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This paper demonstrates the scalability of open-source GPU-accelerated nonlinear programming (NLP) frameworks -- ExaModels.jl and MadNLP.jl -- for solving multi-period alternating current (AC) optimal power flow (OPF) problems on GPUs with high memory capacities (e.g., NVIDIA GH200 with 480 GB of unified memory). There has been a growing interest in solving multi-period AC OPF problems, as the increasingly fluctuating electricity market requires operation planning over multiple periods. These problems, formerly deemed intractable, are now becoming technologically feasible to solve thanks to the advent of high-memory GPU hardware and accelerated NLP tools. This study evaluates the capability of these tools to tackle previously unsolvable multi-period AC OPF instances. Our numerical experiments, run on an NVIDIA GH200, demonstrate that we can solve a multi-period OPF instance with more than 10 million variables up to $10^{-4}$ precision in less than 10 minutes. These results demonstrate the efficacy of the GPU-accelerated NLP frameworks for the solution of extreme-scale multi-period OPF. We provide ExaModelsPower.jl, an open-source modeling tool for multi-period AC OPF models for GPUs.

This paper tackles optimal sensor placement for Bayesian linear inverse problems, a popular version of the more general Optimal Experiment Design (OED) problem, using the D-optimality criterion. This is done by establishing connections between sensor placement and Column Subset Selection Problem (CSSP), which is a well-studied problem in Numerical Linear Algebra (NLA). In particular, we use the Golub-Klema-Stewart (GKS) approach which involves computing the truncated Singular Value Decomposition (SVD) followed by a pivoted QR factorization on the right singular vectors. The algorithms are further accelerated by using randomization to compute the low-rank approximation as well as for sampling the indices. The resulting algorithms are robust, computationally efficient, amenable to parallelization, require virtually no parameter tuning, and come with strong theoretical guarantees. One of the proposed algorithms is also adjoint-free which is beneficial in situations, where the adjoint is expensive to evaluate or is not available. Additionally, we develop a method for data completion without solving the inverse problem. Numerical experiments on model inverse problems involving the heat equation and seismic tomography in two spatial dimensions demonstrate the performance of our approaches., Comment: 31 pages, 9 figures

Strong Constraint 4D Variational (SC-4DVAR) is a data assimilation method that is widely used in climate and weather applications. SC-4DVAR involves solving a minimization problem to compute the maximum a posteriori estimate, which we tackle using the Gauss-Newton method. The computation of the descent direction is expensive since it involves the solution of a large-scale and potentially ill-conditioned linear system, solved using the preconditioned conjugate gradient (PCG) method. To address this cost, we efficiently construct scalable preconditioners using three different randomization techniques, which all rely on a certain low-rank structure involving the Gauss-Newton Hessian. The proposed techniques come with theoretical (probabilistic) guarantees on the condition number, and at the same time, are amenable to parallelization. We also develop an adaptive approach to estimate the sketch size and to choose between the reuse or recomputation of the preconditioner. We demonstrate the performance and effectiveness of our methodology on two representative model problems -- the Burgers and barotropic vorticity equation -- showing a drastic reduction in both the number of PCG iterations and the number of Gauss-Newton Hessian products (after including the preconditioner construction cost)., Comment: 28 pages

Efficiently approximating the probability of system failure has gained increasing importance as expensive simulations begin to play a larger role in reliability quantification tasks in areas such as structural design, power grid design, and safety certification among others. This work derives credible intervals on the probability of failure for a simulation which we assume is a realizations of a Gaussian process. We connect these intervals to binary classification error and comment on their applicability to a broad class of iterative schemes proposed throughout the literature. A novel iterative sampling scheme is proposed which can suggest multiple samples per batch for simulations with parallel implementations. We empirically test our scalable, open-source implementation on a variety simulations including a Tsunami model where failure is quantified in terms of maximum wave hight.

