Sensitivity and Uncertainty Analysis



Introduction

    Usually when we talk about a model , it consists of 3 parts: the input x, output y and the function y = m(x). In our case, this function is given by a differential-algebraic equation (DAE) system. If the equation and the input (parameters and initial values) are given, we can solve the equation and get the solution though DASSL or DASPK. But not all the information in the input or the model are known. The unknown information leads to uncertainty.

    " Uncertainty in a model can have different origins in different decision problems. It may be due to either incomplete information, or fluctuations inhere in the problem, or unpredictable changes in future. " (from the webpage: http://home.ubalt.edu/ntsbarsh/senanaly/SenAnaly.htm)

    When we deal with uncertainty, there are two ways:



Sensitivity:
        Sensitivity gives the rate of changes in the output with respect to the changes of the parameters in the input or in the model. When there are only a few parameters, it is good to use forward mode to compute the sensitivity; When there are a lot of parameters but only a few outputs, it is better to use backward mode to compute; When both the numbers of parameter and output are large, we can only estimate the condition of the system.
 

Forward Sensitivity:
 

    Basic DASSL and DASPK
    New DASPK code with sensitivity capacity  and its design document.


Backward Sensitivity
    Backward sensitivity analysis is also referred as adjoint sensitivity analysis. The sensitivity is not computed at the same time when the DAE is solved. Instead, After the solution trajectory of the DAEs is obtained, the adjoint sensitivity DAEs are constructed and solved backwards from the end time to the start time. The theory and application of the adjoint sensitivity analysis can be found in the following two papers:    
  1. Yang Cao, Shengtai Li, Linda Petzold and Radu Serban, Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution(PS file), SIAM J. Sci. Comput. 24(3), 1076--1089, 2002, 
  2. Yang Cao, Shengtai Li and Linda Petzold, Adjoint Sensitivity Analysis for Differential-Algebraic Equations: Algorithms and Software(PS file), J. Comp. Appl. Math. 149 (1), 171--191, 2002. 

DASPKAdjoint
    The adjoint sensitivity analysis has been implemented in DASPK as a subroutine called DASPKAdjoint. Interested readers can read the DASPK manual for details.


Condition

Several Links:
1. http://sensitivity-analysis.jrc.cec.eu.int/default2.asp?page=wwh
2. http://home.ubalt.edu/ntsbarsh/senanaly/SenAnaly.htm