# Presentation: 183794

## 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**

## Differential and Nonlinear Solvers @ CASC

**CVODE**** – explicit ODE solver**
**IDA**** – implicit DAE solver**
**KINSOL**** – Krylov Inexact Newton solver**

## Sensitivity Analysis: What for?

**Model evaluation**
**Most and/or least influential parameters**
**Model reduction**
**Uncertainty quantification**
**Optimization **
**design optimization**
**optimal control**
**parameter estimation**
**...**
**...**

## Forward Sensitivity Analysis

**Explicit ODE (****CVODE****)**
**i-th sensitivity equation**
**Gradient of a derived function**

**Remarks:**
**Sensitivity r.h.s. can be user-defined, AD-generated, or FD-approximated**
**Sensitivity equations are independent of g!**

## Adjoint Sensitivity Analysis

**Explicit ODE (****CVODE****)**
**For a derived function**
**Adjoint ODE**
**Gradient of derived function**

**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!**

## Forward Sensitivity Variants of CASC Solvers

**CVODES**** currently available**

## Adjoint Sensitivity Variants of CASC Solvers

**CVODEA**** currently available**

## 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

## 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

## 2-D Advection-Diffusion

**2-D Advection-Diffusion**

*u**0*

*****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**

## 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

## 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?**