A machine learning technique is proposed for quantifying uncertainty in power system dynamics with spatiotemporally correlated stochastic forcing. We learn one-dimensional linear partial differential equations for the probability density functions of real-valued quantities of interest. The method is suitable for high-dimensional systems and helps to alleviate the curse of dimensionality., Comment: 5 pages, 2 figures, Accepted to 2022 IEEE PES GM

The classification of structural phase transitions as displacive or order-disorder in character is usually based on spectroscopic data above the transition. We use single crystal x-ray diffraction to investigate structural correlations in the quasi-skutterudites, (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$, which have a quantum phase transition at $x\sim0.9$. Three-dimensional pair distribution functions show that the amplitudes of local atomic displacements are temperature-independent below the transition and persist to well above the transition, a signature of order-disorder behavior. The implications for the associated electronic transitions are discussed., Comment: 5 pages, 4 figures

Estimating probability of failure in aerospace systems is a critical requirement for flight certification and qualification. Failure probability estimation involves resolving tails of probability distribution, and Monte Carlo sampling methods are intractable when expensive high-fidelity simulations have to be queried. We propose a method to use models of multiple fidelities that trade accuracy for computational efficiency. Specifically, we propose the use of multifidelity Gaussian process models to efficiently fuse models at multiple fidelity, thereby offering a cheap surrogate model that emulates the original model at all fidelities. Furthermore, we propose a novel sequential \emph{acquisition function}-based experiment design framework that can automatically select samples from appropriate fidelity models to make predictions about quantities of interest in the highest fidelity. We use our proposed approach in an importance sampling setting and demonstrate our method on the failure level set estimation and probability estimation on synthetic test functions as well as two real-world applications, namely, the reliability analysis of a gas turbine engine blade using a finite element method and a transonic aerodynamic wing test case using Reynolds-averaged Navier--Stokes equations. We demonstrate that our method predicts the failure boundary and probability more accurately and computationally efficiently while using varying fidelity models compared with using just a single expensive high-fidelity model., Comment: 35 page, 16 figures

Data assimilation (DA) in the geophysical sciences remains the cornerstone of robust forecasts from numerical models. Indeed, DA plays a crucial role in the quality of numerical weather prediction, and is a crucial building block that has allowed dramatic improvements in weather forecasting over the past few decades. DA is commonly framed in a variational setting, where one solves an optimization problem within a Bayesian formulation using raw model forecasts as a prior, and observations as likelihood. This leads to a DA objective function that needs to be minimized, where the decision variables are the initial conditions specified to the model. In traditional DA, the forward model is numerically and computationally expensive. Here we replace the forward model with a low-dimensional, data-driven, and differentiable emulator. Consequently, gradients of our DA objective function with respect to the decision variables are obtained rapidly via automatic differentiation. We demonstrate our approach by performing an emulator-assisted DA forecast of geopotential height. Our results indicate that emulator-assisted DA is faster than traditional equation-based DA forecasts by four orders of magnitude, allowing computations to be performed on a workstation rather than a dedicated high-performance computer. In addition, we describe accuracy benefits of emulator-assisted DA when compared to simply using the emulator for forecasting (i.e., without DA).

In this work we present a novel technique, based on a trust-region optimization algorithm and second-order trajectory sensitivities, to compute the extreme trajectories of power system dynamic simulations given a bounded set that represents parametric uncertainty. We show how this method, while remaining computationally efficient compared with sampling-based techniques, overcomes the limitations of previous sensitivity-based techniques to approximate the bounds of the trajectories, when the local approximation loses validity because of the nonlinearity. In addition, we show how this method can be adapted to account for those cases in which the initial conditions depend on the uncertain parameter. To conclude, we present several numerical experiments that showcase the accuracy and scalability of the technique, including a demonstration on the IEEE New England test system.

Chance constraints provide a principled framework to mitigate the risk of high-impact extreme events by modifying the controllable properties of a system. The low probability and rare occurrence of such events, however, impose severe sampling and computational requirements on classical solution methods that render them impractical. This work proposes a novel sampling-free method for solving rare chance constrained optimization problems affected by uncertainties that follow general Gaussian mixture distributions. By integrating modern developments in large deviation theory with tools from convex analysis and bilevel optimization, we propose tractable formulations that can be solved by off-the-shelf solvers. Our formulations enjoy several advantages compared to classical methods: their size and complexity is independent of event rarity, they do not require linearity or convexity assumptions on system constraints, and under easily verifiable conditions, serve as safe conservative approximations or asymptotically exact reformulations of the true problem. Computational experiments on linear, nonlinear and PDE-constrained problems from applications in portfolio management, structural engineering and fluid dynamics illustrate the broad applicability of our method and its advantages over classical sampling-based approaches in terms of both accuracy and efficiency.

