Karhunen-Loeve Decomposition Research Articles

Stochastic Modeling of Two-Phase Transport in Fractured Porous Media Under Geological Uncertainty Using an Improved Probabilistic Collocation Method

Summary Simulation of multiphase transport through fractured porous media is highly affected by the uncertainty in fracture distribution and matrix block size that arises from inherent heterogeneity. To quantify the effect of such uncertainties on displacement performance in porous media, the probabilistic collocation method (PCM) has been applied as a feasible and accurate approach. However, propagation of uncertainty during the simulation of unsteady-state transport through porous media could not be computed by this method or even by the direct-sampling Monte Carlo (MC) approach. Therefore, with this research, we implement a novel numerical modeling workflow that improves PCM on sparse grids and combines it with the Smolyak algorithm for selection of collocation points sets, Karhunen-Loeve (KL) decomposition, and polynomial chaos expansion (PCE) to compute the uncertainty propagation in oil-gas flow through fractured porous media in which gravity drainage force is enabled. The effect of uncertainty in the vertical dimension of matrix blocks, which are frequently an uncertain and history-matching parameter, on simulation results of randomly synthetic 3D fractured media is explored. The developed numerical model is innovatively coupled with solving governing deterministic partial differential equations (PDEs) to compute uncertainty propagation from the first timestep to the last timestep of the simulation. The uncertainty interval and aggregation of uncertainty in ultimate recovery are quantified, and statistical moments for simulation outputs are presented at each timestep. The results reveal that the model properly quantifies uncertainty and extremely reduces central processing unit (or CPU) time in comparison with MC simulation.

The chaotic milling behaviors of interacting swarms after collision

We consider the problem of characterizing the dynamics of interacting swarms after they collide and form a stationary center of mass. Modeling efforts have shown that the collision of near head-on interacting swarms can produce a variety of post-collision dynamics including coherent milling, coherent flocking, and scattering behaviors. In particular, recent analysis of the transient dynamics of two colliding swarms has revealed the existence of a critical transition whereby the collision results in a combined milling state about a stationary center of mass. In the present work, we show that the collision dynamics of two swarms that form a milling state transitions from periodic to chaotic motion as a function of the repulsive force strength and its length scale. We used two existing methods as well as one new technique: Karhunen–Loeve decomposition to show the effective modal dimension chaos lives in, the 0-1 test to identify chaos, and then constrained correlation embedding to show how each swarm is embedded in the other when both swarms combine to form a single milling state after collision. We expect our analysis to impact new swarm experiments which examine the interaction of multiple swarms.

Analysis of the Model Characteristics in the North Atlantic Simulated by the NEMO Model with Data Assimilation

The main aim of this work is to study the spatial–temporal variability of the model’s physical and spectral characteristics in the process of assimilation of observed ocean surface height data from the AVISO (Archiving, Validating and Interpolation Satellite Observation) archive in combination with the NEMO (Nucleus for European Modeling of the Ocean) ocean circulation model for a period of two months. For data assimilation, the GKF (Generalized Kalman filter) method, previously developed by the authors, is used. The purpose of this work is to study the spatial–temporal structure of the simulated characteristics using decomposition into eigenvalues and eigenvectors (Karhunen–Loeve decomposition method). The feature of the GKF method is the fact that the constructed Kalman weight matrix multiplied by the vector of observational data can be represented as a weighted sum of eigenvectors and eigenvalues (spectral characteristics of the matrix), which describe the spatial and temporal structure of corrections to the model. The main investigations are focused on the North Atlantic. Their variability in time and space is estimated in this study. Calculations of the main ocean characteristics, such as the surface height, temperature, salinity, and the current velocities on the surface and in the depths, both with and without assimilation of observational data, over a time interval of 60 days, were performed by using a high-performance computing system. The calculation results have shown that the main spatial variability of characteristics after data assimilation is consistent with the localization of the currents in the North Atlantic.

Interference Mitigation for Metro Train Location Based on Karhunen–Loeve Decomposition

We propose a communication positioning integrated signal (CPIS) to achieve sub-meter positioning of metro trains. However, the electromagnetic environment of metro trains is complex, and the interference of radio frequency signals will reduce the accuracy of train positioning and affect the safe operation of trains. In this paper, an interference mitigation method for metro train positioning based on Karhunen–Loeve decomposition is proposed. The signal is orthogonally expanded, and the eigenvalues of the interference signal are obtained according to the distribution characteristics of the eigenvalues. The reconstructed interference signal is then subtracted from the received signal to achieve interference suppression. The experimental results demonstrate the effectiveness of the method in interference scenarios.

