Application of the Maximum Entropy Method to Multifunctional Materials for Data Fusion and Uncertainty Quantification

2018 ◽  
Author(s):  
Wei Gao ◽  
William S. Oates ◽  
Paul R. Miles ◽  
Ralph C. Smith
Keyword(s):  
Maximum Entropy ◽  
Smart Materials ◽  
Intelligent Systems ◽  
Density Functional ◽  
Electric Displacement ◽  
Phase Field Model ◽  
Density Functional Theory Calculations ◽  
Heterogeneous Data ◽  
Entropy Method ◽  
Model Parameter Uncertainty

Bayesian statistics is a quintessential tool for model validation in many applications including smart materials, adaptive structures, and intelligent systems. It typically uses either experimental data or high-fidelity simulations to infer model parameter uncertainty of reduced order models due to experimental noise and homogenization of quantum or atomistic behavior. When heterogeneous data is available for Bayesian inference, open questions remain on appropriate methods to fuse data and avoid inappropriate weighting on individual data sets. To address this issue, we implement a Bayesian statistical method that begins with maximizing entropy. We show how this method can weight heterogeneous data automatically during the inference process through the error covariance. This Maximum Entropy (ME) method is demonstrated by quantifying uncertainty in 1) a ferroelectric domain structure model and 2) a finite deforming electrostrictive membrane model. The ferroelectric phase field model identifies continuum parameters from multiple density functional theory calculations. In the case of the electrostrictive membrane, parameters are estimated from both mechanical and electric displacement experimental measurements.

Charge density study with the Maximum Entropy Method on model data of silicon. A search for non-nuclear attractors

1996 ◽  
Vol 74 (6) ◽  
pp. 1054-1058 ◽  
Author(s):  
R.Y. de Vries ◽  
W.J. Briels ◽  
D. Fell ◽  
G. te Velde ◽  
E.J. Baerends
Keyword(s):  
Charge Density ◽  
Maximum Entropy ◽  
Density Functional ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
X Ray Diffraction ◽  
Functional Theory ◽  
Structure Factors ◽  
The Density Functional Theory ◽  
Structure Calculations

In 1990 Sakata and Sato applied the maximum entropy method (MEM) to a set of structure factors measured earlier by Saka and Kato with the Pendellösung method. They found the presence of non-nuclear attractors, i.e., maxima in the density between two bonded atoms. We applied the MEM to a limited set of Fourier data calculated from a known electron density distribution (EDD) of silicon. The EDD of silicon was calculated with the program ADF-BAND. This program performs electronic structure calculations, including periodicity, based on the density functional theory of Hohenberg and Kohn. No non-nuclear attractor between two bonded silicon atoms was observed in this density. Structure factors were calculated from this density and the same set of structure factors that was measured by Saka and Kato was used in the MEM analysis. The EDD obtained with the MEM shows the same non-nuclear attractors that were later obtained by Sakata and Sato. This means that the non-nuclear attractors in silicon are really an artefact of the MEM. Key words: Maximum Entropy Method, non-nuclear attractors, charge density. X-ray diffraction.

Charge density of 4-methyl-3-[(tetrahydro-2H-pyran-2-yl)oxy]thiazole-2(3H)-thione. A comprehensive multipole refinement, maximum entropy method and density functional theory study

2020 ◽  
Vol 76 (3) ◽  
pp. 450-468
Author(s):  
Barbora Vénosová ◽  
Julia Koziskova ◽  
Jozef Kožíšek ◽  
Peter Herich ◽  
Karol Lušpai ◽  
...  
Keyword(s):  
Density Functional Theory ◽  
Charge Density ◽  
Maximum Entropy ◽  
Density Functional ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Experimental Results ◽  
Basis Set ◽  
Functional Theory ◽  
Radial Functions

