arXiv subjects

Adrian Sandu

Publications and source records attributed to Adrian Sandu.

At least 37 records · Page 2Linked to original sources

arXiv2022

Physics-informed neural networks for PDE-constrained optimization and control

A fundamental problem in science and engineering is designing optimal control policies that steer a given system towards a desired outcome. This work proposes Control Physics-Informed Neural Networks (Control PINNs) that simultaneously solve for a given system state, and for the optimal control signal, in a one-stage framework that conforms to the underlying physical laws. Prior approaches use a two-stage framework that first models and then controls a system in sequential order. In contrast, a Control PINN incorporates the required optimality conditions in its architecture and in its 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.

cs.LG↗

arXiv2022

A Meta-learning Formulation of the Autoencoder Problem for Non-linear Dimensionality Reduction

A rapidly growing area of research is the use of machine learning approaches such as autoencoders for dimensionality reduction of data and models in scientific applications. We show that the canonical formulation of autoencoders suffers from several deficiencies that can hinder their performance. Using a meta-learning approach, we reformulate the autoencoder problem as a bi-level optimization procedure that explicitly solves the dimensionality reduction task. We prove that the new formulation corrects the identified deficiencies with canonical autoencoders, provide a practical way to solve it, and showcase the strength of this formulation with a simple numerical illustration.

cs.LG↗

arXiv2022

A Stochastic Covariance Shrinkage Approach in Ensemble Transform Kalman Filtering

The Ensemble Kalman Filters (EnKF) employ a Monte-Carlo approach to represent covariance information, and are affected by sampling errors in operational settings where the number of model realizations is much smaller than the model state dimension. To alleviate the effects of these errors EnKF relies on model-specific heuristics such as covariance localization, which takes advantage of the spatial locality of correlations among the model variables. This work proposes an approach to alleviate sampling errors that utilizes a locally averaged-in-time dynamics of the model, described in terms of a climatological covariance of the dynamical system. We use this covariance as the target matrix in covariance shrinkage methods, and develop a stochastic covariance shrinkage approach where synthetic ensemble members are drawn to enrich both the ensemble subspace and the ensemble transformation. We additionally provide for a way in which this methodology can be localized similar to the state-of-the-art LETKF method, and that for a certain model setup, our methodology significantly outperforms it.

stat.ME↗

arXiv2022

Eliminating Order Reduction on Linear, Time-Dependent ODEs with GARK Methods

When applied to stiff, linear differential equations with time-dependent forcing, Runge-Kutta methods can exhibit convergence rates lower than predicted by the classical order condition theory. Commonly, this order reduction phenomenon is addressed by using an expensive, fully implicit Runge-Kutta method with high stage order or a specialized scheme satisfying additional order conditions. This work develops a flexible approach of augmenting an arbitrary Runge-Kutta method with a fully implicit method used to treat the forcing such as to maintain the classical order of the base scheme. Our methods and analyses are based on the general-structure additive Runge-Kutta framework. Numerical experiments using diagonally implicit, fully implicit, and even explicit Runge-Kutta methods confirm that the new approach eliminates order reduction for the class of problems under consideration, and the base methods achieve their theoretical orders of convergence.

math.NA↗

arXiv2022

A fast time-stepping strategy for dynamical systems equipped with a surrogate model

Simulation of complex dynamical systems arising in many applications is computationally challenging due to their size and complexity. Model order reduction, machine learning, and other types of surrogate modeling techniques offer cheaper and simpler ways to describe the dynamics of these systems but are inexact and introduce additional approximation errors. In order to overcome the computational difficulties of the full complex models, on one hand, and the limitations of surrogate models, on the other, this work proposes a new accelerated time-stepping strategy that combines information from both. This approach is based on the multirate infinitesimal general-structure additive Runge-Kutta (MRI-GARK) framework. The inexpensive surrogate model is integrated with a small timestep to guide the solution trajectory, and the full model is treated with a large timestep to occasionally correct for the surrogate model error and ensure convergence. We provide a theoretical error analysis, and several numerical experiments, to show that this approach can be significantly more efficient than using only the full or only the surrogate model for the integration.