Optimal stochastic modeling in random media propagation: Dynamically orthogonal parabolic equations?

Reliable underwater acoustic propagation is challenging due to complex ocean dynamics such as internal-waves and to the uncertain larger-scale ocean physics, acoustics, bathymetry, and seabed fields. For accurate acoustic propagation, capturing the important environmental uncertainties and variabilities and predicting the probability distributions of the acoustic pressure field is then what matters. Prior works towards addressing this goal include (i) wave propagation in random media techniques such as perturbation methods, path integral theory, and coupled-mode transport theory, and (ii) probabilistic modeling techniques such as Monte Carlo sampling and Polynomial Chaos expansions. Recently, we developed a novel technique called the Dynamically Orthogonal Parabolic Equations (DO-ParEq) which represent the sound speed, density, bathymetry, and acoustic pressure fields using optimal dynamic Karhunen-Loeve decompositions. The DO-ParEq are range-evolving partial and stochastic differential equations preserving acoustic nonlinearities and non-Gaussian properties. In this presentation, we showcase the theoretical and computational advantages of the DO-ParEq framework compared to the state-of-the-art techniques in the Pekeris waveguide and wedge benchmark problems, in addition to a realistic ocean example in the New York Bight region.

Modeling of random processes based on Karhunen-Loeve decomposition

The problem of digital modeling of random processes with given either correlation function or spectral density of the process is considered. These functions of a random process are interconnected by the Wiener–Khinchin theorem. The solution of one function can be used to solve another. The development of a mathematical representation of a stationary random process with a given correlation function based on the Karhunen-Loeve transformation, which is most often used to decorrelate the original process in order to describe it more concisely (data compression problem), has been completed. It is proposed to use the Karhunen–Loev transformation to impart the required correlation properties to the original uncorrelated random process by inverting (converting) this transformation. The form of the required transformation for a discrete (in time) representation of input and output processes of various lengths and methods for ensuring the required modeling accuracy are substantiated. A procedure for obtaining a correlation function from a given spectral density of a simulated random process is presented. An experimental study of the proposed method was carried out in the course of computer simulation in the Mathcad package which simplified the solution of the required computational problems. The initial random process was obtained as a sequence of independent (and, therefore, uncorrelated) random numbers, and the output process, as a result of the transformation, was obtained in the work. The calculated approximate correlation function is compared with the given one and the error variance is determined. The results of modeling random processes with given correlation functions and a homogeneous Markov process with a given transition probability are given as well as an example of the transition from a given spectral density of a random process to its correlation function. The results obtained confirm the effectiveness and feasibility of the developed modeling methods which will allow them to be used in computer research and design of various systems.

Efficient GNSS Jamming Mitigation Using the Marcenko Pastur Law and Karhunen–Loeve Decomposition

Efficient GNSS Jamming Mitigation Using the Marcenko Pastur Law and Karhunen–Loeve Decomposition

Dissimilarity Analysis-Based Multimode Modeling for Complex Distributed Parameter Systems

For complex distributed parameter systems (DPSs) with strong nonlinearities and time-varying dynamics, the conventional spatiotemporal modeling methods become ill-suited since the elementary assumption that the process data follow a unimodal Gaussian distribution usually becomes invalid. In this paper, a multimode method is proposed for modeling of such systems. First, the original operating space is partitioned along the time dimension into several subspaces via modified dissimilarity analysis. Each subspace represents the local spatiotemporal characteristics of the original system. Second, the Karhunen–Loeve decomposition (KLD)-based spatiotemporal modeling approach is applied to approximate the local dynamics of each subspace. Finally, an ensemble model is obtained using the soft weighting sum of the local ones, where the corresponding weights are calculated by principal component regression. By properly decomposing the original space into several local parts, the ensemble model is capable of handling the strong nonlinearities and time-varying dynamics of the system. The validity and efficiency of the proposed method are verified on two representative applications: 1) a one-dimensional parabolic catalytic rod and 2) a two-dimensional curing thermal process. The experimental results show that the proposed method provides a superior performance regarding modeling accuracy compared to several baselines.

