Numerical Aspects of the CMAQ Adjoint

1. Numerical Aspects of the CMAQ Adjoint Kumaresh Singh, Adrian Sandu (Virginia Tech) Amir Hakami, John H. Seinfeld (Caltech) Daewon Byun, Violeta Coarfa, Peter Percell (Univ of Houston) Acknowledgement HARC H45 CMAS Conference 10/18/06

2. Introduction • GOAL: To Perform Adjoint Sensitivity Analysis and Data Assimilation tests for CMAQ v4.5. So far the Models-3 CMAQ does not have an adjoint model. • Adjoint Models Discrete: 1. Solve numerically the PDE 2. Take adjoint of numerical discretization Continuous: 1. Develop (analytically) the adjoint PDE (e.g., reverse wind fields) 2. Solve numerically the adjoint PDE CMAS Conference 10/18/06

3. Automatic Differentiation • Tangent Linear Model: Accumulates the derivatives of intermediate variables with respect to the independent variables. • Adjoint Model: Propagates the derivatives of final values with respect to intermediate variables. In either case, the Automatic Differentiation produces code that computes the values of the analytical derivatives accurate to machine precision. CMAS Conference 10/18/06

4. Procedure to perform Data Assimilation • Rewrite Parts of Original Code. Special Functions such as min(), max(), abs() are not compatible with the TAMC interface. • Generate Tangent-Linear and Adjoint models for transport processes such as VDIFF, HDIFF, VPPM, HPPM. • Develop interface codes for advection files and TAMC generated AD files. • Validate the Automatic Differentiated subroutines. • Apply b-LBFGS optimization method to perform data-assimilation tests. • Implement two-levels of check-pointing to carry-out adjoint calculations. CMAS Conference 10/18/06

5. Rotating Cone Test for Advectionhorizontal adv (hppm), tangent linear, adjoint Fig 1: t = 0, Cinitial Fig 3: t = 0, Ctlminitial Fig 5: t = T, λfinal Fig 2: t = T, Cfinal Fig 4: t = T, Ctlmfinal Fig 6: t = 0, λinitial CMAS Conference 10/18/06

6. Validation Results (Advection)TLM vs Finite Difference Fig 7: t = Tinitial, Finite Diff, TLM, Direct Fig 8: t = Tfinal, Finite Diff, TLM, Direct CMAS Conference 10/18/06

7. Validation Results (Advection)ADJ vs TLM Test1: Transfer Matrix i=1 j=1 F=0.543906 A^=0.543906 i=4 j=5 F=0.207375 A^=0.207375 i=10 j=5 F=0.141969 A^=0.141969 Fig 9: Transfer Matrix: TLM vs ADJ Test2: Relative Difference Validation procedure is based on the fact that where, λ are the adjoint variables and δc are perturbations of the forward solution. The relative difference obtained is: relative difference = -0.0944 CMAS Conference 10/18/06

8. Validation Test (Diffusion)TLM vs Finite Difference Fig 2: t = Tfinal, Finite Diff, TLM, Direct Fig 1: t = Tinitial, Finite Diff, TLM, Direct CMAS Conference 10/18/06

9. Validation Test (Diffusion)ADJ vs TLM Fig 3: Transfer Matrix: TLM vs ADJ Test2: Relative Difference Test1: Transfer Matrix i=6 j=1 F=0.103583 A^=0.103583 i=6 j=6 F=0.584716 A^=0.584716 i=6 j=7 F=0.103722 A^=0.103722 Fig 3: TLM vs ADJ, relative error-time plot CMAS Conference 10/18/06

10. Data Assimilation Results Fig 1: Discrete Adjoint (perturbation in amplitude only) Fig 2: Continuous Adjoint (perturbation in amplitude only) CMAS Conference 10/18/06

11. Data Assimilation Results (Continued) Fig 3: Discrete Adjoint (perturbation in position & amplitude) Fig 4: Continuous Adjoint (perturbation in position & amplitude) CMAS Conference 10/18/06

12. CMAQ Continuous Adjoint Results CMAS Conference 10/18/06

13. Future Work • Resolving the issues with Discrete Adjoints • Couple them with the model-3 CMAQ v4.5 • Validate the results CMAS Conference 10/18/06