Scholarly Works, Computational Science Laboratory
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Browsing Scholarly Works, Computational Science Laboratory by Content Type "Article"
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- Adjoint-Matching Neural Network Surrogates for Fast 4D-Var Data AssimilationChennault, Austin; Popov, Andrey A.; Subrahmanya, Amit N.; Cooper, Rachel; Karpatne, Anuj; Sandu, Adrian (2021-11-16)The data assimilation procedures used in many operational numerical weather forecasting systems are based around variants of the 4D-Var algorithm. The cost of solving the 4D-Var problem is dominated by the cost of forward and adjoint evaluations of the physical model. This motivates their substitution by fast, approximate surrogate models. Neural networks offer a promising approach for the data-driven creation of surrogate models. The accuracy of the surrogate 4D-Var problem’s solution has been shown to depend explicitly on accurate modeling of the forward and adjoint for other surrogate modeling approaches and in the general nonlinear setting. We formulate and analyze several approaches to incorporating derivative information into the construction of neural network surrogates. The resulting networks are tested on out of training set data and in a sequential data assimilation setting on the Lorenz-63 system. Two methods demonstrate superior performance when compared with a surrogate network trained without adjoint information, showing the benefit of incorporating adjoint information into the training process.
- Alternating directions implicit integration in a general linear method frameworkSarshar, Arash; Roberts, Steven; Sandu, Adrian (Elsevier, 2021-05-15)Alternating Directions Implicit (ADI) integration is an operator splitting approach to solve parabolic and elliptic partial differential equations in multiple dimensions based on solving sequentially a set of related one-dimensional equations. Classical ADI methods have order at most two, due to the splitting errors. Moreover, when the time discretization of stiff one-dimensional problems is based on Runge–Kutta schemes, additional order reduction may occur. This work proposes a new ADI approach based on the partitioned General Linear Methods framework. This approach allows the construction of high order ADI methods. Due to their high stage order, the proposed methods can alleviate the order reduction phenomenon seen with other schemes. Numerical experiments are shown to provide further insight into the accuracy, stability, and applicability of these new methods.
- Efficient methods for computing observation impact in 4D-Var data assimilationCioaca, Alexandru; Sandu, Adrian; de Sturler, Eric (Springer, 2013-12-01)This paper presents a practical computational approach to quantify the effect of individual observations in estimating the state of a system. Such an analysis can be used for pruning redundant measurements, and for designing future sensor networks. The mathematical approach is based on computing the sensitivity of the reanalysis (unconstrained optimization solution) with respect to the data. The computational cost is dominated by the solution of a linear system, whose matrix is the Hessian of the cost function, and is only available in operator form. The right hand side is the gradient of a scalar cost function that quantities the forecast error of the numerical model. The use of adjoint models to obtain the necessary first and second order derivatives is discussed. We study various strategies to accelerate the computation, including matrix-free iterative solvers, preconditioners, and an in-house multigrid solver. Experiments are conducted on both a small-size shallow-water equations model, and on a large-scale numerical weather prediction model, in order to illustrate the capabilities of the new methodology.
- An Ensemble Variational Fokker-Planck Method for Data AssimilationSubrahmanya, Amit N.; Popov, Andrey A.; Sandu, Adrian (2021-11-27)Particle flow filters that aim to smoothly transform particles from samples from a prior distribution to samples from a posterior are a major topic of active research. In this work we introduce a generalized framework which we call the the Variational Fokker-Planck method for filtering and smoothing in data assimilation that encompasses previous methods such as the mapping particle filter and the particle flow filter. By making use of the properties of the optimal Ito process that solves the underlying Fokker-Planck equation we can explicitly include heuristics methods such as rejuvenation and regularization that fit into this framework. We also extend our framework to higher dimensions using localization and covariance shrinkage, and provide a robust implicit-explicit method for solving the stochastic initial value problem describing the Ito process. The effectiveness of the variational Fokker-Planck method is demonstrated on three test problems, namely the Lorenz '63, Lorenz '96 and the quasi-geostrophic equations.
- Investigation of Nonlinear Model Order Reduction of the Quasigeostrophic Equations through a Physics-Informed Convolutional AutoencoderCooper, Rachel; Popov, Andrey A.; Sandu, Adrian (2021-08-27)Reduced order modeling (ROM) is a field of techniques that approximates complex physics-based models of real-world processes by inexpensive surrogates that capture important dynamical characteristics with a smaller number of degrees of freedom. Traditional ROM techniques such as proper orthogonal decomposition (POD) focus on linear projections of the dynamics onto a set of spectral features. In this paper we explore the construction of ROM using autoencoders (AE) that perform nonlinear projections of the system dynamics onto a low dimensional manifold learned from data. The approach uses convolutional neural networks (CNN) to learn spatial features as opposed to spectral, and utilize a physics informed (PI) cost function in order to capture temporal features as well. Our investigation using the quasi-geostrophic equations reveals that while the PI cost function helps with spatial reconstruction, spatial features are less powerful than spectral features, and that construction of ROMs through machine learning-based methods requires significant investigation into novel non-standard methodologies.
