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Adjoint Formulation for an EmbeddedBoundary Cartesian Method
, 2005
"... A discreteadjoint formulation is presented for the threedimensional Euler equations discretized on a Cartesian mesh with embedded boundaries. The solution algorithm for the adjoint and flowsensitivity equations leverages the Runge–Kutta timemarching scheme in conjunction with the parallel multig ..."
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Cited by 6 (2 self)
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A discreteadjoint formulation is presented for the threedimensional Euler equations discretized on a Cartesian mesh with embedded boundaries. The solution algorithm for the adjoint and flowsensitivity equations leverages the Runge–Kutta timemarching scheme in conjunction with the parallel multigrid method of the flow solver. The matrixvector products associated with the linearization of the flow equations are computed onthefly, thereby minimizing the memory requirements of the algorithm at a computational cost roughly equivalent to a flow solution. Threedimensional test cases, including a wingbody geometry at transonic flow conditions and an entry vehicle at supersonic flow conditions, are presented. These cases verify the accuracy of the linearization and demonstrate the efficiency and robustness of the adjoint algorithm for complexgeometry problems.
A quasiminimal residual method for simultaneous primaldual solutions and superconvergent functional estimates
 SIAM Journal on Scientific Computing
"... Abstract. The adjoint solution has found many uses in computational simulations where the quantities of interest are the functionals of the solution, including design optimization, error estimation, and control. In those applications where both the solution and the adjoint are desired, the conventio ..."
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Abstract. The adjoint solution has found many uses in computational simulations where the quantities of interest are the functionals of the solution, including design optimization, error estimation, and control. In those applications where both the solution and the adjoint are desired, the conventional approach is to apply iterative methods to solve the primal and dual problems separately. However, we show that there is an advantage associated with iterating the primal and dual problem simultaneously since this enables the construction of iterative methods where both the primal and the dual iterates may be chosen so that they provide functional estimates that are “superconvergent” in that the error converges at twice the order of the optimal global solution error norm. In particular, we show that the structure of the Lanczos process allows for this superconvergence property and propose a modified QMR method which uses the same Lanczos process to simultaneously solve the primal and dual problems. Thus both the primal and the dual systems are solved at essentially the same computational cost as the conventional QMR method applied to the primal problem alone. Numerical experiments show that our proposed method does indeed exhibit superconvergence behavior.
A Gridenabled Problem Solving Environment for Parallel Computational Engineering Design
 Advances in Engineering Software
"... This paper describes the development and application of a piece of engineering software that provides a Problem Solving Environment (PSE) capable of launching, and interfacing with, computational jobs executing on remote resources on a computational Grid. In particular it is demonstrated how a comp ..."
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Cited by 5 (2 self)
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This paper describes the development and application of a piece of engineering software that provides a Problem Solving Environment (PSE) capable of launching, and interfacing with, computational jobs executing on remote resources on a computational Grid. In particular it is demonstrated how a complex, serial, engineering optimisation code may be efficiently parallelised, Gridenabled and embedded within a PSE. The environment is highly flexible, allowing remote users from different sites to collaborate, and permitting computational tasks to be executed in parallel across multiple Grid resources, each of which may be a parallel architecture. A full working prototype has been built and successfully applied to a computationally demanding engineering optimisation problem. This particular problem stems from elastohydrodynamic lubrication and involves optimising the computational model for a lubricant based on the match between simulation results and experimentally observed data.
Monte Carlo evaluation of sensitivities in computational finance
, 2007
"... In computational finance, Monte Carlo simulation is used to compute the correct prices for financial options. More important, however, is the ability to compute the socalled “Greeks”, the first and second order derivatives of the prices with respect to input parameters such as the current asset pri ..."
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Cited by 5 (1 self)
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In computational finance, Monte Carlo simulation is used to compute the correct prices for financial options. More important, however, is the ability to compute the socalled “Greeks”, the first and second order derivatives of the prices with respect to input parameters such as the current asset price, interest rate and level of volatility. This paper discusses the three main approaches to computing Greeks: finite difference, likelihood ratio method (LRM) and pathwise sensitivity calculation. The last of these has an adjoint implementation with a computational cost which is independent of the number of first derivatives to be calculated. We explain how the practical development of adjoint codes is greatly assisted by using Algorithmic Differentiation, and in particular discuss the performance achieved by the FADBAD++ software package which is based on templates and operator overloading within C++. The pathwise approach is not applicable when the financial payoff function is not differentiable, and even when the payoff is differentiable, the use of scripting in realworld implementations means it can be very difficult in practice to evaluate the derivative of very complex financial products. A new idea is presented to address these limitations by combining the adjoint pathwise approach for the stochastic path evolution with LRM for the payoff evaluation. I.
On the Iterative Solution of Adjoint Equations
, 2000
"... This paper considers the iterative solution of the adjoint equations which arise in the context of design optimisation. It is shown that naive adjoining of the iterative solution of the original linearised equations results in an adjoint code which cannot be interpreted as an iterative solution of t ..."
