SUMMARYAdjoint formulation is employed for the optimal control of ow around a rotating cylinder, governed by the unsteady Navier-Stokes equations. The main objective consists of suppressing Karman vortex shedding in the wake of the cylinder by controlling the angular velocity of the rotating body, which can be constant in time or time-dependent. Since the numerical control problem is ill-posed, regularization is employed. An empirical logarithmic law relating the regularization coe cient to the Reynolds number was derived for 606Re6140. Optimal values of the angular velocity of the cylinder are obtained for Reynolds numbers ranging from Re = 60 to Re = 1000. The results obtained by the computational optimal control method agree with previously obtained experimental and numerical observations. A signiÿcant reduction of the amplitude of the variation of the drag coe cient is obtained for the optimized values of the rotation rate.

Abstract. The use of reduced-order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most important, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error, while the derived bounds are within an order of magnitude.Key words. model reduction, proper orthogonal decomposition, small sample statistical condition estimation, adjoint method AMS subject classifications. 65L10, 65L99DOI. 10.1137/0706843921. Introduction. Model reduction of dynamical systems described by differential equations is ubiquitous in science and engineering [2]. Reduced-order models (ROMs) are used for efficient simulation [17,32] and control [18,28]. Moreover, the process of creating low-order models forces the researcher to isolate and quantify the dominant physical mechanisms, revealing effective design decisions that would not have been identified through numerical simulation, experiments, or "black box" optimization methods [31].The proper orthogonal decomposition (POD) method has been used extensively in a variety of fields including fluid dynamics [23], identification of coherent structures [12,21], and control [27] and inverse problems [19]. The method has been em-

Abstract. The use of reduced order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors, by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most importantly, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error while the derived bounds are within an order of magnitude. Key words. model reduction, proper orthogonal decomposition, small sample statistical condition estimation, adjoint method AMS subject classifications. 65L10, 65L991. Introduction. Model reduction of dynamical systems described by differential equations is ubiquitous in science and engineering [1]. Reduced models are used for efficient simulation [12,24] and control [13,22]. Moreover, the process of creating low-order models forces the researcher to isolate and quantify the dominant physical mechanisms, revealing effective design decisions that would not have been identified through numerical simulation, experiments or "black box" optimization methods [23].The Proper Orthogonal Decomposition (POD) method has been used extensively in a variety of fields including fluid dynamics [18], identification of coherent structures [8,16] , and study of the dynamic wind pressures acting on buildings [11], to name only a few. A detailed description [8] of the POD approach as a reduction method shows that, for a given number of modes, POD is the most efficient choice among all linear decompositions in the sense that it retains, on average, the greatest possible kinetic energy.As soon as one contemplates the use of a reduced model, questions concerning the quality of the approximation become paramount. To judge the quality of the reduced model, it is important to estimate its error. An algorithm for estimating the error of a class of reduction methods based on projection techniques was presented in [25]. In this approach, the original problem is linearized around the initial time. The resulting first-order error estimates are valid for only a small number of time steps (during which the Jacobian matrix can be considered constant). First-order estimates

Abstract. The use of reduced order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors, by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most importantly, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error while the derived bounds are within an order of magnitude. Key words. model reduction, proper orthogonal decomposition, small sample statistical condition estimation, adjoint method AMS subject classifications. 65L10, 65L991. Introduction. Model reduction of dynamical systems described by differential equations is ubiquitous in science and engineering [1]. Reduced models are used for efficient simulation [12,24] and control [13,22]. Moreover, the process of creating low-order models forces the researcher to isolate and quantify the dominant physical mechanisms, revealing effective design decisions that would not have been identified through numerical simulation, experiments or "black box" optimization methods [23].The Proper Orthogonal Decomposition (POD) method has been used extensively in a variety of fields including fluid dynamics [18], identification of coherent structures [8,16] , and study of the dynamic wind pressures acting on buildings [11], to name only a few. A detailed description [8] of the POD approach as a reduction method shows that, for a given number of modes, POD is the most efficient choice among all linear decompositions in the sense that it retains, on average, the greatest possible kinetic energy.As soon as one contemplates the use of a reduced model, questions concerning the quality of the approximation become paramount. To judge the quality of the reduced model, it is important to estimate its error. An algorithm for estimating the error of a class of reduction methods based on projection techniques was presented in [25]. In this approach, the original problem is linearized around the initial time. The resulting first-order error estimates are valid for only a small number of time steps (during which the Jacobian matrix can be considered constant). First-order estimates

7Optimal control of the 1-D Riemann problem of Euler equations is studied, with the initial values for pressure and 8 density as control parameters. The least-squares type cost functional employs either distributed observations in time or 9 observations calculated at the end of the assimilation window. Existence of solutions for the optimal control problem is 10 proven. Smooth and nonsmooth optimization methods employ the numerical gradient (respectively, a subgradient) of 11 the cost functional, obtained from the adjoint of the discrete forward model. The numerical flow obtained with the 12 optimal initial conditions obtained from the nonsmooth minimization matches very well with the observations. The 13 algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the 14 longer time horizon, except when the observations corresponding to shocks are detected and removed. 20Optimal control methods are presently employed for various applications in different fields: aerody-21 namics, meteorology, acoustics, financial mathematics and chemistry to mention but a few. Since the vast 22 majority of applications consist of the minimization of a cost functional derived from continuous models, 23 we have solved an optimization problem involving a cost functional for a discontinuous model. Our results 24 show that, for the example considered, nonsmooth optimization methods provide very good results in 25 combination with the adjoint method for subgradient computation. 26Nondifferentiable optimization algorithms employing subgradients were introduced following the sem-27 inal work of Lemarechal [45] (e.g. [11,46,53,55,66] to cite but a few).

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Two of the most important areas in computational finance: Greeks and, respectively, calibration, are based on efficient and accurate computation of a large number of sensitivities. This paper gives an overview of adjoint and automatic differentiation (AD), also known as algorithmic differentiation, techniques to calculate these sensitivities. When compared to finite difference approximation, this approach can potentially reduce the computational cost by several orders of magnitude, with sensitivities accurate up to machine precision. Examples and a literature survey are also provided. *