The structure of 4-methyl-3-[(tetrahydro-2H-pyran-2-yl)oxy]thiazole-2(3H)-thione (MTTOTHP) was investigated using X-ray diffraction and computational chemistry methods for determining properties of the nitrogen—oxygen bond, which is the least stable entity upon photochemical excitation. Experimentally measured structure factors have been used to determine and characterize charge density via the multipole model (MM) and the maximum entropy method (MEM). Theoretical investigation of the electron density and the electronic structure has been performed in the finite basis set density functional theory (DFT) framework. Quantum Theory of Atoms In Molecules (QTAIM), deformation densities and Laplacians maps have been used to compare theoretical and experimental results. MM experimental results and predictions from theory differ with respect to the sign and/or magnitude of the Laplacian at the N—O bond critical point (BCP), depending on the treatment of n values of the MM radial functions. Such Laplacian differences in the N—O bond case are discussed with respect to a lack of flexibility in the MM radial functions also reported by Rykounov et al. [Acta Cryst. (2011), B67, 425–436]. BCP Hessian eigenvalues show qualitatively matching results between MM and DFT. In addition, the theoretical analysis used domain-averaged fermi holes (DAFH), natural bond orbital (NBO) analysis and localized (LOC) orbitals to characterize the N—O bond as a single σ bond with marginal π character. Hirshfeld atom refinement (HAR) has been employed to compare to the MM refinement results and/or neutron dataset C—H bond lengths and to crystal or single molecule geometry optimizations, including considerations of anisotropy of H atoms. Our findings help to understand properties of molecules like MTTOTHP as progenitors of free oxygen radicals.

A Maximum Entropy Approach for Uncertainty Quantification and Analysis of Multifunctional Materials

2017 ◽  
Author(s):  
Wei Gao ◽  
William S. Oates ◽  
Ralph C. Smith
Keyword(s):  
Maximum Entropy ◽  
Likelihood Function ◽  
Constitutive Relations ◽  
Model Development ◽  
Electric Displacement ◽  
Heterogeneous Data ◽  
Multifunctional Materials ◽  
Data Set ◽  
Entropy Functional ◽  
Moment Constraint

The Maximum Entropy (ME) method is shown to provide a new approach for quantifying model uncertainty in the presence of complex, heterogeneous data. This is important in model validation of a variety of multifunctional constitutive relations. For example, multifunctional materials contain field-coupled material parameters that should be self-consistent regardless of the measurement. A classical example is piezoelectricity which may be quantified from charge induced by stress or strain induced by an electric field. The proposed tools provide new statistical information to address measurement discrepancies, guide model development, and catalyze materials discovery for data fusion problems. The error between the model outputs and heterogeneous data is quantified and used to formulate a second moment constraint within the entropy functional. This leads to an augmented likelihood function that weights each individual set of data by its respective variance and covariance between each data set. As a first step, the method is evaluated on a piezoelectric ceramic to illustrate how the covariance matrix influences piezoelectric parameter estimation from heterogeneous electric displacement and strain data.

scholarly journals Применение метода максимальной энтропии к временным рядам, получаемым в реальном времени в рамках нестационарной теории функционала плотности

2021 ◽  
pp. 64-73
Author(s):  
Y. Zempo ◽  
S.S. Kano
Keyword(s):  
Density Functional Theory ◽  
Spectral Analysis ◽  
Dipole Moment ◽  
Maximum Entropy ◽  
Density Functional ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Data Set ◽  
Functional Theory ◽  
Raw Data

The maximum entropy method is one of the key techniques for spectral analysis. The main feature is to describe spectra in low frequency with short timeseries data. We adopted the maximum entropy method to analyze the spectrum from the dipole moment obtained by the timedependent density functional theory calculation in real time, which is intensively studied and applied to computing optical properties. In the maximum entropy method analysis, we proposed that we use the concatenated data set made from severaltimes repeated raw data together with the phase. We have applied this technique to spectral analysis of the dynamic dipole moment obtained from timedependent density functional theory dipole moment of several molecules such as oligofluorene with n = 8. As a result, the higher resolution can be obtained without any peak shift due to the phase jump. The peak position is in good agreement to that of FT with just raw data. This paper presents the efficiency and characteristic features of this technique. Метод максимальной энтропии — один из основных в спектральном анализе. Его главная особенность — описание низкочастотных спектров короткими временными рядами данных. Авторы применили метод максимальной энтропии для анализа спектров дипольного момента, полученных расчетами в реальном времени по нестационарной теории функционала плотности. Данный вопрос интенсивно изучается и находит практическое применение при расчетах оптических свойств. При анализе методом максимальной энтропии предложено использовать объединенные наборы данных, включающие несколько повторяющихся последовательностей исходных данных с учетом фазы. Данный метод был применен при проведении спектрального анализа динамического дипольного момента, рассчитанного по нестационарной теории функционала плотности на основе дипольного момента нескольких молекул — в частности, молекул олигофлуорена при n = 8. В итоге удалось повысить разрешение без смещения максимумов из-за скачка фазы. Положение максимумов хорошо согласуется с результатами применения преобразования Фурье к необработанным исходным данным. В настоящей статье представлены особенности данного метода и показатели его эффективности.

