## A Nonlinear Programming Perspective on Sensitivity Calculations for Systems Governed by State Equations, Volume 201659Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1997 - 35 pages |

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adjoint approach adjoint equations adjoint problem analytical formulae Banach space bidx,v bilinear form bilinear maps boundary value problem boundedly invertible calculation of sensitivities choice of norm coefficient computing a direction computing derivatives condition error conjugate gradient algorithm constraints corresponds costate defined denote derivative F'(a derivative of F descent with respect dh/du is boundedly differential equations differential operator direction of steepest domain dual equality constrained optimization finite-difference approximations form 36 governed by differential Hilbert space identification implicit differentiation Implicit Function Theorem infinite-dimensional inner product interpretation iterative Jacobian Jn Jn L2 norm Lagrange multiplier linear functional matrix minimize F(a nonbasic variables nonlinear programming obtain the gradient optimization algorithms optimization problems perturbations problem minimize problems governed reduced gradient reduced Hessian sensitivity calculations sensitivity equations approach Sobolev norm Sobolev space solve space adjoint steepest descent Suppose systems governed Theorem 2.2 transpose truncation error weak solution