math.NA↗

arXiv2022

A Class of Multirate Infinitesimal GARK Methods

Differential equations arising in many practical applications are characterized by multiple time scales. Multirate time integration seeks to solve them efficiently by discretizing each scale with a different, appropriate time step, while ensuring the overall accuracy and stability of the numerical solution. In a seminal paper Knoth and Wolke (APNUM, 1998) proposed a hybrid solution approach: discretize the slow component with an explicit Runge-Kutta method, and advance the fast component via a modified fast differential equation. The idea led to the development of multirate infinitesimal step (MIS) methods by Wensch et al. (BIT, 2009.)Günther and Sandu (BIT, 2016) explained MIS schemes as a particular case of multirate General-structure Additive Runge-Kutta (MR-GARK) methods. The hybrid approach offers extreme flexibility in the choice of the numerical solution process for the fast component. This work constructs a family of multirate infinitesimal GARK schemes (MRI-GARK) that extends the hybrid dynamics approachin multiple ways. Order conditions theory and stability analyses are developed, and practical explicit and implicit methods of up to order four are constructed. Numerical results confirm the theoretical findings. We expect the new MRI-GARK family to be most useful for systems of equations with widely disparate time scales, where the fast process is dispersive, and where the influence of the fast component on the slow dynamics is weak.

math.NA↗

arXiv2021

A unified formulation of splitting-based implicit time integration schemes

Splitting-based time integration approaches such as fractional steps, 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 and stability analysis 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.

math.NA↗

arXiv2021

Design of High-Order Decoupled Multirate GARK Schemes

Multirate time integration methods apply different step sizes to resolve different components of the system based on the local activity levels. This local selection of step sizes allows increased computational efficiency while achieving the desired solution accuracy. While the multirate idea is elegant and has been around for decades, multirate methods are not yet widely used in applications. This is due, in part, to the difficulties raised by the construction of high order multirate schemes. Seeking to overcome these challenges, this work focuses on the design of practical high-order multirate methods using the theoretical framework of generalized additive Runge-Kutta (MrGARK) methods, which provides the generic order conditions and the linear and nonlinear stability analyses. A set of design criteria for practical multirate methods is defined herein: method coefficients should be generic in the step size ratio, but should not depend strongly on this ratio; unnecessary coupling between the fast and the slow components should be avoided; and the step size controllers should adjust both the micro- and the macro-steps. Using these criteria, we develop MrGARK schemes of up to order four that are explicit-explicit (both the fast and slow component are treated explicitly), implicit-explicit (implicit in the fast component and explicit in the slow one), and explicit-implicit (explicit in the fast component and implicit in the slow one). Numerical experiments illustrate the performance of these new schemes.

math.NA↗

arXiv2021

A Stochastic Covariance Shrinkage Approach to Particle Rejuvenation in the Ensemble Transform Particle Filter

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.

math.ST↗

arXiv2021

Investigation of Nonlinear Model Order Reduction of the Quasigeostrophic Equations through a Physics-Informed Convolutional Autoencoder

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.

cs.LG↗

arXiv2021

Multifidelity Ensemble Kalman Filtering Using Surrogate Models Defined by Physics-Informed Autoencoders

Data assimilation is a Bayesian inference process that obtains an enhanced understanding of a physical system of interest by fusing information from an inexact physics-based model, and from noisy sparse observations of reality. The multifidelity ensemble Kalman filter (MFEnKF) recently developed by the authors combines a full-order physical model and a hierarchy of reduced order surrogate models in order to increase the computational efficiency of data assimilation. The standard MFEnKF uses linear couplings between models, and is statistically optimal in case of Gaussian probability densities. This work extends MFEnKF to work with non-linear couplings between the models. Optimal nonlinear projection and interpolation operators are obtained by appropriately trained physics-informed autoencoders, and this approach allows to construct reduced order surrogate models with less error than conventional linear methods. Numerical experiments with the canonical Lorenz '96 model illustrate that nonlinear surrogates perform better than linear projection-based ones in the context of multifidelity filtering.

math.OC↗

arXiv2021

Multirate Linearly-Implicit GARK Schemes

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.

math.NA↗

arXiv2020

Implicit multirate GARK methods

This work considers multirate generalized-structure additively partitioned Runge-Kutta (MrGARK) methods for solving stiff systems of ordinary differential equations (ODEs) with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate coupling strategies, problems for assessing linear stability, and techniques to efficiently implement Newton iterations for stage equations. Unlike much of the existing multirate stability analysis which is limited in scope to particular methods, we present general statements on stability and describe fundamental limitations for certain types of multirate schemes. New implicit multirate methods up to fourth order are derived, and their accuracy and efficiency properties are verified with numerical tests.

