Report on Sensitivity Analysis

Report on Sensitivity Analysis TOPS SciDAC Project All-hands Meeting 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 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 Dense Linear Solver Preconditioned GMRES Linear Solver General Preconditioner Modules Vector Kernels CASC 2 Sensitivity Analysis: What for? • Model evaluation • Most and/or least influential parameters • Model reduction • Uncertainty quantification • Optimization • • • • design optimization optimal control parameter estimation … • … CASC 3 Forward Sensitivity Analysis • Explicit ODE (CVODE) y  f t , y , p  y t0   y 0 p  Remarks: • Sensitivity r.h.s. can be user-defined, AD-generated, or FD-approximated • i-th sensitivity equation • Sensitivity equations are independent of g! dy  f  dy f    dp i  y  dp i p i dy t0   dy 0 dp i dp i • Gradient of a derived function g (t , y , p) dg  g  dy g    dp  y  dp p CASC Computational effort: y R y , p R p , g R N N Ng s  dy / dp   R N y  Np 1  N  N p y 4 Adjoint Sensitivity Analysis • Explicit ODE (CVODE) y  f t , y , p  y t0   y 0 p  Remarks: • Formulation can be extended to find gradients of g(tf,y,p) • For a derived function G (p)  t f g(t , y, p)dt 0 • No FD approximation of the adjoint r.h.s. • Adjoint equations are independent of p! t • Adjoint ODE  f   g  λ     λ     y   y  λ t f   0 T T Computational effort: y R y , p R p , g R N N Ng • Gradient of derived function dG t  t g p  λ T fp dt  λ T (t0 )y 0p dp f 0 λ R Ny 1  N  N g y 5 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 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 CVODES ODE Integrator IDAS DAE Integrator KINSOLS Nonlinear Solver Band Linear Solver Dense Linear Solver Preconditioned GMRES Linear Solver General Preconditioner Modules Vector Kernels CASC 6 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 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 CVODEA ODE Integrator IDAA DAE Integrator KINSOLA Nonlinear Solver Band Linear Solver Dense Linear Solver Preconditioned GMRES Linear Solver General Preconditioner Modules Modified Vector Kernels 7 CASC Effects of Aerosols on Cloud Properties* 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 Problem dimensions • • CASC *K. Ny 300 Np=2 8 Grant, C. Chuang, S. Lee, C. Woodward 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 • • Ny=19000 Np=3 CASC *C. Woodward, K. Grant, R. Maxwell 9 2-D Advection-Diffusion Problem description • 2-D time-dependent PDEs with homogeneous Dirichlet B.C.  Explicit ODEs u0 ( x,y,t )  Ω  [0,2]  [0,1]  [0,1] ut  u xx  3 / 2 u x  u yy u ( x,y,t )  0 , ( x,y )  Ω u ( x,y,t )  u 0 ( x,y ) , t  0 G   u ( x,y,t )dxdydt  λt   λxx  3 / 2 λx  λ yy  1 λ( x,y,t )  0 , ( x,y )  Ω λ( x,y,t )  0 , t  1 l  dG for du0=d(x-x’,y-y’) δG   λ( x,y,0)δu0 ( x,y)dxdy x,y Problem dimensions • • CASC Nu=800 Np=800 10 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 • • CASC *L. Ny 500 Np=24 Raja, R. Kee, R. Serban, L. Petzold 11 Future Developments • Code development: • IDAS and IDAA • KINSOLA • SciDAC collaboration: • • • • Terrascale Supernova Initiative: Sensitivity analysis for radiation hydrodynamics (CVODES/IDAS) Other? Time-dependent DE constrained optimization Other? • TOPS collaborations? CASC 12