Про моделювання гауссового процесу із точністю та надійністю в просторі $L_p([0,T])$

Стаття присвячена моделюванню випадкового процесу із наперд заданоюВ останнi часи теорiя стохастичних процесiв та полiв широко використовується в рiзних галузях науки i не тiльки в природничих сферах, а саме її використання є важливим у фiзицi, радiофiзицi, iнформатицi, програмнiй iнженерiї, соцiологiї, бiологiї, океаналогiї, метеорологiї, фiнансовiй математицi, теорiї прийняття рiшень, системах масового обслуговування тощо. Тому актуальною проблемою для ймовiрносникiв є побудова математичної моделi випадкового процесу або поля та вивчення її аналiтичних властивостей. Проблеми чисельного моделювання стають особливо важливими завдяки потужним можливостям комп’ютерних технологiй, що дозволяють створювати програмнi засоби для моделювання та для передбачення поведiнки випадкового процесу. Пiд статистичним моделюванням ми розумiємо комп’ютерну реалiзацiю спочатку випадкової величини, а потiм вже випадкового процесу або поля при заданих характеристиках даних об’єктiв моделювання. Стаття присвячена моделюванню випадкового процесу iз наперед заданою точнiстю та надiйнiстю в банаховому просторi Lp([0, T]). Припускається, що випадковий процес є стацiонарним гауссовим iз вiдомою скiнченною коварiацiйною функцiєю. Якщо випадковий процес подано як збiжний у середньому квадратичному ряд iз випадковими доданками, то, зазвичай, у якостi моделi можна розглядати скiнченну суми перших доданкiв, тобто зрiзку ряду. Тому, перша проблема, яка виникає у статтi, як розкласти випадковий процес у ряд при вiдомiй коварiацiйнiй функцiї. Для цього у статтi використовується Теорема Карунена-Лоєва i для побудови моделi застосовуємо розклад Карунена-Лоєва випадкового процесу. У данiй роботi особливу увагу придiлено точностi та надiйностi побудованої моделi. Це означає, що спочатку ми будуємо модель, а потiм її перевiряємо за допомогою певних тестiв на адекватнiсть iз заданими вхiдними параметрами. Отже, знаючи наперед точнiсть та надiйнiсть та з використанням доведених у статтi результатiв для перевiрки адекватностi, можна стверджувати, що побудова модель буде гарно описувати початковий випадковий процес.

Study of the energy convergence of the Karhunen-Loeve decomposition applied to the large-eddy simulation of a high-Reynolds-number pressure-driven boundary layer

The convergence of the Karunen-Lo\`eve (KL) decomposition in high-Reynolds-number boundary layers is investigated using large-eddy simulations. It is found that the KL dimension, namely, the number of KL modes necessary to represent 90% of the turbulent kinetic energy, is up to 3 orders of magnitude higher than values commonly reported in earlier studies. This indicates that more caution should be exercised when considering the convergence of a proper orthogonal decomposition basis in high-Reynolds-number boundary layers, and it illustrates once more the challenges associated with representing turbulence in a low-dimensional basis.

Spatial–Temporal Variability of the Calculated Characteristics of the Ocean in the Arctic Zone of Russia by Using the NEMO Model with Altimetry Data Assimilation

The spatial–temporal variability of the calculated characteristics of the ocean in the Arctic zone of Russia is studied. In this study, the known hydrodynamic model of the ocean Nucleus for European Modelling of the Ocean (NEMO) is used with assimilation of observation data on the sea surface height taken from the Archiving, Validating and Interpolation Satellite Observation (AVISO) archive. We use the Generalized Kalman filter (GKF) method, developed earlier by the authors of this study, in conjunction with the method of decomposition of symmetric matrices into empirical orthogonal functions (EOF, Karhunen–Loeve decomposition). The investigations are focused mostly on the northern seas of Russia. The main characteristics of the ocean, such as the current velocity, sea surface height, and sea surface temperature are calculated with data assimilation (DA) and without DA (the control calculation). The calculation results are analyzed and their spatial–temporal variability over a time period of 14 days is studied. It is shown that the main spatial variability of characteristics after DA is in good agreement with the localization of currents in the North Atlantic and in the Arctic zone of Russia. The contribution of each of the eigenvectors and eigenvalues of the covariation matrix to the spatial–temporal variability of the calculated characteristics is shown by using the EOF analysis.

Analysis of the Stochastic Quarter-Five Spot Problem Using Polynomial Chaos.

