Discrete Realization Research Articles

Using Pattern Counts to Quantify the Difference Between a Pair of Three-Dimensional Realizations

When comparing different ways of modeling discrete three-dimensional realizations such as facies, it is useful to have a measure of difference (or similarity) in the geometry of these realizations. We propose a method for evaluating such difference by comparing pattern counts for a small template. Tests on synthetic datasets demonstrate that the proposed difference effectively differentiates between realizations of a Boolean model and those generated using multiple-point statistics with the Boolean realizations as training images. We also observed that multiple-point statistics realizations based on similar training images yield smaller differences to one another compared to those based on training images from dissimilar concepts. This suggests that the proposed difference is a useful tool for comparing discrete three-dimensional realizations.

Differentiable stochastic halo occupation distribution

ABSTRACT In this work, we demonstrate how differentiable stochastic sampling techniques developed in the context of deep reinforcement learning can be used to perform efficient parameter inference over stochastic, simulation-based, forward models. As a particular example, we focus on the problem of estimating parameters of halo occupation distribution (HOD) models that are used to connect galaxies with their dark matter haloes. Using a combination of continuous relaxation and gradient re-parametrization techniques, we can obtain well-defined gradients with respect to HOD parameters through discrete galaxy catalogue realizations. Having access to these gradients allows us to leverage efficient sampling schemes, such as Hamiltonian Monte Carlo, and greatly speed up parameter inference. We demonstrate our technique on a mock galaxy catalogue generated from the Bolshoi simulation using a standard HOD model and find near-identical posteriors as standard Markov chain Monte Carlo techniques with an increase of ∼8× in convergence efficiency. Our differentiable HOD model also has broad applications in full forward model approaches to cosmic structure and cosmological analysis.

Treating detector systematics via a likelihood free inference method

Estimating the impact of systematic uncertainties in particle physics experiments is challenging, especially since the detector response is unknown analytically in most situations and needs to be estimated through Monte Carlo (MC) simulations. Typically, detector property variations are parameterized in ways that implicitly assume a specific physics model, which can introduce biases on quantities measured by an analysis. In this paper, we present a method to recover a model-independent, event-wise estimation of the detector response variation by applying a likelihood-free inference method to a set of MC simulations representing discrete detector realizations. The method provides a re-weighting scheme for every event, which can be used to apply the effects of detector property variations fully decoupled from the assumed physics model. Using a toy MC example inspired by fixed-baseline neutrino oscillation experiments, we demonstrate the performance of our method. We show that it fully decouples the modeling of the detector response from the physics parameters to be measured in a MC forward-folding analysis.

Space-time variational material modeling: a new paradigm demonstrated for thermo-mechanically coupled wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage

We formulate variational material modeling in a space-time context. The starting point is the description of the space-time cylinder and the definition of a thermodynamically consistent Hamilton functional which accounts for all boundary conditions on the cylinder surface. From the mechanical perspective, the Hamilton principle then yields thermo-mechanically coupled models by evaluation of the stationarity conditions for all thermodynamic state variables which are displacements, internal variables, and temperature. Exemplary, we investigate in this contribution elastic wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage. Therein, one key novel aspect are initial and end time velocity conditions for the wave equation, replacing classical initial conditions for the displacements and the velocities. The motivation is intensively discussed and illustrated with the help of a prototype numerical simulation. From the mathematical perspective, the space-time formulations are formulated within suitable function spaces and convex sets. The unified presentation merges engineering and applied mathematics due to their mutual interactions. Specifically, the chosen models are of high interest in many state-of-the art developments in modeling and we show the impact of this holistic physical description on space-time Galerkin finite element discretization schemes. Finally, we study a specific discrete realization and show that the resulting system using initial and end time conditions is well-posed.

LMI-Based robust controller for load frequency control of three area interconnected system

The purpose of this research is to create a robust predictive load frequency control (RPLFC) for power systems with unpredictable parameters and temporal delays in communication networks. The study looks closely at the communication of delay constraints across the various control regions as well as the relationship between delay restrictions and control gains. In addition, the usage of delay restrictions as a new performance index to steer controller design is discussed, along with the regulation of the controller for a trade-off between delay tolerance and dynamic reaction, and opting for the upper bound of the sampling period of a discrete realization of the controller, and the upper bound of the fault counter of the communication channel. All of these topics are brought up in the context of determining how to steer controller design. To determine whether or not the proposed technique is useful, simulation studies have all been outfitted with PID-type controllers and have been run for both single-area and three-area LFC schemes. MATLAB/SIMULINK served as the platform upon which the simulations were carried out.

