Variational Data Assimilation - Adjoint Sensitivity Analysis Yan Ding, Ph.D. National Center for Computational Hydroscience and Engineering The University of Mississippi, University, MS Presentation for ENGR , Numerical Optimization, Summer Session,

Outline Dynamical and Data Error A Simple Dynamic System (a pendulum) Performance Function (Measure of Dynamic and Data Error) Calculus of Variations Lagrangian Multiplier Approach Remarks

Objective Introduce variational data assimilation using a simple dynamic system ( a mass spring oscillator) Derive adjoint equation of a dynamic system Make “optimal” estimates of the location of the mass-spring given both data and dynamics The mass-spring oscillator system is ideal to study because (1) it is the most familiar dynamic system in existence (2) analytical Solutions may be written, and (3) it is linear

Dynamic and Data Error A perfect mass-spring will oscillate back and forth forever. However, no oscillator is perfect and some damping will occur whether due to a rusty spring or simply air friction. Therefore the dynamics have error and predictions based on a perfect mass spring oscillator dynamic will be in error. Now suppose one has measured data of the position of the mass as it is oscillating back and forth. This lecture will demonstrate how to identify the damping parameter by minimizing the errors between model results and measurements in terms of adjoint sensitivity approach. An animation of a pendulum showing the velocity and acceleration vectors (v and A).

A Simple Damped Oscillator X : displacement from origin α : damping parameter (unknown) Initial conditions Initially released at rest:

Measurement and Error Suppose there are M position data d i measured at various times, errors ε i represent the discrepancies between measurements d i and model results x(t i ): : measurements

Dynamic Error Dynamic system is not perfect before the “true” values of the unknown parameter have been identified. Allow for errors in the dynamics during the parameter searching process, i.e. λ(t) is errors in the dynamics

Combine Dynamics, Data, and Parameter Performance Function Interested in position of mass for 0 ≤ t ≤T In order to find the “true” values of the parameter, minimize a performance function which combines dynamic, data, and parameter errors α * : Estimated parameter value W: Weighting factor δ(t-t i ): Dirac delta function

How to Minimize the J ? The optimization is to find the true parameter value α satisfying a dynamic system such that where x is satisfied with the dynamic equation of the oscillator Local minimum theory : Necessary Condition: If α is the true value, then  J(α)=0; In other words, the first variation of J should be vanished, namely: δx: variation of x (an infinitesimal), δα: variation of α

Calculus of Variations H.O.T. which contains all the terms related to δ 2 x, δ 2 α, and δxδα can be neglected. Integrate by parts:

Calculus of Variations (cont.) According to chain rule

Optimal Conditions: Adjoint Equation Necessity Conditions: Adjoint Equation Optimal parameter condition:

Transversality (Final) Conditions of Adjoint Equation Recall initial conditions And initially release velocity: Then Recall To make the first variation of J vanished, there should be Transversality (Final) Conditions of λ

Summary Prediction Inverse calculation Parameter update 0 ≤ t ≤T Evaluation

Summary (cont.) Dynamic error λ(t) is an adjoint variable of the dynamic equation. Dynamic error λ(t) is also called as Lagrangian multiplier if one forms an augmented perfomance function Dynamic equation Then, one may directly take variation of J, i.e. δJ to obtain the adjoint equation and transversality conditions.