Plasmid Preparation:Article Title: Towards Overcoming the Curse of Dimensionality: The Third-Order Adjoint Method for Sensitivity Analysis of Response-Coupled Linear Forward/Adjoint Systems, with Applications to Uncertainty Quantification and Predictive Modeling
Article Snippet: .. Predicted Best-Estimate Expected Values for the Responses and Parameters in the Joint Phase-Space of Responses and Parameters The MaxEnt Gaussian p2 ( x|x0, C ) has a quadratic form and hence possesses a single saddle point, which is determined by requiring the first-variation δQ(αD , rD; δα, δr) of the exponential term in Equation (121) to vanish at the saddle point xD , (αD, rD), namely: δQ(αD , rD; δα, δr) , { d dε Q(αD + εδα, rD + εδr) } ε=0 = ∂Q ∂α δα+ ∂Q ∂r δr = 0 . (127) Imposing the requirement indicated in Equation (127) while taking Equation (122) into account yields the following system of equations: ∂ ∂α −12 ( α−α0 r− E(rc) )†( Cα Cαr Crα Cr )−1( α−α0 r− E(rc) ) − 1 2 [r− E(rm)]†C−1m [r− E(rm)] = 0, (128) ∂ ∂r −12 ( α−α0 r− E(rc) )†( Cα Cαr Crα Cr )−1( α−α0 r− E(rc) ) − 1 2 [r− E(rm)]†C−1m [r− E(rm)] = 0. (129) Recalling that: ∂ ( x†Ax ) ∂x = x† ( A + A† ) (130) and applying the result shown in Equation (130) to evaluate the expressions provided in Equations (128) and (129), respectively, yields the following system of equations at the saddle point, denoted as (αD, rD):( Cα Cαr Crα Cr )−1( αD −α0 rD − E(rc) ) + ( 0 0 0 C−1m )( αD −α0 rD − E(rc) ) + ( 0 C−1m [E(rc) − E(rm)] ) = ( 0 0 ) . (131) Solving Equation (131) yields the following intermediate expressions:( αD −α0 + CαrC−1m [rD − E(rm)] rD − E(rc) + CrC−1m [rD − E(rm)] ) = ( 0 0 ) . (132) For computational purposes, the results obtained in Equation (132) are recast into the following forms: rD = E(rm) + Cm(Cm + Cr) −1[E(rc) − E(rm)], (133) αD = α 0 −Cαr(Cm + Cr)−1[E(rc) − E(rm)]. (134) In view of Equation (125), it follows that “best-estimate predicted” results for the posterior expectation values for the responses and parameters, which will be denoted as ( rbe,αbe ) , will have the same expressions as shown in Equations (133) and (134), namely: rbe , ∫ D rp2 ( x|x0, C ) dα dr = rD = rm + Cm(Cm + Cr) −1[E(rc) − E(rm)], (135) αbe , ∫ D αp2 ( x|x0, C ) dα dr = αD = α0 −Cαr(Cm + Cr)−1[E(rc) − E(rm)]. (136) Since the components of the vector E(rc), and the components of the matrices Cr and Cαr contain 2nd-order and 3rd-order sensitivities, the formulas presented in Equations (135) and (136) generalize all of the previous formulas of this type found in data assimilation/assimilation procedures published to date (which contain at most first-order sensitivities). ..
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