We develop a novel computational method for evaluating the extreme excursion probabilities arising from random initialization of nonlinear dynamical systems. The method uses excursion probability theory to formulate a sequence of Bayesian inverse problems that, when solved, yields the biasing distribution. Solving multiple Bayesian inverse problems can be expensive; more so in higher dimensions. To alleviate the computational cost, we build machine-learning-based surrogates to solve the Bayesian inverse problems that give rise to the biasing distribution. This biasing distribution can then be used in an importance sampling procedure to estimate the extreme excursion probabilities., Comment: arXiv admin note: text overlap with arXiv:2001.11904

In this paper we focus on the parameter estimation of dynamic load models with stochastic terms, in particular, load models where protection settings are uncertain, such as in aggregated air conditioning units. We show how the uncertainty in the aggregated protection characteristics can be formulated as a stochastic differential equation with process noise. We cast the parameter inversion within a Bayesian parameter estimation framework, and we present methods to include process noise. We demonstrate the benefits of considering stochasticity in the parameter estimation and the risks of ignoring it., Comment: 7 pages, 9 figures, to be published in Electric Power Systems Research PSCC 2020

We develop a novel computational method for evaluating the extreme excursion probabilities arising for random initialization of nonlinear dynamical systems. The method uses a Markov chain Monte Carlo or a Laplace approximation approach to construct a biasing distribution that in turn is used in an importance sampling procedure to estimate the extreme excursion probabilities. The prior and likelihood of the biasing distribution are obtained by using Rice's formula from excursion probability theory. We use Gaussian mixture biasing distributions and approximate the non-Gaussian initial excitation by the method of moments to circumvent the linearity and Gaussianity assumptions needed by excursion probability theory. We demonstrate the effectiveness of this computational framework for nonlinear dynamical systems of up to 100 dimensions.

Fluid flow in the transonic regime finds relevance in aerospace engineering, particularly in the design of commercial air transportation vehicles. Computational fluid dynamics models of transonic flow for aerospace applications are computationally expensive to solve because of the high degrees of freedom as well as the coupled nature of the conservation laws. While these issues pose a bottleneck for the use of such models in aerospace design, computational costs can be significantly minimized by constructing special, structure-preserving surrogate models called reduced-order models. Such models are known to incur huge off-line costs, however, which can sometimes outweigh their potential benefits. Furthermore, their prediction accuracy is known to be poor under transonic flow conditions. In this work, we propose a machine learning method to construct reduced-order models via deep neural networks, and we demonstrate its ability to preserve accuracy with significantly lower offline and online costs. In addition, our machine learning methodology is physics-informed and constrained through the utilization of an interpretable encoding by way of proper orthogonal decomposition. Application to the inviscid transonic flow past the RAE2822 airfoil under varying freestream Mach numbers and angles of attack, as well as airfoil shape parameters with a deforming mesh, shows that the proposed approach adapts to high-dimensional parameter variation well. Notably, the proposed framework precludes knowledge of numerical operators utilized in the data generation phase, thereby demonstrating its potential utility in fast exploration of design space for diverse engineering applications.

Rapid simulations of advection-dominated problems are vital for multiple engineering and geophysical applications. In this paper, we present a long short-term memory neural network to approximate the nonlinear component of the reduced-order model (ROM) of an advection-dominated partial differential equation. This is motivated by the fact that the nonlinear term is the most expensive component of a successful ROM. For our approach, we utilize a Galerkin projection to isolate the linear and the transient components of the dynamical system and then use discrete empirical interpolation to generate training data for supervised learning. We note that the numerical time-advancement and linear-term computation of the system ensure a greater preservation of physics than does a process that is fully modeled. Our results show that the proposed framework recovers transient dynamics accurately without nonlinear term computations in full-order space and represents a cost-effective alternative to solely equation-based ROMs.

We present an adaptive grid matrix-free operator approximation scheme based on a "product-convolution" interpolation of convolution operators. This scheme is appropriate for operators that are locally translation-invariant, even if these operators are high-rank or full-rank. Such operators arise in Schur complement methods for solving partial differential equations (PDEs), as Hessians in PDE-constrained optimization and inverse problems, as integral operators, as covariance operators, and as Dirichlet-to-Neumann maps. Constructing the approximation requires computing the impulse responses of the operator to point sources centered on nodes in an adaptively refined grid of sample points. A randomized a-posteriori error estimator drives the adaptivity. Once constructed, the approximation can be efficiently applied to vectors using the fast Fourier transform. The approximation can be efficiently converted to hierarchical matrix ($H$-matrix) format, then inverted or factorized using scalable $H$-matrix arithmetic. The quality of the approximation degrades gracefully as fewer sample points are used, allowing cheap lower quality approximations to be used as preconditioners. This yields an automated method to construct preconditioners for locally translation-invariant Schur complements. We directly address issues related to boundaries and prove that our scheme eliminates boundary artifacts. We test the scheme on a spatially varying blurring kernel, on the non-local component of an interface Schur complement for the Poisson operator, and on the data misfit Hessian for an advection dominated advection-diffusion inverse problem. Numerical results show that the scheme outperforms existing methods., Comment: Submitted to SISC