Neural network surrogate for flow prediction and robust optimization in fractured reservoir systems

Embedded discrete fracture models are widely used to model flow in naturally fractured systems. These models are, however, time-consuming to simulate, and this limits their use for computationally demanding applications such as optimization. In this work we present a deep-learning-based surrogate model for fast prediction of time-varying well flow rates over multiple fracture realizations, for given well bottom-hole pressure schedules. Our surrogate model applies convolutional and recurrent neural networks. The convolutional neural network is used to capture the spatial variability of fractures in different realizations, though the direct input of the fracture geometry is not feasible because structured data are required. We show, however, that the use of the single-phase steady-state pressure solution defined on the structured rock-matrix grid, for each realization, provides the requisite input to the convolutional neural network. The resulting proxy is shown to give accurate flow rate predictions for different discrete fracture realizations. The proxy is then applied for robust optimization (expected net present value computed over an ensemble of realizations is optimized) subject to nonlinear output constraints. Proxy-based optimization provides an 18% improvement in net present value, with an overall speedup of ∼500 relative to simulation-based optimization. The optimized solutions are validated through comparison to simulation results, and agreement is within 1%.

Objective trajectories in hybrid classical-quantum dynamics

Consistent dynamics which couples classical and quantum degrees of freedom exists, provided it is stochastic. This dynamics is linear in the hybrid state, completely positive and trace preserving. One application of this is to study the back-reaction of quantum fields on space-time which does not suffer from the pathologies of the semi-classical equations. Here we introduce several toy models in which to study hybrid classical-quantum evolution, including a qubit coupled to a particle in a potential, and a quantum harmonic oscillator coupled to a classical one. We present an unravelling approach to calculate the dynamics, and provide code to numerically simulate it. Unlike the purely quantum case, the trajectories (or histories) of this unravelling can be unique, conditioned on the classical degrees of freedom for discrete realisations of the dynamics, when different jumps in the classical degrees of freedom are accompanied by the action of unique operators on the quantum system. As a result, the “measurement postulate'' of quantum theory is not needed; quantum systems become classical because they interact with a fundamentally classical field.

A computationally informed realisation algorithm for lithium-ion batteries implemented with LiiBRA.jl

Real-time battery modelling advancements have quickly become required as the adoption of battery electric vehicles (BEVs) has rapidly increased. In this paper an open-source, improved discrete realisation algorithm, implemented in Julia for the creation and simulation of reduced-order, real-time capable physics-based models is presented. This work reduces the Doyle–Fuller–Newman electrochemical model into continuous-form transfer functions and introduces a computationally informed discrete realisation algorithm (CI-DRA) to generate the reduced-order representation. Further improvements in conventional offline model creation are obtained as well as achieving in-vehicle capable model creation for ARM-based computing architectures. Furthermore, a parametric sensitivity analysis of the presented architecture is completed as well as experimental validation of a worldwide harmonised light vehicle test procedure (WLTP) for an LG Chem. M50 21700 parameterisation. A performance comparison to a MATLAB implementation is completed showcasing a mean computational time improvement of 3.51 times for LiiBRA.jl on x86 hardware. Finally, an ARM-based implementation showcases full system model generation within three minutes for potential in-vehicle updates.

On local path behavior of Surgailis multifractional processes

Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the classical Multifractional Brownian Motion (MBM) { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter H {\mathcal {H}} of the well-known Fractional Brownian Motion by a deterministic function H ( t ) {\mathcal {H}}(t) having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two non-classical Gaussian multifactional processes denoted by { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} . In our article, under a rather weak condition on the functional parameter H ( ⋅ ) {\mathcal {H}}(\cdot ) , we show that { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} and { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} as well as { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} only differ by a part which is locally more regular than { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} itself. On one hand this result implies that the two non-classical multifractional processes { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} have exactly the same local path behavior as that of the classical MBM { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} . On the other hand it allows to obtain from discrete realizations of { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} strongly consistent statistical estimators for values of their functional parameter.