- Linearly Implicit General Linear MethodsSarshar, Arash; Roberts, Steven; Sandu, Adrian (2021-12-01)Linearly implicit Runge–Kutta methods provide a fitting balance of implicit treat- ment of stiff systems and computational cost. In this paper we extend the class of linearly implicit Runge–Kutta methods to include multi-stage and multi-step methods. We provide the order con- dition theory to achieve high stage order and overall accuracy while admitting arbitrary Jacobians. Several classes of linearly implicit general linear methods (GLMs) are discussed based on existing families such as type 2 and type 4 GLMs, two-step Runge–Kutta methods, parallel IMEX GLMs, and BDF methods. We investigate the stability implications for stiff problems and provide numerical studies for the behavior of our methods compared to linearly implicit Runge–Kutta methods. Our experiments show nominal order of convergence in test cases where Rosenbrock methods suffer from order reduction.
- Machine learning based algorithms for uncertainty quantification in numerical weather prediction modelsMoosavi, Azam; Rao, Vishwas; Sandu, Adrian (Elsevier, 2021-03-01)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. In order to address these questions we employ two machine learning approaches, random forests and artificial neural networks. 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.
- MATLODE: A MATLAB ODE solver and sensitivity analysis toolboxD'Augustine, Anthony; Sandu, Adrian (2016)
- Multirate linearly-implicit GARK schemesGuenther, Michael; Sandu, Adrian (Springer, 2021-12-28)Many complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to their dynamics, in order to achieve increased computational efficiency. The stiff components of the system, fast or slow, are best discretized with implicit base methods in order to ensure numerical stability. To this end, linearly implicit methods are particularly attractive as they solve only linear systems of equations at each step. This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework for linearly-implicit multirate time integration. The order conditions theory considers both exact and approximative Jacobians. The effectiveness of implicit multirate methods depends on the coupling between the slow and fast computations; an array of efficient coupling strategies and the resulting numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit methods, that allow arbitrarily small micro-steps and offer extreme computational flexibility, are constructed. The new unifying framework includes existing multirate Rosenbrock(-W) methods as particular cases, and opens the possibility to develop new classes of highly effective linearly implicit multirate integrators.
- A non-smooth dynamics framework for simulating frictionless spatial joints with clearancesReal-world multibody systems do not have ideal joints; most joints have some clearance. The clearance allows the connected bodies to undergo a misalignment, and the resulting dynamics is governed by the contacts thus formed. Two approaches are typically taken to deal with con- tacts: the commonly used continuous dynamics approaches assume the Hertzian nature of the contact modeled by nonlinear unilateral spring- damper elements; while the non-smooth dynamics approach results in a complementarity problem. This paper employs a non-smooth dynam- ics approach to develop a coherent framework for the simulation of multibody systems having frictionless joints with clearances. Because clearances are of small magnitude relative to the dimensions of the mechanical components, the nature of the contact in the joints is assumed to be inelastic. Using this assumption and the general non-smooth dynamics framework, the parametric formulations for cylindrical, prismatic, and revolute joints with clearances are derived. The equations of motion are formulated, and their time-discretized counterparts are cast as a nonlinear programming problem. The proposed scheme also enforces normalization constraint on Euler parameters, in contrast to state-of- the-art methods, that is conducive to stability of the solution, for a suitable range of step-sizes. In addition, a variable time-stepping scheme is introduced, that includes the step-size as an extra variable in the opti- mization and its stability properties are discussed. The versatility of the proposed framework is demonstrated through numerical experiments.
- A Numerical Investigation of Matrix-Free Implicit Time-Stepping Methods for Large CFD SimulationsSarshar, Arash; Tranquilli, Paul; Pickering, Brent P.; McCall, Andrew; Sandu, Adrian; Roy, Christopher J. (2016)This paper is concerned with the development and testing of advanced time-stepping methods suited for the integration of time-accurate, real-world applications of computational fluid dynamics (CFD). The performance of several time discretization methods is studied numerically with regards to computational efficiency, order of accuracy, and stability, as well as the ability to treat effectively stiff problems. We consider matrix-free implementations, a popular approach for time-stepping methods applied to large CFD applications due to its adherence to scalable matrix-vector operations and a small memory footprint. We compare explicit methods with matrix-free implementations of implicit, linearly-implicit, as well as Rosenbrock-Krylov methods. We show that Rosenbrock-Krylov methods are competitive with existing techniques excelling for a number of prob- lem types and settings.
