arXiv subjects

Ramakrishna Tipireddy

Publications and source records attributed to Ramakrishna Tipireddy.

14 recordsLinked to original sources

arXiv2023

Conditional Korhunen-Loéve regression model with Basis Adaptation for high-dimensional problems: uncertainty quantification and inverse modeling

We propose a methodology for improving the accuracy of surrogate models of the observable response of physical systems as a function of the systems' spatially heterogeneous parameter fields with applications to uncertainty quantification and parameter estimation in high-dimensional problems. Practitioners often formulate finite-dimensional representations of spatially heterogeneous parameter fields using truncated unconditional Karhunen-Loéve expansions (KLEs) for a certain choice of unconditional covariance kernel and construct surrogate models of the observable response with respect to the random variables in the KLE. When direct measurements of the parameter fields are available, we propose improving the accuracy of these surrogate models by representing the parameter fields via conditional Karhunen-Loéve expansions (CKLEs). CKLEs are constructed by conditioning the covariance kernel of the unconditional expansion on the direct measurements via Gaussian process regression and then truncating the corresponding KLE. We apply the proposed methodology to constructing surrogate models via the Basis Adaptation (BA) method of the stationary hydraulic head response, measured at spatially discrete observation locations, of a groundwater flow model of the Hanford Site, as a function of the 1,000-dimensional representation of the model's log-transmissivity field. We find that BA surrogate models of the hydraulic head based on CKLEs are more accurate than BA surrogate models based on unconditional expansions for forward uncertainty quantification tasks. Furthermore, we find that inverse estimates of the hydraulic transmissivity field computed using CKLE-based BA surrogate models are more accurate than those computed using unconditional BA surrogate models.

cs.LG↗

arXiv2023

Extending Conformal Prediction to Hidden Markov Models with Exact Validity via de Finetti's Theorem for Markov Chains

Conformal prediction is a widely used method to quantify the uncertainty of a classifier under the assumption of exchangeability (e.g., IID data). We generalize conformal prediction to the Hidden Markov Model (HMM) framework where the assumption of exchangeability is not valid. The key idea of the proposed method is to partition the non-exchangeable Markovian data from the HMM into exchangeable blocks by exploiting the de Finetti's Theorem for Markov Chains discovered by Diaconis and Freedman (1980). The permutations of the exchangeable blocks are viewed as randomizations of the observed Markovian data from the HMM. The proposed method provably retains all desirable theoretical guarantees offered by the classical conformal prediction framework in both exchangeable and Markovian settings. In particular, while the lack of exchangeability introduced by Markovian samples constitutes a violation of a crucial assumption for classical conformal prediction, the proposed method views it as an advantage that can be exploited to improve the performance further. Detailed numerical and empirical results that complement the theoretical conclusions are provided to illustrate the practical feasibility of the proposed method.

stat.ME↗

arXiv2021

Lorenz System State Stability Identification using Neural Networks

Nonlinear dynamical systems such as Lorenz63 equations are known to be chaotic in nature and sensitive to initial conditions. As a result, a small perturbation in the initial conditions results in deviation in state trajectory after a few time steps. The algorithms and computational resources needed to accurately identify the system states vary depending on whether the solution is in transition region or not. We refer to the transition and non-transition regions as unstable and stable regions respectively. We label a system state to be stable if it's immediate past and future states reside in the same regime. However, at a given time step we don't have the prior knowledge about whether system is in stable or unstable region. In this paper, we develop and train a feed forward (multi-layer perceptron) Neural Network to classify the system states of a Lorenz system as stable and unstable. We pose this task as a supervised learning problem where we train the neural network on Lorenz system which have states labeled as stable or unstable. We then test the ability of the neural network models to identify the stable and unstable states on a different Lorenz system that is generated using different initial conditions. We also evaluate the classification performance in the mismatched case i.e., when the initial conditions for training and validation data are sampled from different intervals. We show that certain normalization schemes can greatly improve the performance of neural networks in especially these mismatched scenarios. The classification framework developed in the paper can be a preprocessor for a larger context of sequential decision making framework where the decision making is performed based on observed stable or unstable states.