math.NA↗

arXiv2020

Linearly Implicit Multistep Methods for Time Integration

Time integration methods for solving initial value problems are an important component of many scientific and engineering simulations. Implicit time integrators are desirable for their stability properties, significantly relaxing restrictions on timestep size. However, implicit methods require solutions to one or more systems of nonlinear equations at each timestep, which for large simulations can be prohibitively expensive. This paper introduces a new family of linearly implicit multistep methods (LIMM), which only requires the solution of one linear system per timestep. Order conditions and stability theory for these methods are presented, as well as design and implementation considerations. Practical methods of order up to five are developed that have similar error coefficients, but improved stability regions, when compared to the widely used BDF methods. Numerical testing of a self-starting variable stepsize and variable order implementation of the new LIMM methods shows measurable performance improvement over a similar BDF implementation.

math.NA↗

arXiv2020

Linearly implicit GARK schemes

Systems driven by multiple physical processes are central to many areas of science and engineering. Time discretization of multiphysics systems is challenging, since different processes have different levels of stiffness and characteristic time scales. The multimethod approach discretizes each physical process with an appropriate numerical method; the methods are coupled appropriately such that the overall solution has the desired accuracy and stability properties. The authors developed the general-structure additive Runge-Kutta (GARK) framework, which constructs multimethods based on Runge-Kutta schemes. This paper constructs the new GARK-ROS/GARK-ROW families of multimethods based on linearly implicit Rosenbrock/Rosenbrock-W schemes. For ordinary differential equation models, we develop a general order condition theory for linearly implicit methods with any number of partitions, using exact or approximate Jacobians. We generalize the order condition theory to two-way partitioned index-1 differential-algebraic equations. Applications of the framework include decoupled linearly implicit, linearly implicit/explicit, and linearly implicit/implicit methods. Practical GARK-ROS and GARK-ROW schemes of order up to four are constructed.

math.NA↗

arXiv2020

A Multifidelity Ensemble Kalman Filter with Reduced Order Control Variates

This work develops a new multifidelity ensemble Kalman filter (MFEnKF) algorithm based on linear control variate framework. The approach allows for rigorous multifidelity extensions of the EnKF, where the uncertainty in coarser fidelities in the hierarchy of models represent control variates for the uncertainty in finer fidelities. Small ensembles of high fidelity model runs are complemented by larger ensembles of cheaper, lower fidelity runs, to obtain much improved analyses at only small additional computational costs. We investigate the use of reduced order models as coarse fidelity control variates in the MFEnKF, and provide analyses to quantify the improvements over the traditional ensemble Kalman filters. We apply these ideas to perform data assimilation with a quasi-geostrophic test problem, using direct numerical simulation and a corresponding POD-Galerkin reduced order model. Numerical results show that the two-fidelity MFEnKF provides better analyses than existing EnKF algorithms at comparable or reduced computational costs.

math.NA↗

arXiv2020

Parallel implicit-explicit general linear methods

High-order discretizations of partial differential equations (PDEs) necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner. Implicit-explicit (IMEX) integration based on general linear methods (GLMs) offers an attractive solution due to their high stage and method order, as well as excellent stability properties. The IMEX characteristic allows stiff terms to be treated implicitly and nonstiff terms to be efficiently integrated explicitly. This work develops two systematic approaches for the development of IMEX GLMs of arbitrary order with stages that can be solved in parallel. The first approach is based on diagonally implicit multistage integration methods (DIMSIMs) of types 3 and 4. The second is a parallel generalization of IMEX Euler and has the interesting feature that the linear stability is independent of the order of accuracy. Numerical experiments confirm the theoretical rates of convergence and reveal that the new schemes are more efficient than serial IMEX GLMs and IMEX Runge-Kutta methods.

math.NA↗

arXiv2020

Convergence Results for Implicit--Explicit General Linear Methods

This paper studies fixed-step convergence of implicit-explicit general linear methods. We focus on a subclass of schemes that is internally consistent, has high stage order, and favorable stability properties. Classical, index-1 differential algebraic equation, and singular perturbation convergence analyses results are given. For all these problems IMEX GLMs from the class of interest converge with the full theoretical orders under general assumptions. The convergence results require the time steps to be sufficiently small, with upper bounds that are independent on the stiffness of the problem.

math.NA↗