Complex numerical weather prediction models incorporate a variety of physical processes, each described by multiple alternative physical schemes with specific parameters. The selection of the physical schemes and the choice of the corresponding physical parameters during model configuration can significantly impact the accuracy of model forecasts. There is no combination of physical schemes that works best for all times, at all locations, and under all conditions. It is therefore of considerable interest to understand the interplay between the choice of physics and the accuracy of the resulting forecasts under different conditions. This paper demonstrates the use of machine learning techniques to study the uncertainty in numerical weather prediction models due to the interaction of multiple physical processes. The first problem addressed herein is the estimation of systematic model errors in output quantities of interest at future times, and the use of this information to improve the model forecasts. The second problem considered is the identification of those specific physical processes that contribute most to the forecast uncertainty in the quantity of interest under specified meteorological conditions. The discrepancies between model results and observations at past times are used to learn the relationships between the choice of physical processes and the resulting forecast errors. Numerical experiments are carried out with the Weather Research and Forecasting (WRF) model. The output quantity of interest is the model precipitation, a variable that is both extremely important and very challenging to forecast. The physical processes under consideration include various micro-physics schemes, cumulus parameterizations, short wave, and long wave radiation schemes. The experiments demonstrate the strong potential of machine learning approaches to aid the study of model errors., Comment: 23 pages, 5 figures, 4 tables

Data assimilation is the process to fuse information from priors, observations of nature, and numerical models, in order to obtain best estimates of the parameters or state of a physical system of interest. Presence of large errors in some observational data, e.g., data collected from a faulty instrument, negatively affect the quality of the overall assimilation results. This work develops a systematic framework for robust data assimilation. The new algorithms continue to produce good analyses in the presence of observation outliers. The approach is based on replacing the traditional $\L_2$ norm formulation of data assimilation problems with formulations based on $\L_1$ and Huber norms. Numerical experiments using the Lorenz-96 and the shallow water on the sphere models illustrate how the new algorithms outperform traditional data assimilation approaches in the presence of data outliers., Comment: 25 pages, Submitted to SISC

This paper constructs an ensemble-based sampling smoother for four-dimensional data assimilation using a Hybrid/Hamiltonian Monte-Carlo approach. The smoother samples efficiently from the posterior probability density of the solution at the initial time. Unlike the well-known ensemble Kalman smoother, which is optimal only in the linear Gaussian case, the proposed methodology naturally accommodates non-Gaussian errors and non-linear model dynamics and observation operators. Unlike the four-dimensional variational met\-hod, which only finds a mode of the posterior distribution, the smoother provides an estimate of the posterior uncertainty. One can use the ensemble mean as the minimum variance estimate of the state, or can use the ensemble in conjunction with the variational approach to estimate the background errors for subsequent assimilation windows. Numerical results demonstrate the advantages of the proposed method compared to the traditional variational and ensemble-based smoothing methods., Comment: 33 Pages

A parallel-in-time algorithm based on an augmented Lagrangian approach is proposed to solve four-dimensional variational (4D-Var) data assimilation problems. The assimilation window is divided into multiple sub-intervals that allows to parallelize cost function and gradient computations. Solution continuity equations across interval boundaries are added as constraints. The augmented Lagrangian approach leads to a different formulation of the variational data assimilation problem than weakly constrained 4D-Var. A combination of serial and parallel 4D-Vars to increase performance is also explored. The methodology is illustrated on data assimilation problems with Lorenz-96 and the shallow water models., Comment: 22 Pages

Inverse problems use physical measurements along with a computational model to estimate the parameters or state of a system of interest. Errors in measurements and uncertainties in the computational model lead to inaccurate estimates. This work develops a methodology to estimate the impact of different errors on the variational solutions of inverse problems. The focus is on time evolving systems described by ordinary differential equations, and on a particular class of inverse problems, namely, data assimilation. The computational algorithm uses first-order and second-order adjoint models. In a deterministic setting the methodology provides a posteriori error estimates for the inverse solution. In a probabilistic setting it provides an a posteriori quantification of uncertainty in the inverse solution, given the uncertainties in the model and data. Numerical experiments with the shallow water equations in spherical coordinates illustrate the use of the proposed error estimation machinery in both deterministic and probabilistic settings., Comment: Contains a total of 51 pages