Analysis of fluids in porous media is of great importance in many applications. There are many mathematical models that can be used in the analysis. More realistic models should account for the stochastic variations of the model parameters due to the nature of the porous material and/or the properties of the fluid. In this paper, the standard porous media problem with random permeability is considered. Both the deterministic and stochastic problems are analyzed using the finite volume technique. The solution statistics of the stochastic problem are computed using both Polynomial Chaos Expansion (PCE) and the Karhunen-Loeve (KL) decomposition with an exponential correlation function. The results of both techniques are compared with the Monte Carlo sampling to verify the efficiency. Results have shown that PCE with first order polynomials provides higher accuracy for lower (less than 20%) permeability variance. For higher permeability variance, using higher-order PCE considerably improves the accuracy of the solution. The PCE is also combined with KL decomposition and faster convergence is achieved. The KL-PCE combination should carefully choose the number of KL decomposition terms based on the correlation length of the random permeability. The suggested techniques are successfully applied to the quarter-five spot problem.

Spatiotemporal Modeling for Nonlinear Distributed Thermal Processes Based on KL Decomposition, MLP and LSTM Network

Estimation of absolute temperature distributions is crucial for many thermal processes in the nonlinear distributed parameter systems, such as predicting the curing temperature distribution of the chip, the temperature distribution of the catalytic rod, and so on. In this work, a spatiotemporal model based on the Karhunen-Loeve (KL) decomposition, the multilayer perceptron (MLP), and the long short-term memory (LSTM) network, named KL-MLP-LSTM, is developed for estimating temperature distributions with a three-step procedure. Firstly, the infinite-dimensional model is transformed into a finite-dimensional model, where the KL decomposition method is used for dimension reduction and spatial basis functions extraction. Secondly, a novel MLP-LSTM hybrid time series model is constructed to deal with the two inherently coupled nonlinearities. Finally, the spatiotemporal temperature distribution model can be reconstructed through spatiotemporal synthesis. The effectiveness of the proposed model is validated by the data from a snap curing oven thermal process. Satisfactory agreement between the results of the current model and the other well-established model shows that the KL-MLP-LSTM model is reliable for estimating the temperature distributions during the thermal process.

A qualitative recognition method based on Karhunen–Loeve decomposition for near-infrared (NIR) and mid infrared (MIR) spectroscopy analysis

Qualitative recognition is an important research area of NIR and MIR spectroscopy analyses.

A Karhunen-Loeve Galerkin Online Modeling Approach for the Thermal Dynamics of Li-Ion Batteries

The thermal dynamics of Li-ion batteries are very complicated, and the battery temperature is spatially distributed imbalanced from the battery interior to the surface. The thermal dynamics are commonly modeled by partial differential equations (PDEs); however, parameter identification for PDEs is difficult and time consuming. In this work, a Karhunen-Loeve Galerkin method is proposed to obtain a simple but effective low-order model for the distributed thermal dynamics of Li-ion batteries. The Karhunen-Loeve decomposition method is applied to capture the most representative spatial modes of the dynamics, while the Galerkin method is used to obtain the corresponding temporal modes. All the uncertain physical parameters in the temporal modes are identified by the Levenberg-Marquardt algorithm. The updated temporal modes synthesized with spatial modes can offer a fast estimation of the temperature distribution on the battery surface and thus has the potential to provide distributed temperature prediction for the battery management system. The proposed modeling scheme is tested on a 60Ah Li-ion battery cell, and the simulation result shows an excellent match in the temperature distribution and a faster computing speed than the rigorous physical model.

Cosmological discordances. III. More on measure properties, large-scale-structure constraints, the Hubble constant andPlanckdata

Consistency between cosmological data sets is essential for ongoing and future cosmological analyses. We first investigate the questions of stability and applicability of some moment-based inconsistency measures to multiple data sets. We show that the recently introduced index of inconsistency (IOI) is numerically stable while it can be applied to multiple data sets. We use an illustrative construction of constraints as well as an example with real data sets (i.e. WMAP versus Planck) to show some limitations of the application of the Karhunen-Loeve decomposition to discordance measures. Second, we perform various consistency analyzes using IOI between multiple current data sets while \textit{working with the entire common parameter spaces}. We find current Large-Scale-Structure (LSS) data sets (Planck CMB lensing, DES lensing-clustering and SDSS RSD) all to be consistent with one another. This is found to be not the case for Planck temperature (TT) versus polarization (TE,EE) data, where moderate inconsistencies are present. Noteworthy, we find a strong inconsistency between joint LSS probes and Planck with IOI=5.27, and a moderate tension between DES and Planck with IOI=3.14. Next, using the IOI metric, we compare the Hubble constant from five independent probes. We confirm previous strong tensions between local measurement (SH0ES) and Planck as well as between H0LiCOW and Planck, but also find new strong tensions between SH0ES measurement and the joint LSS probes with IOI=6.73 (i.e. 3.7-$\sigma$ in 1D) as well as between joint LSS and combined probes SH0ES+H0LiCOW with IOI=8.59 (i.e. 4.1-$\sigma$ in 1D). Whether due to systematic effects in the data sets or problems with the underlying model, sources of these old and new tensions need to be identified and dealt with.