math.DS↗

arXiv2021

Time-dependent stochastic basis adaptation for uncertainty quantification

We extend stochastic basis adaptation and spatial domain decomposition methods to solve time varying stochastic partial differential equations (SPDEs) with a large number of input random parameters. Stochastic basis adaptation allows the determination of a low dimensional stochastic basis representation of a quantity of interest (QoI). Extending basis adaptation to time-dependent problems is challenging because small errors introduced in the previous time steps of the low dimensional approximate solution accumulate over time and cause divergence from the true solution. To address this issue we have introduced an approach where the basis adaptation varies at every time step so that the low dimensional basis is adapted to the QoI at that time step. We have coupled the time-dependent basis adaptation with domain decomposition to further increase the accuracy in the representation of the QoI. To illustrate the construction, we present numerical results for one-dimensional time varying linear and nonlinear diffusion equations with random space-dependent diffusion coefficients. Stochastic dimension reduction techniques proposed in the literature have mainly focused on quantifying the uncertainty in time independent and scalar QoI. To the best of our knowledge, this is the first attempt to extend dimensional reduction techniques to time varying and spatially dependent quantities such as the solution of SPDEs.

math.NA↗

arXiv2020

An efficient epistemic uncertainty quantification algorithm for a class of stochastic models: A post-processing and domain decomposition framework

Partial differential equations (PDEs) are fundamental for theoretically describing numerous physical processes that are based on some input fields in spatial configurations. Understanding the physical process, in general, requires computational modeling of the PDE. Uncertainty in the computational model manifests through lack of precise knowledge of the input field or configuration. Uncertainty quantification (UQ) in the output physical process is typically carried out by modeling the uncertainty using a random field, governed by an appropriate covariance function. This leads to solving high-dimensional stochastic counterparts of the PDE computational models. Such UQ-PDE models require a large number of simulations of the PDE in conjunction with samples in the high-dimensional probability space, with probability distribution associated with the covariance function. Those UQ computational models having explicit knowledge of the covariance function are known as aleatoric UQ (AUQ) models. The lack of such explicit knowledge leads to epistemic UQ (EUQ) models, which typically require solution of a large number of AUQ models. In this article, using a surrogate, post-processing, and domain decomposition framework with coarse stochastic solution adaptation, we develop an offline/online algorithm for efficiently simulating a class of EUQ-PDE models.

math.NA↗

arXiv2020

Physics-Informed Gaussian Process Regression for Probabilistic States Estimation and Forecasting in Power Grids

Real-time state estimation and forecasting is critical for efficient operation of power grids. In this paper, a physics-informed Gaussian process regression (PhI-GPR) method is presented and used for probabilistic forecasting and estimating the phase angle, angular speed, and wind mechanical power of a three-generator power grid system using sparse measurements. In standard data-driven Gaussian process regression (GPR), parameterized models for the prior statistics are fit by maximizing the marginal likelihood of observed data, whereas in PhI-GPR, we compute the prior statistics by solving stochastic equations governing power grid dynamics. The short-term forecast of a power grid system dominated by wind generation is complicated by the stochastic nature of the wind and the resulting uncertain mechanical wind power. Here, we assume that the power-grid dynamic is governed by the swing equations, and we treat the unknown terms in the swing equations (specifically, the mechanical wind power) as random processes, which turns these equations into stochastic differential equations. We solve these equations for the mean and variance of the power grid system using the Monte Carlo simulations method. We demonstrate that the proposed PhI-GPR method can accurately forecast and estimate both observed and unobserved states, including the mean behavior and associated uncertainty. For observed states, we show that PhI-GPR provides a forecast comparable to the standard data-driven GPR, with both forecasts being significantly more accurate than the autoregressive integrated moving average (ARIMA) forecast. We also show that the ARIMA forecast is much more sensitive to observation frequency and measurement errors than the PhI-GPR forecast.

