1. Report on Sensitivity Analysis Radu Serban Keith Grant, Alan Hindmarsh, Steven Lee, Carol Woodward Center for Applied Scientific Computing, LLNL Work performed under the auspices of the U.S. Department of Energy By Lawrence Livermore National Laboratory under Contract W-7405-Eng-48 TOPS SciDAC Project All-hands Meeting

2. 2 CASC Differential and Nonlinear Solvers @ CASC CVODE – explicit ODE solver IDA – implicit DAE solver KINSOL – Krylov Inexact Newton solver User main routine User problem-defining function User preconditioner function CVODE ODE Integrator IDA DAE Integrator KINSOL Nonlinear Solver Band Linear Solver Preconditioned GMRES Linear Solver General Preconditioner Modules Vector Kernels Dense Linear Solver

3. 3 CASC Sensitivity Analysis: What for? Model evaluation Most and/or least influential parameters Model reduction Uncertainty quantification Optimization design optimization optimal control parameter estimation …

4. 4 CASC Forward Sensitivity Analysis Explicit ODE ( CVODE ) i-th sensitivity equation Gradient of a derived function Computational effort: Remarks: Sensitivity r.h.s. can be user-defined, AD-generated, or FD-approximated Sensitivity equations are independent of g!

5. 5 CASC Adjoint Sensitivity Analysis Explicit ODE ( CVODE ) For a derived function Adjoint ODE Gradient of derived function Computational effort: Remarks: Formulation can be extended to find gradients of g(t f,y,p) No FD approximation of the adjoint r.h.s. Adjoint equations are independent of p!

6. 6 CASC Forward Sensitivity Variants of CASC Solvers CVODES currently available User main routine Specification of problem parameters Activation of sensitivity computation User problem-defining function User preconditioner function CVODES ODE Integrator IDAS DAE Integrator KINSOLS Nonlinear Solver Band Linear Solver Preconditioned GMRES Linear Solver General Preconditioner Modules Vector Kernels Dense Linear Solver Options - sensitivity approach (simultaneous or staggered) - user-defined, FD, or AD-generated sensitivity r.h.s. - error control on sensitivity variables - user-defined tolerances for sensitivity variables

7. 7 CASC Adjoint Sensitivity Variants of CASC Solvers CVODEA currently available User main routine Activation of sensitivity computation User problem-defining function User reverse function User preconditioner function User reverse preconditioner function CVODEA ODE Integrator IDAA DAE Integrator KINSOLA Nonlinear Solver Band Linear Solver Preconditioned GMRES Linear Solver General Preconditioner Modules Modified Vector Kernels Dense Linear Solver Implementation - check point approach; total cost is 2 forward solutions + 1 backward solution - integrate any system backwards in time - may require modifications to some user-defined vector kernels

8. 8 CASC Effects of Aerosols on Cloud Properties * Problem dimensions N y  300 N p =2 Problem description Condensation-evaporation eqs. coupled with eqs. of parcel motion and properties Implicit ODEs Sensitivity of cloud liquid water to temperature and water vapor profiles * K. Grant, C. Chuang, S. Lee, C. Woodward

9. 9 CASC Groundwater Flow Problem description Variably saturated flow nonlinear elliptic PDEs Nonlinear eqs. Study influence of permeability field on solution (pressure) Quantify uncertainty in solution due to uncertainty in relative permeability and saturation curves Problem dimensions N y =19000 N p =3 * C. Woodward, K. Grant, R. Maxwell

10. 10 CASC 2-D Advection-Diffusion u0u0  G for  u 0 =  (x-x’,y-y’) Problem dimensions N u =800 N p =800 Problem description 2-D time-dependent PDEs with homogeneous Dirichlet B.C. Explicit ODEs

11. 11 CASC CVD of Superconducting Thin Films (YBaCuO) * Problem description Compressible, chemically reacting, stagnation-flow equations 1-D time-varying PDEs Hessenberg index-2 DAEs Control film stoichiometry through inlet composition Problem dimensions N y  500 N p =24 * L. Raja, R. Kee, R. Serban, L. Petzold

12. 12 CASC Future Developments Code development: IDAS and IDAA KINSOLA SciDAC collaboration: Terrascale Supernova Initiative: Sensitivity analysis for radiation hydrodynamics ( CVODES / IDAS ) Other? TOPS collaborations? Time-dependent DE constrained optimization Other?