Dark Energy Survey Year 3 results: marginalization over redshift distribution uncertainties using ranking of discrete realizations

ABSTRACT Cosmological information from weak lensing surveys is maximized by sorting source galaxies into tomographic redshift subsamples. Any uncertainties on these redshift distributions must be correctly propagated into the cosmological results. We present hyperrank, a new method for marginalizing over redshift distribution uncertainties, using discrete samples from the space of all possible redshift distributions, improving over simple parametrized models. In hyperrank, the set of proposed redshift distributions is ranked according to a small (between one and four) number of summary values, which are then sampled, along with other nuisance parameters and cosmological parameters in the Monte Carlo chain used for inference. This approach can be regarded as a general method for marginalizing over discrete realizations of data vector variation with nuisance parameters, which can consequently be sampled separately from the main parameters of interest, allowing for increased computational efficiency. We focus on the case of weak lensing cosmic shear analyses and demonstrate our method using simulations made for the Dark Energy Survey (DES). We show that the method can correctly and efficiently marginalize over a wide range of models for the redshift distribution uncertainty. Finally, we compare hyperrank to the common mean-shifting method of marginalizing over redshift uncertainty, validating that this simpler model is sufficient for use in the DES Year 3 cosmology results presented in companion papers.

Validation Methods for Energy Time Series Scenarios From Deep Generative Models

The design and operation of modern energy systems are heavily influenced by time-dependent and uncertain parameters, e.g., renewable electricity generation, load-demand, and electricity prices. These are typically represented by a set of discrete realizations known as scenarios. A popular scenario generation approach uses deep generative models (DGM) that allow scenario generation without prior assumptions about the data distribution. However, the validation of generated scenarios is difficult, and a comprehensive discussion about appropriate validation methods is currently lacking. To start this discussion, we provide a critical assessment of the currently used validation methods in the energy scenario generation literature. In particular, we assess validation methods based on probability density, auto-correlation, and power spectral density. Furthermore, we propose using the multifractal detrended fluctuation analysis (MFDFA) as an additional validation method for non-trivial features like peaks, bursts, and plateaus. As representative examples, we train generative adversarial networks (GANs), Wasserstein GANs (WGANs), and variational autoencoders (VAEs) on two renewable power generation time series (photovoltaic and wind from Germany in 2013 to 2015) and an intra-day electricity price time series form the European Energy Exchange in 2017 to 2019. We apply the four validation methods to both the historical and the generated data and discuss the interpretation of validation results as well as common mistakes, pitfalls, and limitations of the validation methods. Our assessment shows that no single method sufficiently characterizes a scenario but ideally validation should include multiple methods and be interpreted carefully in the context of scenarios over short time periods.

Modelling of Electric Vehicle with PID Controller Transfer Function using GA and Model-Reduced Order DRA Algorithm

In this paper, a model of an electric vehicle transfer function using GA and a model-reduced order discrete time realization (DRA) algorithm is presented. The electric vehicle (EV) control system regulates vehicle speed according to the driver’s command signal and brings the vehicle to its equilibrium point, i.e., the desired speed under any abnormal conditions. The controller transfer function is designed based on EV's dynamic differential equations. An infinite-order transcendental transfer function for the EV model is approximated to find high-reliability discrete-time state-space reduced-order models (ROMs). Reduced the robustness and increased the predictions' accuracy of ROM output in terms of accurate numerical simulations of the EV transfer function. Also, the design of the PID controller parameter using optimization techniques (OT) for the tuning of conventional controllers for the desired speed is done using a linearized model. In this study, the performance of a conventional PID controller is compared with that of a GA-based PID controller. In terms of maximum overshoot, peak time, rising time, settling time, and steady state error, the simulation results show that the suggested controller is preferable.

Localization of plane wave expansion using the method of quasisolution searching

The localization of the plane wave expansion is considered for the analysis of wave beam transformations on transversely inhomogeneous structures. To implement compact localization in the spatial domain, it is proposed to use a system of non-intersecting rectangular windows. Problems of discrete realization of local plane wave expansion are analyzed. An algorithm for the numerical implementation of the localization of the plane wave expansion based on the quasisolution search method is proposed. Its advantage over the algorithm based on the discrete Fourier transform is demonstrated. The efficiency of its use has been demonstrated by the example of analyzing the wave beam reflection from the secondary aperture with transverse inhomogeneity.