stat.ML↗

arXiv2020

Bayesian phase estimation with adaptive grid refinement

We introduce a novel Bayesian phase estimation technique based on adaptive grid refinement method. This method automatically chooses the number particles needed for accurate phase estimation using grid refinement and cell merging strategies such that the total number of particles needed at each step is minimal. The proposed method provides a powerful alternative to traditional sampling based sequential Monte Carlo method which tend to fail in certain instances such as when the posterior distribution is bimodal. We also combine grid based and sampling based methods as hybrid particle filter where grid based method can be used to estimate a small but dominant set of parameters and Liu-West (LW) based SMC for the remaining set of parameters. Principal kurtosis analysis can be used to decide the choice of parameters for grid refinement method and for sampling based methods. We provide numerical results comparing the performance of the proposed grid refinement method with Liu-West resampling based SMC. Numerical results suggest that the proposed method is quite promising for quantum phase estimation. It can be easily adapted to Hamiltonian learning which is a very useful technique for estimating unknown parameters of a Hamiltonian and for characterizing unknown quantum devices.

quant-ph↗

arXiv2020

FPDetect: Efficient Reasoning About Stencil Programs Using Selective Direct Evaluation

We present FPDetect, a low overhead approach for detecting logical errors and soft errors affecting stencil computations without generating false positives. We develop an offline analysis that tightly estimates the number of floating-point bits preserved across stencil applications. This estimate rigorously bounds the values expected in the data space of the computation. Violations of this bound can be attributed with certainty to errors. FPDetect helps synthesize error detectors customized for user-specified levels of accuracy and coverage. FPDetect also enables overhead reduction techniques based on deploying these detectors coarsely in space and time. Experimental evaluations demonstrate the practicality of our approach.

cs.DC↗

arXiv2019

Electric Load and Power Forecasting Using Ensemble Gaussian Process Regression

We propose a new forecasting method for predicting load demand and generation scheduling. Accurate week-long forecasting of load demand and optimal power generation is critical for efficient operation of power grid systems. In this work, we use a synthetic data set describing a power grid with 700 buses and 134 generators over a 365-days period with data synthetically generated at an hourly rate. The proposed approach for week-long forecasting is based on the Gaussian process regression (GPR) method, with prior covariance matrices of the quantities of interest (QoI) computed from ensembles formed by up to twenty preceding weeks of QoI observations. Then, we use these covariances within the GPR framework to forecast the QoIs for the following week. We demonstrate that the the proposed ensemble GPR (EGPR) method is capable of accurately forecasting weekly total load demand and power generation profiles. The EGPR method is shown to outperform traditional forecasting methods including the standard GPR and autoregressive integrated moving average (ARIMA) methods.

cs.LG↗

arXiv2019

Conditional Karhunen-Loève expansion for uncertainty quantification and active learning in partial differential equation models

We use a conditional Karhunen-Loève (KL) model to quantify and reduce uncertainty in a stochastic partial differential equation (SPDE) problem with partially-known space-dependent coefficient, $Y(x)$. We assume that a small number of $Y(x)$ measurements are available and model $Y(x)$ with a KL expansion. We achieve reduction in uncertainty by conditioning the KL expansion coefficients on measurements. We consider two approaches for conditioning the KL expansion: In Approach 1, we condition the KL model first and then truncate it. In Approach 2, we first truncate the KL expansion and then condition it. We employ the conditional KL expansion together with Monte Carlo and sparse grid collocation methods to compute the moments of the solution of the SPDE problem. Uncertainty of the problem is further reduced by adaptively selecting additional observation locations using two active learning methods. Method 1 minimizes the variance of the PDE coefficient, while Method 2 minimizes the variance of the solution of the PDE. We demonstrate that conditioning leads to dimension reduction of the KL representation of $Y(x)$. For a linear diffusion SPDE with uncertain log-normal coefficient, we show that Approach 1 provides a more accurate approximation of the conditional log-normal coefficient and solution of the SPDE than Approach 2 for the same number of random dimensions in a conditional KL expansion. Furthermore, Approach 2 provides a good estimate for the number of terms of the truncated KL expansion of the conditional field of Approach 1. Finally, we demonstrate that active learning based on Method 2 is more efficient for uncertainty reduction in the SPDE's states (i.e., it leads to a larger reduction of the variance) than active learning using Method 2.

math.PR↗

arXiv2019

Solvers and precondtioners based on Gauss-Seidel and Jacobi algorithms for non-symmetric stochastic Galerkin system of equations