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This paper considers the iterative solution of the adjoint equations which arise in the context of design optimisation. It is shown that naive adjoining of the iterative solution of the original linearised equations results in an adjoint code which cannot be interpreted as an iterative solution of the adjoint equations. However, this can be achieved through appropriate algebraic manipulations. This is important in design optimisation because one can reduce the computational cost by starting the adjoint iteration from the adjoint solution obtained in the previous design step.
Approximation of the scattering amplitude and linear systems
 ELECTRON T. NUMER. ANA
, 2008
"... The simultaneous solution of Ax = b and A T y = g, where A is a nonsingular matrix, is required in a number of situations. Darmofal and Lu have proposed a method based on the QuasiMinimal Residual algorithm (QMR). We will introduce a technique for the same purpose based on the LSQR method and sh ..."
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Cited by 4 (0 self)
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The simultaneous solution of Ax = b and A T y = g, where A is a nonsingular matrix, is required in a number of situations. Darmofal and Lu have proposed a method based on the QuasiMinimal Residual algorithm (QMR). We will introduce a technique for the same purpose based on the LSQR method and show how its performance can be improved when using the generalized LSQR method. We further show how preconditioners can be introduced to enhance the speed of convergence and discuss different preconditioners that can be used. The scattering amplitude g T x, a widely used quantity in signal processing for example, has a close connection to the above problem since x represents the solution of the forward problem and g is the righthand side of the adjoint system. We show how this quantity can be efficiently approximated using Gauss quadrature and introduce a blockLanczos process that approximates the scattering amplitude, and which can also be used with preconditioning.
Aerodynamic Shape Optimization Using a Cartesian Adjoint Method and CAD Geometry
"... We present a new approach for the computation of shape sensitivities using the discrete adjoint and flowsensitivity methods on Cartesian meshes with general polyhedral cells (cutcells) at the wall boundaries. By directly linearizing the cutcell geometric constructors of the mesh generator, an effi ..."
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We present a new approach for the computation of shape sensitivities using the discrete adjoint and flowsensitivity methods on Cartesian meshes with general polyhedral cells (cutcells) at the wall boundaries. By directly linearizing the cutcell geometric constructors of the mesh generator, an efficient and robust computation of shape sensitivities is achieved. We show that the error convergence rate of the flow solution and its sensitivity, as well as the objective function and its gradient is consistent with the secondorder spatial discretization of the threedimensional Euler equations. The performance of the approach is demonstrated for an airfoil optimization problem in transonic flow, and a CADbased shape optimization of a reentry capsule in hypersonic flow. The approach is wellsuited for conceptual design studies where fast turnaround time is required. I.
Smoking Adjoints: fast evaluation of Greeks in Monte Carlo Calculations
"... This paper presents an adjoint method to accelerate the calculation of Greeks by Monte Carlo simulation. The method calculates price sensitivities along each path; but in contrast to a forward pathwise calculation, it works backward recursively using adjoint variables. Along each path, the forward a ..."
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This paper presents an adjoint method to accelerate the calculation of Greeks by Monte Carlo simulation. The method calculates price sensitivities along each path; but in contrast to a forward pathwise calculation, it works backward recursively using adjoint variables. Along each path, the forward and adjoint implementations produce the same values, but the adjoint method rearranges the calculations to generate potential computational savings. The adjoint method outperforms a forward implementation in calculating the sensitivities of a small number of outputs to a large number of inputs. This applies, for example, in estimating the sensitivities of an interest rate derivatives book to multiple points along an initial forward curve or the sensitivities of an equity derivatives book to multiple points on a volatility surface. We illustrate the application of the method in the setting of the LIBOR market model. Numerical results confirm that the computational advantage of the adjoint method grows in proportion to the number of initial forward rates.
Adjointbased aerodynamic shape optimization
, 2003
"... An adjoint system of the Euler equations of gas dynamics is derived in order to solve aerodynamic shape optimization problems with gradientbased methods. The derivation is based on the fully discrete flow model and involves differentiation and transposition of the system of equations obtained by an ..."
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Cited by 2 (2 self)
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An adjoint system of the Euler equations of gas dynamics is derived in order to solve aerodynamic shape optimization problems with gradientbased methods. The derivation is based on the fully discrete flow model and involves differentiation and transposition of the system of equations obtained by an unstructured and nodecentered finitevolume discretization. Solving the adjoint equations allows an efficient calculation of gradients, also when the subject of optimization is described by hundreds or thousands of design parameters. Such a fine geometry description may cause wavy or otherwise irregular designs during the optimization process. Using the onetoone mapping defined by a Poisson problem is a known technique that produces smooth design updates while keeping a fine resolution of the geometry. This technique is extended here to combine the smoothing effect with constraints on the geometry, by defining the design updates as solutions of a quadratic programming problem associated with the Poisson problem. These methods