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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 EnergyBy Lawrence Livermore National Laboratory under Contract W-7405-Eng-48
TOPS SciDAC Project All-hands Meeting
2CASC
Differential and Nonlinear Solvers @ CASC
CVODE – explicit ODE solver
IDA – implicit DAE solver
KINSOL – Krylov Inexact Newton solver
User main routineUser problem-defining functionUser preconditioner function
CVODEODE
Integrator
IDADAE
Integrator
KINSOLNonlinear
Solver
BandLinearSolver
PreconditionedGMRES
Linear Solver
GeneralPreconditioner
Modules
VectorKernels
DenseLinearSolver
3CASC
Sensitivity Analysis: What for?
• Model evaluation• Most and/or least influential parameters
• Model reduction
• Uncertainty quantification
• Optimization • design optimization• optimal control• parameter estimation• …
• …
4CASC
Forward Sensitivity Analysis
• Explicit ODE (CVODE)
• i-th sensitivity equation
• Gradient of a derived function
pyy
pyfy
0
0
,,
t
t
ii
iii
tp
y
p
y
p
f
p
y
y
f
p
y
0
d
d
d
d
d
d
d
d
0
p
g
p
y
y
g
p
g
pyg
d
d
d
d
),,(t
yp
py
gpy
pys
gpy
NN
NN
NNN
1
d/d
,,
R
RRR
Computational effort:
Remarks:
• Sensitivity r.h.s. can be user-defined, AD-generated, or FD-approximated
• Sensitivity equations are independent of g!
5CASC
Adjoint Sensitivity Analysis
• Explicit ODE (CVODE)
• For a derived function
• Adjoint ODE
• Gradient of derived function
pyy
pyfy
0
0
,,
t
t
ft
ttt
0)d,,()( pygpG
0λ
y
gλ
y
fλ
ft
TT
yg
y
gpy
λ
gpy
NN
N
NNN
1
,,
R
RRR
Computational effort:
ft
t p tt0
00TT )(d
d
dpp yλfλg
p
G
Remarks:
• Formulation can be extended to find gradients of g(tf,y,p)
• No FD approximation of the adjoint r.h.s.
• Adjoint equations are independent of p!
6CASC
Forward Sensitivity Variants of CASC Solvers
• CVODES currently available
User main routineSpecification of problem parametersActivation of sensitivity computationUser problem-defining functionUser preconditioner function
CVODESODE
Integrator
IDASDAE
Integrator
KINSOLSNonlinear
Solver
BandLinearSolver
PreconditionedGMRES
Linear Solver
GeneralPreconditioner
Modules
VectorKernels
DenseLinearSolver
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
7CASC
Adjoint Sensitivity Variants of CASC Solvers
• CVODEA currently available
User main routineActivation of sensitivity computationUser problem-defining functionUser reverse functionUser preconditioner functionUser reverse preconditioner function
CVODEAODE
Integrator
IDAADAE
Integrator
KINSOLANonlinear
Solver
BandLinearSolver
PreconditionedGMRES
Linear Solver
GeneralPreconditioner
Modules
ModifiedVectorKernels
DenseLinearSolver
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
8CASC
Effects of Aerosols on Cloud Properties*
Problem dimensions• Ny 300
• Np=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
9CASC
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
*C. Woodward, K. Grant, R. Maxwell
10CASC
2-D Advection-Diffusion
u0
G for u0=(x-x’,y-y’)
10)(
)(0)(
123
t,x,y,tλ
Ωx,y,x,y,tλ
λλ/λλ yyxxxt
0)()(
)(0)(
23
0
t,x,yux,y,tu
Ωx,y,x,y,tu
uu/uu yyxxxt
dxdydtx,y,tuG )( dxdyx,yδux,y,λδGx,y
)()0( 0
]10[]10[]20[)( ,,,Ωx,y,t
Problem dimensions• Nu=800
• Np=800
Problem description
• 2-D time-dependent PDEs with homogeneous Dirichlet B.C. Explicit ODEs
11CASC
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• Ny 500
• Np=24
*L. Raja, R. Kee, R. Serban, L. Petzold
12CASC
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?