Quantifying forcing uncertainties in the hydrodynamics of the Gironde estuary

High tide, combined with high meteorological surge levels and discharges in the Garonne and Dordogne rivers in the Gironde estuary (South-West France), may lead to high water levels and flooding near the Blayais Nuclear Power Plant and the city of Bordeaux, with significant economic and social impacts. A global sensitivity analysis (GSA) was performed with a Telemac2D numerical model currently used for operational water level forecasts. The major sources of uncertainties were identified by computing the Sobol’ indices for uncertain inputs with an analysis of variance (ANOVA) approach for a 7-day storm event in 2003. The generation of the GSA ensemble of simulations consists of sampling scalar and field random variables: constant and uniform friction coefficients, as well as time-varying hydrological and maritime forcings. The temporal perturbation of time-dependent upstream hydrological and downstream maritime forcings is assumed to be represented by a Gaussian process characterized by a correlation time scale calculated from observations. A Karhunen-Loeve decomposition was then applied to retain a limited number of eigenmodes. The GSA is performed for 20 random variables using GENCI HPC computational resources for task parallelism and domain decomposition. This requires the use of 250,000 runs for an elapsed simulation time of 101 days on 32,768 cores. The performance of the ensemble was assessed with a rank diagram and a reliability curve in comparison with a set of measured water levels at 12 observing stations along the estuary. It was shown that, for this event, the maritime boundary conditions and the Strickler coefficients have a predominant role along the length of the estuary with an influence driven by the tidal cycle. In the upstream fluvial areas, the friction coefficient and hydrological inputs are predominant.

Incremental Spatiotemporal Learning for Online Modeling of Distributed Parameter Systems

An incremental spatiotemporal learning scheme is proposed for online modeling of distributed parameter systems (DPSs). A novel incremental learning method is developed to recursively update the spatial basis functions and the corresponding temporal model based on the Karhunen–Loeve decomposition for time-space separation. The time-space synthesis continually evolves by adding new increment data with more updated information and revising the existing parameters of the dynamic system. In this way, the spatiotemporal structure is inherited and updated efficiently as output data increases over time. The adaptive nature of this evolving structure makes it promising for online modeling of DPSs under streaming data environment. The proposed incremental modeling scheme is evaluated on the classical benchmark of a catalytic rod problem. The simulation results demonstrate the viability and efficiency of the proposed method for online modeling of DPSs.

Review for order reduction based on proper orthogonal decomposition and outlooks of applications in mechanical systems

Abstract This paper presents a review of proper orthogonal decomposition (POD) methods for order reduction in a variety of research areas. The historical development and basic mathematical formulation of the POD method are introduced. POD for parametric dynamic systems is introduced, and a physical interpretation of the POD approach based on the proper orthogonal modes (POMs) is presented. The equivalence between POD and three other order reduction methods is discussed: the first alternative method is singular value decomposition (SVD), the second is principal component analysis (PCA), and the third is Karhunen-Loeve decomposition (KLD). A classification of POD methods is described based on the parameter adaptation and sampling. Actual applications of POD methods for order reduction in engineering systems are illustrated. Finally, outlooks on the use of POD methods in high-dimensional nonlinear dynamic systems are presented in more detail to provide direct guidance for researchers in various areas of engineering.

Discrimination of Mild Alzheimer’s Disease Patients using Cluster Analysis of Information Transmission in EEG

A description method of information transmission was presented and applied to EEGs recorded from groups of mild Alzheimer’s disease (AD) patients and normal controls. To monitor the general interdependence of EEG channels, time-delayed mutual information was computed using the Fraser- Swinney algorithm. To manage the stochastic nature of mutual information and global connectivity of the brain, the modified Karhunen-Loeve decomposition (information transmission map: ITM), which was conducted as a quantification method of the resulting stochastic mutual information, was used. The integration of mutual information with Omega complexity elevated the discrimination power for mild AD. This provides a good procedure, capable of capturing the fine structure in the global information transmission between brain parts, compared with mutual information calculation alone. Compared with normal aged controls, mild AD patients showed significantly different short-time clustered behavior in the information transmission from the EEG. They also show a peculiar pattern of information flows from the successive snapshot monitoring. The observations suggest a discrimination of the mild AD patients from the normal aged controls.