scholarly journals Location of Cu2+in CHA zeolite investigated by X-ray diffraction using the Rietveld/maximum entropy method

IUCrJ ◽  
2014 ◽  
Vol 1 (6) ◽  
pp. 382-386 ◽  
Author(s):  
Casper Welzel Andersen ◽  
Martin Bremholm ◽  
Peter Nicolai Ravnborg Vennestrøm ◽  
Anders Bank Blichfeld ◽  
Lars Fahl Lundegaard ◽  
...  
Keyword(s):  
Maximum Entropy ◽  
Density Functional ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
Membered Ring ◽  
Zeolite Catalysts ◽  
Structural Description ◽  
X Ray Diffraction ◽  
X Ray ◽  
Catalytically Active

Accurate structural models of reaction centres in zeolite catalysts are a prerequisite for mechanistic studies and further improvements to the catalytic performance. The Rietveld/maximum entropy method is applied to synchrotron powder X-ray diffraction data on fully dehydrated CHA-type zeolites with and without loading of catalytically active Cu2+for the selective catalytic reduction of NOxwith NH3. The method identifies the known Cu2+sites in the six-membered ring and a not previously observed site in the eight-membered ring. The sum of the refined Cu occupancies for these two sites matches the chemical analysis and thus all the Cu is accounted for. It is furthermore shown that approximately 80% of the Cu2+is located in the new 8-ring site for an industrially relevant CHA zeolite with Si/Al = 15.5 and Cu/Al = 0.45. Density functional theory calculations are used to corroborate the positions and identity of the two Cu sites, leading to the most complete structural description of dehydrated silicoaluminate CHA loaded with catalytically active Cu2+cations.

The maximum entropy method for data fusion and uncertainty quantification in multifunctional materials and structures

2021 ◽  
pp. 1045389X2110482
Author(s):  
Wei Gao ◽  
Paul R Miles ◽  
Ralph C Smith ◽  
William S Oates
Keyword(s):  
Maximum Entropy ◽  
Heterogeneous Data ◽  
Entropy Method ◽  
Data Sets ◽  
Multiple Model ◽  
Domain Structures ◽  
Multiple Data ◽  
Multiphysics Coupling ◽  
Multiple Data Sets ◽  
Quantities Of Interest

The quantification of uncertainty in intelligent material systems and structures requires methods to objectively compare complex models to measurements, where the majority of cases include multiple model outputs and quantities of interests given multiphysics coupling. This creates questions about constructing appropriate measures of uncertainty during fusion of data and comparisons between data and models. Novel materials with complex or poorly understood coupling can benefit from advanced statistical analysis to judge models in light of multiphysics data. Here, we apply the Maximum Entropy (ME) method to more complicated ferroelectric single crystals containing domain structures and soft electrostrictive membranes under both mechanical and electrical loading. Multiple quantities of interest are considered, which requires fusing heterogeneous information together when quantifying the uncertainty of lower fidelity models. We find that parameters, which were initially unidentifiable using a single quantity of interest, become identifiable using multiple quantities of interest. We also show that posterior densities may broaden or narrow when multiple data sets are fused together. This is likely due to conflict or agreement, respectively, between the different quantities of interest and the multiple model outputs. Such information is important to advance our predictions of intelligent materials and structures from multi-model inputs and heterogeneous data.

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