Data-Space Inversion With a Recurrent Autoencoder for Naturally Fractured Systems

Data-space inversion (DSI) is a data assimilation procedure that directly generates posterior flow predictions, for time series of interest, without calibrating model parameters. No forward flow simulation is performed in the data assimilation process. DSI instead uses the prior data generated by performingO(1000)simulations on prior geomodel realizations. Data parameterization is useful in the DSI framework as it enables representation of the correlated time-series data quantities in terms of low-dimensional latent-space variables. In this work, a recently developed parameterization based on a recurrent autoencoder (RAE) is applied with DSI for a real naturally fractured reservoir. The parameterization, involving the use of a recurrent neural network and an autoencoder, is able to capture important correlations in the time-series data. RAE training is accomplished using flow simulation results for 1,350 prior model realizations. An ensemble smoother with multiple data assimilation (ESMDA) is applied to provide posterior DSI data samples. The modeling in this work is much more complex than that considered in previous DSI studies as it includes multiple 3D discrete fracture realizations, three-phase flow, tracer injection and production, and complicated field-management logic leading to frequent well shut-in and reopening. Results for the reconstruction of new simulation data (not seen in training), using both the RAE-based parameterization and a simpler approach based on principal component analysis (PCA) with histogram transformation, are presented. The RAE-based procedure is shown to provide better accuracy for these data reconstructions. Detailed posterior DSI results are then presented for a particular “true” model (which is outside the prior ensemble), and summary results are provided for five additional “true” models that are consistent with the prior ensemble. These results again demonstrate the advantages of DSI with RAE-based parameterization for this challenging fractured reservoir case.

Modeling of Retinal Optical Coherence Tomography Based on Stochastic Differential Equations: Application to Denoising.

In this paper a statistical modeling, based on stochastic differential equations (SDEs), is proposed for retinal Optical Coherence Tomography (OCT) images. In this method, pixel intensities of image are considered as discrete realizations of a Levy stable process. This process has independent increments and can be expressed as response of SDE to a white symmetric alpha stable (s [Formula: see text]) noise. Based on this assumption, applying appropriate differential operator makes intensities statistically independent. Mentioned white stable noise can be regenerated by applying fractional Laplacian operator to image intensities. In this way, we modeled OCT images as s [Formula: see text] distribution. We applied fractional Laplacian operator to image and fitted s [Formula: see text] to its histogram. Statistical tests were used to evaluate goodness of fit of stable distribution and its heavy tailed and stability characteristics. We used modeled s [Formula: see text] distribution as prior information in maximum a posteriori (MAP) estimator in order to reduce the speckle noise of OCT images. Such a statistically independent prior distribution simplified denoising optimization problem to a regularization algorithm with an adjustable shrinkage operator for each image. Alternating Direction Method of Multipliers (ADMM) algorithm was utilized to solve the denoising problem. We presented visual and quantitative evaluation results of the performance of this modeling and denoising methods for normal and abnormal images. Applying parameters of model in classification task as well as indicating effect of denoising in layer segmentation improvement illustrates that the proposed method describes OCT data more accurately than other models that do not remove statistical dependencies between pixel intensities.

Dual evolution mixed variational analysis of multiphase flow in the subsurface

Dual evolution mixed variational formulations of multiphase flow models in the subsurface are established and analyzed. Specifically, compositional multiphase flow of a triphasic water–oil–gas multicomponent flow is treated, with a dual Gibbs oil–gas mass transfer model. Qualitative analysis of the mixed variational models is performed applying composition duality principles and methods. For multiphysics as well as parallel computing, macro-hybrid variational reformulations are considered. Dual semi-implicit time-marching schemes are presented as semi-discrete approximations for the flow-transport variational models, as well as for a precondition augmented exactly penalized dual Gibbs optimality variational formulation, which are all implementable as parallel proximal penalty–duality algorithms. In this manner, bases for fully discrete non-conforming macro-hybrid mixed finite element numerical realizations are rendered.

