Alexandru Cioaca

Second-order adjoints for solving PDE-constrained optimization problems

Inverse problems are of the utmost importance in many fields of science and engineering. In the variational approach, inverse problems are formulated as partial differential equation-constrained optimization problems, where the optimal estimate of the uncertain parameters is the minimizer of a certain cost functional subject to the constraints posed by the model equations. The numerical solution of such optimization problems requires the computation of derivatives of the model output with respect to model parameters. The first-order derivatives of a cost functional (defined on the model output) with respect to a large number of model parameters can be calculated efficiently through first-order adjoint (FOA) sensitivity analysis. Second-order adjoint (SOA) models give second derivative information in the form of matrix–vector products between the Hessian of the cost functional and user-defined vectors. Traditionally, the construction of second-order derivatives for large-scale models has been considered too costly. Consequently, data assimilation applications employ optimization algorithms that use only first-order derivative information, such as nonlinear conjugate gradients and quasi-Newton methods. In this paper, we discuss the mathematical foundations of SOA sensitivity analysis and show that it provides an efficient approach to obtain Hessian-vector products. We study the benefits of using second-order information in the numerical optimization process for data assimilation applications. The numerical studies are performed in a twin experiment setting with a two-dimensional shallow water model. Different scenarios are considered with different discretization approaches, observation sets, and noise levels. Optimization algorithms that employ second-order derivatives are tested against widely used methods that require only first-order derivatives. Conclusions are drawn regarding the potential benefits and the limitations of using high-order information in large-scale data assimilation problems.

An optimization framework to improve 4D-Var data assimilation system performance

This paper develops a computational framework for optimizing the parameters of data assimilation systems in order to improve their performance. The approach formulates a continuous meta-optimization problem for parameters; the meta-optimization is constrained by the original data assimilation problem. The numerical solution process employs adjoint models and iterative solvers. The proposed framework is applied to optimize observation values, data weighting coefficients, and the location of sensors for a test problem. The ability to optimize a distributed measurement network is crucial for cutting down operating costs and detecting malfunctions.

Efficient methods for computing observation impact in 4D-Var data assimilation

This paper presents a practical computational approach to quantify the effect of individual observations in estimating the state of a system. Such a methodology can be used for pruning redundant measurements and for designing future sensor networks. The mathematical approach is based on computing the sensitivity of the analyzed model states (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 quantifies 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.

Adjoint sensitivity analysis for numerical weather prediction: applications to power grid optimization

We present an approach to estimate adjoint sensitivities of economic metrics of relevance in the power grid with respect to physical weather variables using numerical weather prediction models. We demonstrate that this capability can significantly enhance planning and operations. We illustrate the method using a large-scale computational study where we compute sensitivities of the regional generation cost in the state of Illinois with respect to wind speed and temperature fields inside and outside the state.

Low-rank approximations for computing observation impact in 4D-Var data assimilation

Abstract We present an efficient computational framework to quantify the impact of individual observations in four dimensional variational data assimilation. The proposed methodology uses first and second order adjoint sensitivity analysis, together with matrix-free algorithms to obtain low-rank approximations of observation impact matrix. This novel technique is illustrated in what follows on important applications such as data pruning and the identification of faulty sensors for a two dimensional shallow water test system.

Obtaining and using second order derivative information in the solution of large scale inverse problems

Inverse problems are of utmost importance in many fields of science and engineering. In the variational approach inverse problems are formulated as constrained optimization problems, where the optimal estimate of the uncertain parameters is the minimizer of a certain cost function subject to the model constraints. The numerical solution of such optimization problems requires the derivatives of a chosen cost function I dependent on the model parameters. Given that the parameter space is large in real-life problems, the derivatives of I can be calculated efficiently through first order adjoint sensitivity analysis. Second order adjoint models give second derivative information in the form of products between the Hessian of the cost functional and a user defined vector. In this paper we review the mathematical foundations of the second order adjoint sensitivity method. We then evaluate their performance in several data assimilation, sensitivity analysis, and uncertainty quantification scenarios, for a two dimensional shallow water flow simulation. In the data assimilation problem, we compare the performance of several well-known optimization methods that make use of first and second order information.

Dynamic Sensor Network Configuration in InfoSymbiotic Systems Using Model Singular Vectors

Abstract Data assimilation is an important data-driven application (DDDAS) where measurements of the real system are used to con- strain simulation results. This paper describes a methodology for dynamically configuring sensor networks in data assimilation systems based on numerical models of time-evolving differential equations. The proposed methodology uses the dominant model singular vectors, which reveal the directions of maximal error growth. New sensors are dynamically placed such as to minimize an estimation error energy norm. A shallow water test problem is used to illustrate our approach.

Efficient Computation of Observation Impact in 4D-Var Data Assimilation

Data assimilation combines information from an imperfect model, sparse and noisy observations, and error statistics, to produce a best estimate of the state of a physical system. Different observational data points have different contributions to reducing the uncertainty with which the state is estimated. Quantifying the observation impact is important for analyzing the effectiveness of the assimilation system, for data pruning, and for designing future sensor networks. This paper is concerned with quantifying observation impact in the context of four dimensional variational data assimilation. The main computational challenge is posed by the solution of linear systems, where the system matrix is the Hessian of the variational cost function. This work discusses iterative strategies to efficiently solve this system and compute observation impacts.