences in spectral quality are primarily driven by the strength of bias in the PDF. In addition, a sampling coverage threshold is observed that appears to be connected to the dependence of each NUS method on its random seed. The differences between sampling schemes and biases are most relevant below 20% coverage where seed-dependence is high, whereas at higher coverages, the performance metrics for all of the sampling schemes begin to converge and approach a seed-independent regime. The results presented here show that aggressive sampling at low coverage can produce high-quality spectra by employing a sampling scheme that adheres to a decaying PDF with a bias to a broad range of short evolution times and includes relatively few FIDs at long evolution times.

Polarized neutron diffraction on a diamagnetic bismuth single crystal

Abstract A large single crystal of bismuth has been studied between 2 K and 150 K by means of polarized neutron diffraction in fields up 6.2 T applied along the trigonal axis. The recorder flipping ratios are very small in agreement with the diamagnetic response of bismuth. Maximum entropy reconstruction shows that the leading contribution to the magnetic field induced magnetization is indeed associated with electron states centered at bismuth. However, small contribution stems from states placed outside bismuth atoms.

On the Five-Moment Hamburger Maximum Entropy Reconstruction

We consider the Maximum Entropy Reconstruction (MER) as a solution to the five-moment truncated Hamburger moment problem in one dimension. In the case of five monomial moment constraints, the probability density function (PDF) of the MER takes the form of the exponential of a quartic polynomial. This implies a possible bimodal structure in regions of moment space. An analytical model is developed for the MER PDF applicable near a known singular line in a centered, two-component, third- and fourth-order moment (μ_3, μ_4) space, consistent with the general problem of five moments. The model consists of the superposition of a perturbed, centered Gaussian PDF and a small-amplitude packet of PDF-density, called the outlying moment packet (OMP), sitting far from the mean. Asymptotic solutions are obtained which predict the shape of the perturbed Gaussian and both the amplitude and position on the real line of the OMP. The asymptotic solutions show that the presence of the OMP gives rise to an MER solution that is singular along a line in ( μ_3, μ_4) space emanating from, but not including, the point representing a standard normal distribution, or thermodynamic equilibrium. We use this analysis of the OMP to develop a numerical regularization of the MER, creating a procedure we call the Hybrid MER (HMER). Compared with the MER, the HMER is a significant improvement in terms of robustness and efficiency while preserving accuracy in its prediction of other important distribution features, such as higher order moments.

Starspot activity and differential rotation in KIC 11560447

AbstractUsing high-precision photometry from the Kepler mission, we investigate patterns of spot activity on the K1-type subgiant component of KIC 11560447, a short-period late-type eclipsing binary. We tested the validity of maximum entropy reconstructions of starspots by numerical simulations. Our procedure successfully captures up to three large spot clusters migrating in longitude. We suggest a way to measure a lower limit for stellar differential rotation, using slopes of spot patterns in the reconstructed time-longitude diagram. We find solar-like differential rotation and recurrent spot activity with a long-term trend towards a dominant axisymmetric spot distribution during the period of observations.

High Order Moment Model for Polydisperse Evaporating Sprays towards Interfacial Geometry Description

In this paper we propose a new Eulerian modeling and related accurate and robust numerical methods, describing polydisperse evaporating sprays, based on high order moment methods in size. The main novelty of this model is its capacity to describe some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. For this purpose, we use fractional size-moments, where the size variable is taken as the droplet surface. In order to evaluate the evaporation of the polydisperse spray, we use a smooth reconstruction which maximizes the Shannon entropy. However, the use of fractional moments introduces some theoretical and numerical difficulties, which need to be tackled. First, relying on a study of the moment space, we extend the Maximum Entropy (ME) reconstruction of the size distribution to the case of fractional moments. Then, we propose a new accurate and realizable algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. This algorithm is based on a mathematical analysis of the kinetic evolution due to evaporation, where it shown that the evolution of some negative order fractional moments have to be properly predicted, a peculiarity related to the use of fractional moments. The present model and numerical schemes yield an accurate and stable evaluationof the moment dynamics with minimal number of variables, as well as a minimal computational cost as with the EMSM model, but with the very interesting additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables, which are essential in oder to describe atomization.

A non-uniform sampling approach enables studies of dilute and unstable proteins.

NMR spectroscopy is a powerful method in structural and functional analysis of macromolecules and has become particularly prevalent in studies of protein structure, function and dynamics. Unique to NMR spectroscopy is the relatively low constraints on sample preparation and the high level of control of sample conditions. Proteins can be studied in a wide range of buffer conditions, e.g. different pHs and variable temperatures, allowing studies of proteins under conditions that are closer to their native environment compared to other structural methods such as X-ray crystallography and electron microscopy. The key disadvantage of NMR is the relatively low sensitivity of the method, requiring either concentrated samples or very lengthy data-acquisition times. Thus, proteins that are unstable or can only be studied in dilute solutions are often considered practically unfeasible for NMR studies. Here, we describe a general method, where non-uniform sampling (NUS) allows for signal averaging to be monitored in an iterative manner, enabling efficient use of spectrometer time, ultimately leading to savings in costs associated with instrument and isotope-labelled protein use. The method requires preparation of multiple aliquots of the protein sample that are flash-frozen and thawed just before acquisition of a short NMR experiments carried out while the protein is stable (12h in the presented case). Non-uniform sampling enables sufficient resolution to be acquired for each short experiment. Identical NMR datasets are acquired and sensitivity is monitored after each co-added spectrum is reconstructed. The procedure is repeated until sufficient signal-to-noise is obtained. We discuss how maximum entropy reconstruction is used to process the data, and propose a variation on the previously described method of automated parameter selection. We conclude that combining NUS with iterative co-addition is a general approach, and particularly powerful when applied to unstable proteins.