Fragmented Rockfall Volume Distribution from Photogrammetry-Based Structural Mapping and Discrete Fracture Networks

The design of rockfall protection structures requires information about the falling block volumes. Computational tools for rockfall trajectory simulation are now capable of modeling block fragmentation, requiring the fragmented volume-relative frequency distribution of rockfalls as input. This can be challenging at locations with scarce or nonexistent rockfall records and where block surveys are not feasible. The work in this paper shows that simple discrete fracture network realizations from structural mapping based on photogrammetric techniques can be used to reliably estimate rock fall block volumes. These estimates can be used for dimensioning rockfall protection structures in cases where data is scarce or not available. The methodology is tested at two sites in the Canadian Cordillera where limestone outcrops have been the source of recurrent rockfalls. The results suggest that fragmentation will largely tend to occur through weak planes and expansion of non-persistent discontinuities, while other block breakage mechanisms exert less influence in the fragmented volume-relative frequency distribution of rockfalls. Therefore, block volume distribution can be estimated using a simple discrete fracture network (DFN) with fully persistent discontinuities. Limitations of the methods are also discussed, as well as potential future research to address such limitations.

The q‐Bannai–Ito algebra and multivariate (−q)‐Racah and Bannai–Ito polynomials

The Gasper and Rahman multivariate $(-q)$-Racah polynomials appear as connection coefficients between bases diagonalizing different abelian subalgebras of the recently defined higher rank $q$-Bannai-Ito algebra $\mathcal{A}_n^q$. Lifting the action of the algebra to the connection coefficients, we find a realization of $\mathcal{A}_n^q$ by means of difference operators. This provides an algebraic interpretation for the bispectrality of the multivariate $(-q)$-Racah polynomials, as was established in [Iliev, Trans. Amer. Math. Soc. 363 (3) (2011), 1577-1598]. Furthermore, we extend the Bannai-Ito orthogonal polynomials to multiple variables and use these to express the connection coefficients for the $q = 1$ higher rank Bannai-Ito algebra $\mathcal{A}_n$, thereby proving a conjecture from [De Bie et al., Adv. Math. 303 (2016), 390-414]. We derive the orthogonality relation of these multivariate Bannai-Ito polynomials and provide a discrete realization for $\mathcal{A}_n$.

A robust nonlinear model predictive controller for a post-combustion CO2 capture absorber unit

This study presents the design of a robust nonlinear model predictive controller (NMPC) for the absorber unit of a post-combustion CO2 capture process operating downstream from a coal-fired power plant. The controller employs a dynamic mechanistic model for this process, which allows for a precise treatment of the sources of uncertainty. In the case of the absorber, the model enables explicit consideration of uncertainty in the process operating conditions dictated by upstream units and in physical property parameters associated with phenomenological descriptions of the process. Using the multi-scenario approach, discrete realizations of the uncertain parameters inside a given uncertainty region can be incorporated into the controller to produce control actions that result in a robust operation in closed-loop. The multi-scenario controller was tested under several disturbance rejection scenarios of varying complexity and their performance was assessed against that of single-scenario NMPCs. The benefits of the robust NMPC approach were most evident for longer simulations and, considering the continuous nature of a power plant and its emissions, make the robust NMPC approach suitable for the operation of this process in the presence of uncertainty and disturbances.

Hyperbolic fracton model, subsystem symmetry, and holography. II. The dual eight-vertex model

The discovery of fracton states of matter opens up an exciting, largely unexplored field of many-body physics. Certain fracton states' similarity to gravity is an intriguing property. In an earlier work [H. Yan, Phys. Rev. B 99, 155126 (2019)], we have demonstrated that a simple fracton model in anti-de Sitter space satisfies several major holographic properties. In this follow-up paper, we study the eight-vertex model dual to the original model. The dual model has the advantage of illuminating the mutual information and subsystem charges pictorially, which helps to reveal its connections to various other topics in the study of holography and fracton phases. At zero temperature, the dual eight-vertex model is a discrete realization of the bit-thread model, a powerful tool developed to visualize holography. The bit-thread picture combined with subsystem charges can give a quantitative account of the isometry between the bulk and the boundary at finite energy, which is also a key issue for holography. The black hole microscopic degrees of freedom can be identified in this picture, which turn out to be encoded non-locally on the horizon. The eight-vertex model proves to be a very helpful venue to improve our understanding of the hyperbolic fracton model as a toy model of holography.