- Parametric formulations of spatial joints with clearances: A non-smooth dynamics approachChaturvedi, Ekansh; Sandu, Adrian; Sandu, Corina (Virginia Tech, 2023-05-15)The conventional approach of simulating multibody dynamic systems treats the joint interfaces as ideal, that means that the bodies are in absolute alignment with each other in the desired relative directions of motion. However, in real life systems the clearances between the bodies allow the bodies to undergo a certain misalignment and the dynamics is governed by the contacts thus formed. Contact detection and evaluation of contact forces is yet another problem that needs to be addressed. Popular approaches assume the Hertzian nature of the contact and thus evaluate contact forces using nonlinear unilateral spring-damper elements. This approach results in very stiff differential algebraic equations and hence make the numerical integration computationally expensive. Furthermore, the Hertzian approach does not address truly elastic or truly inelastic nature of the contact. This work describes the parametric formulations for fundamental spatial joints with clearances and the non-smooth dynamics approach to solve the resulting equations of motion. The sets of spatial joint expressions for cylindrical, prismatic and revolute joints, and the non-smooth dynamics formulations are derived, considering their interdependence with great care. Further, the nature of the contact with clearances is discussed. The formulation is demonstrated through three case-studies and a detailed analysis of the results is presented. Additionally, a differentiation with respect to the ideal joint counterpart of the revolute joint case study is presented using tangent space ordinary differential equation formulation.
- Physics-informed neural networks for PDE-constrained optimization and controlBarry-Straume, Jostein; Sarshar, Arash; Popov, Andrey A.; Sandu, Adrian (2022-05-06)A fundamental problem of science is designing optimal control policies that manipulate a given environment into producing the desired outcome. Control PhysicsInformed Neural Networks simultaneously solve a given system state, and its respective optimal control, in a one-stage framework that conforms to the physical laws of the system. Prior approaches use a two-stage framework that models and controls a system sequentially, whereas Control PINNs incorporate the required optimality conditions in their architecture and loss function. The success of Control PINNs is demonstrated by solving the following open-loop optimal control problems: (i) an analytical problem (ii) a one-dimensional heat equation, and (iii) a two-dimensional predator-prey problem.
- A Stochastic Covariance Shrinkage Approach to Particle Rejuvenation in the Ensemble Transform Particle FilterPopov, Andrey A.; Subrahmanya, Amit N.; Sandu, Adrian (2021-09-20)Rejuvenation in particle filters is necessary to prevent the collapse of the weights when the number of particles is insufficient to sample the high probability regions of the state space. Rejuvenation is often implemented in a heuristic manner by the addition of stochastic samples that widen the support of the ensemble. This work aims at improving canonical rejuvenation methodology by the introduction of additional prior information obtained from climatological samples; the dynamical particles used for importance sampling are augmented with samples obtained from stochastic covariance shrinkage. The ensemble transport particle filter, and its second order variant, are extended with the proposed rejuvenation approach. Numerical experiments show that modified filters significantly improve the analyses for low dynamical ensemble sizes.
- Subspace adaptivity in Rosenbrock-Krylov methods for the time integration of initial value problemsTranquilli, Paul; Glandon, Ross; Sandu, Adrian (Elsevier, 2021-03-15)The Rosenbrock–Krylov family of time integration schemes is an extension of Rosenbrock-W methods that employs a specific Krylov based approximation of the linear system solutions arising within each stage of the integrator. This work proposes an extension of Rosenbrock–Krylov methods to address stability questions which arise for methods making use of inexact linear system solution strategies. Two approaches for improving the stability and efficiency of Rosenbrock–Krylov methods are proposed, one through direct control of linear system residuals and the second through a novel extension of the underlying Krylov space to include stage right hand side vectors. Rosenbrock–Krylov methods employing the new approaches show a substantial improvement in computational efficiency relative to prior implementations.
- Symplectic GARK methods for Hamiltonian systemsGuenther, Michael; Sandu, Adrian; Zanna, Antonella (2021-03-06)Generalized Additive Runge-Kutta schemes have shown to be a suitable tool for solving ordinary differential equations with additively partitioned right-hand sides. This work generalizes these GARK schemes to symplectic GARK schemes for additively partitioned Hamiltonian systems. In a general setting, we derive conditions for symplecticeness, as well as symmetry and time-reversibility. We show how symplectic and symmetric schemes can be constructed based on schemes which are only symplectic. Special attention is given to the special case of partitioned schemes for Hamiltonians split into multiple potential and kinetic energies. Finally we show how symplectic GARK schemes can use efficiently different time scales and evaluation costs for different potentials by using different order for these parts.
- A unified formulation of splitting-based implicit time integration schemesGonzalez-Pinto, Severiano; Hernandez-Abreu, Domingo; Perez-Rodriguez, Maria S.; Sarshar, Arash; Roberts, Steven; Sandu, Adrian (Academic Press – Elsevier, 2022-01-01)Splitting-based time integration approaches such as fractional step, alternating direction implicit, operator splitting, and locally one dimensional methods partition the system of interest into components, and solve individual components implicitly in a cost-effective way. This work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge–Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions for this class, and explain their application to partitioned systems of ordinary differential equations. We show that classical splitting methods belong to the IMIM GARK family, and therefore can be studied in this unified framework. New IMIM-GARK splitting methods are developed and tested using parabolic systems.