Application of the Direct Quadrature Method of Moments for the modelling of the enzymatic hydrolysis of cellulose: II. Case of insoluble substrate

In the second part of this study, the Population Balance Model developed for the enzymatic hydrolysis of cellulose polymer chains is extended to particulate cellulose substrates. In this case, the particle fragmentation process is complex. This phenomenon is induced by the hydrodynamic shear stress when the particle cohesion is lower than the hydrodynamic force. The particle cohesion depends on the mechanical properties of the substrate particles which are affected by the Endoglucanase (EG) activity that hydrolyses accessible glycosidic bonds all along the reaction. These two effects are taken into account in the model. The Exoglucanase (CBH) activity is assimilated to an erosion process as for soluble substrates. The β-glucosidase activity as well as the inhibition effect are also considered in the global population balance-based model. The numerical resolution of the Population Balance Equation (PBE) is based on the Direct Quadrature Method of Moments (DQMOM) coupled with the Maximum Entropy (ME) reconstruction technique. Numerical results in different cases show coherent prediction of the Particle Size Distribution (PSD) evolution during the hydrolysis reaction as well as the kinetics of simple sugars release. The modelling procedure is supported by experimental data which reveal the importance of the particles cohesion and its evolution during the hydrolysis reaction.

Application of random coherence order selection in gradient-enhanced multidimensional NMR

Development of multidimensional NMR is essential to many applications, for example in high resolution structural studies of biomolecules. Multidimensional techniques enable separation of NMR signals over several dimensions, improving signal resolution, whilst also allowing identification of new connectivities. However, these advantages come at a significant cost. The Fourier transform theorem requires acquisition of a grid of regularly spaced points to satisfy the Nyquist criterion, while frequency discrimination and acquisition of a pure phase spectrum require acquisition of both quadrature components for each time point in every indirect (non-acquisition) dimension, adding a factor of 2N-1 to the number of free- induction decays which must be acquired, where N is the number of dimensions. Compressed sensing (CS) ℓ1-norm minimisation in combination with non-uniform sampling (NUS) has been shown to be extremely successful in overcoming the Nyquist criterion. Previously, maximum entropy reconstruction has also been used to overcome the limitation of frequency discrimination, processing data acquired with only one quadrature component at a given time interval, known as random phase detection (RPD), allowing a factor of two reduction in the number of points for each indirect dimension (Maciejewski et al. 2011 PNAS 108 16640). However, whilst this approach can be easily applied in situations where the quadrature components are acquired as amplitude modulated data, the same principle is not easily extended to phase modulated (P-/N-type) experiments where data is acquired in the form exp (iωt) or exp (-iωt), and which make up many of the multidimensional experiments used in modern NMR. Here we demonstrate a modification of the CS ℓ1-norm approach to allow random coherence order selection (RCS) for phase modulated experiments; we generalise the nomenclature for RCS and RPD as random quadrature detection (RQD). With this method, the power of RQD can be extended to the full suite of experiments available to modern NMR spectroscopy, allowing resolution enhancements for all indirect dimensions; alone or in combination with NUS, RQD can be used to improve experimental resolution, or shorten experiment times, of considerable benefit to the challenging applications undertaken by modern NMR.

Convex accelerated maximum entropy reconstruction

Maximum entropy (MaxEnt) spectral reconstruction methods provide a powerful framework for spectral estimation of nonuniformly sampled datasets. Many methods exist within this framework, usually defined based on the magnitude of a Lagrange multiplier in the MaxEnt objective function. An algorithm is presented here that utilizes accelerated first-order convex optimization techniques to rapidly and reliably reconstruct nonuniformly sampled NMR datasets using the principle of maximum entropy. This algorithm – called CAMERA for Convex Accelerated Maximum Entropy Reconstruction Algorithm – is a new approach to spectral reconstruction that exhibits fast, tunable convergence in both constant-aim and constant-lambda modes. A high-performance, open source NMR data processing tool is described that implements CAMERA, and brief comparisons to existing reconstruction methods are made on several example spectra.

Two dimensional IR-FID-CPMG acquisition and adaptation of a maximum entropy reconstruction

By acquiring the FID signal in two-dimensional TD-NMR spectroscopy, it is possible to characterize mixtures or complex samples composed of solid and liquid phases. We have developed a new sequence for this purpose, called IR-FID-CPMG, making it possible to correlate spin–lattice T1 and spin–spin T2 relaxation times, including both liquid and solid phases in samples. We demonstrate here the potential of a new algorithm for the 2D inverse Laplace transformation of IR-FID-CPMG data based on an adapted reconstruction of the maximum entropy method, combining the standard decreasing exponential decay function with an additional term drawn from Abragam’s FID function. The results show that the proposed IR-FID-CPMG sequence and its related inversion model allow accurate characterization and quantification of both solid and liquid phases in multiphasic and compartmentalized systems. Moreover, it permits to distinguish between solid phases having different T1 relaxation times or to highlight cross-relaxation phenomena.