In this work, solvers and preconditioners based on Gauss-Seidel and Jacobi algorithms are explored for stochastic Galerkin discretization of partial differential equations (PDEs) with random input data. Gauss-Seidel and Jacobi algorithms are formulated such that the existing software is leveraged in the computational effort. These algorithms are also used as preconditioners to Krylov iterative methods. The solvers and preconditiners are compared with Krylov based iterative methods with the traditional mean-based preconditioner [13] and Kronecker-product preconditioner [17] by solving a steady state state advection-diffusion equation, which upon discretization, results in a non-symmetric positive definite matrix on left-hand-side. Numerical results show that an approximate version of Gauss-Seidel algorithm is a good preconditioner for GMRES to solve non-symmetric Galerkin system of equations.

math.NA↗

arXiv2019

A comparative study of physics-informed neural network models for learning unknown dynamics and constitutive relations

We investigate the use of discrete and continuous versions of physics-informed neural network methods for learning unknown dynamics or constitutive relations of a dynamical system. For the case of unknown dynamics, we represent all the dynamics with a deep neural network (DNN). When the dynamics of the system are known up to the specification of constitutive relations (that can depend on the state of the system), we represent these constitutive relations with a DNN. The discrete versions combine classical multistep discretization methods for dynamical systems with neural network based machine learning methods. On the other hand, the continuous versions utilize deep neural networks to minimize the residual function for the continuous governing equations. We use the case of a fedbatch bioreactor system to study the effectiveness of these approaches and discuss conditions for their applicability. Our results indicate that the accuracy of the trained neural network models is much higher for the cases where we only have to learn a constitutive relation instead of the whole dynamics. This finding corroborates the well-known fact from scientific computing that building as much structural information is available into an algorithm can enhance its efficiency and/or accuracy.

cs.LG↗

arXiv2018

Physics-informed Machine Learning Method for Forecasting and Uncertainty Quantification of Partially Observed and Unobserved States in Power Grids

We present a physics-informed Gaussian Process Regression (GPR) model to predict the phase angle, angular speed, and wind mechanical power from a limited number of measurements. In the traditional data-driven GPR method, the form of the Gaussian Process covariance matrix is assumed and its parameters are found from measurements. In the physics-informed GPR, we treat unknown variables (including wind speed and mechanical power) as a random process and compute the covariance matrix from the resulting stochastic power grid equations. We demonstrate that the physics-informed GPR method is significantly more accurate than the standard data-driven one for immediate forecasting of generators' angular velocity and phase angle. We also show that the physics-informed GPR provides accurate predictions of the unobserved wind mechanical power, phase angle, or angular velocity when measurements from only one of these variables are available. The immediate forecast of observed variables and predictions of unobserved variables can be used for effectively managing power grids (electricity market clearing, regulation actions) and early detection of abnormal behavior and faults. The physics-based GPR forecast time horizon depends on the combination of input (wind power, load, etc.) correlation time and characteristic (relaxation) time of the power grid and can be extended to short and medium-range times.

eess.SP↗

arXiv2017

Stochastic basis adaptation and spatial domain decomposition for PDEs with random coefficients

We present a novel uncertainty quantification approach for high-dimensional stochastic partial differential equations that reduces the computational cost of polynomial chaos methods by decomposing the computational domain into non-overlapping subdomains and adapting the stochastic basis in each subdomain so the local solution has a lower dimensional random space representation. The local solutions are coupled using the Neumann-Neumann algorithm, where we first estimate the interface solution then evaluate the interior solution in each subdomain using the interface solution as a boundary condition. The interior solutions in each subdomain are computed independently of each other, which reduces the operation count from $O(N^\alpha)$ to $O(M^\alpha),$ where $N$ is the total number of degrees of freedom, $M$ is the number of degrees of freedom in each subdomain, and the exponent $\alpha>1$ depends on the uncertainty quantification method used. In addition, the localized nature of solutions makes the proposed approach highly parallelizable. We illustrate the accuracy and efficiency of the approach for linear and nonlinear differential equations with random coefficients.

math.NA↗