a robust technique for lumped parameter inverse boundary value problems

P. Venkataraman

Mechanical Engineering

P. Venkataraman Rochester Institute of TechnologyDETC2012 – 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem

32nd CIE, Chicago IL, Aug 2012

A Robust Technique for Lumped Parameter Inverse Boundary Value Problems

P. Venkataraman

P. Venkataraman




Todays Presentation

1. Introduction to Lumped Parameter Inverse BVP2. The Solution Procedure3. The Example4. Conclusion

P. Venkataraman




3

Lumped Parameter Inverse BVP

An Example :2

1 1 1 2 1 1 3

2 4 1 2 5

3 6 1 3

0;

0;

0;

y b y b y y b

y b y y b

y b y y

Fluid flow in a long vertical channel with fluid injection

Forward Problem:

Given: 1 1 2 3

1 1 2 3

(0) 0; (0) 0; (0) 0; (0) 0;

(1) 1; (1) 0; (1) 0; (1) 1;

y y y y

y y y y

1 2 3

4 5 6

100; 100; 276;

100; 1; 70;

b b b

b b b

and

Find: 1 2 3y x , y x , y x

Inverse Problem:

Given: 1 2 3y x , y x , y x and1 1 2 3

1 1 2 3

(0) 0; (0) 0; (0) 0; (0) 0;

(1) 1; (1) 0; (1) 0; (1) 1;

y y y y

y y y y

Find: 1 2 3 4 5 6b ,b ,b ,b ,b ,b

P. Venkataraman




4


Forward Problem:Given: 1 1 2 3

1 1 2 3

(0) 0; (0) 0; (0) 0; (0) 0;

(1) 1; (1) 0; (1) 0; (1) 1;

y y y y

y y y y

1 2 3

4 5 6

100; 100; 276;

100; 1; 70;

b b b

b b b

and

Find: 1 2 3y x , y x , y x

Inverse Problem:Given: 1 2 3y x , y x , y x and

1 1 2 3

1 1 2 3

(0) 0; (0) 0; (0) 0; (0) 0;

(1) 1; (1) 0; (1) 0; (1) 1;

y y y y

y y y y

Find: 1 2 3 4 5 6b ,b ,b ,b ,b ,b

• is well posed (solution exist and unique)• assumes perfect measurement of parameters and boundary conditions• error in the solution will vanish as the perturbation in the parameters tends

to zero

• Inverse problems are considered naturally unstable, ill-posed, not unique• cannot be satisfactorily solved mathematically• no valid inverse problem based on smooth or perfect data• all current methods use some sort of regularization (artificial objective

function for minimization)• inverse problem cannot be satisfactorily solved without partial information

P. Venkataraman




5


The solution of inverse BVP in this paper is robust:

is natural based on derivative informationprocedure can be adapted fro the forward problem too

does not require regularization

does not require dimensional control

does not require partial information

P. Venkataraman




6

The Solution Procedure

Some Assumptions :

• we simulate discrete non smooth data to represent measurement error from smooth data of the forward problem

• each data stream is connected to a differential equation• the solution of the inverse BVP is the value of the lumped

parameters• the solution of the inverse BVP is also the trajectory based on

the parameters• quality of solution is determined by closeness of trajectory

determined using the value for parameters to the original smooth trajectory

• final trajectory is obtained using numerical integration (collocation)

• for comparison we assume that the boundary conditions are not perturbed

P. Venkataraman




7


Step 1: Data Smoothing using a recursive Bezier filterBezier filter determines the best order that minimizes the sum of least squared error (LSE) and the sum of the absolute error (LAE)

Step 2: The first optimization procedureObtain the first estimate for the lumped parameters by the minimization of the sum of the residuals over a set of available data points

Step 3: The second optimization procedureObtain the second estimate for the lumped parameters by minimizing the sum of the error between original data and data obtained through numerical integration (collocation) over a reduced region

Step 4: Final numerical integration to generate the trajectory based on the solution in Step 3 Boundary conditions are assumed perfect to compare the trajectoryTrajectory is compared to underlying smooth trajectory

P. Venkataraman




8


MATLAB was used for all calculations

A combination of symbolic and numerical processing was used to postpone round-off errors

The following MATLAB functions was used in the implementation• fminunc : unconstrained function minimization• matlabFunction : conversion of symbolic objects• bvp4c :

No special programming techniques were necessary

Computations were performed using a standard laptop

P. Venkataraman




9

The Example

Example of fluid flow in a long vertical channel with fluid injection on one side

R is the Reynolds number and Pe is the Peclet number. A is an unknown parameter which is determined through the extra boundary condition. The example is defined for a Reynolds number of 100 for which the value for A is 2.76.

21 1 1 1

2 1 2

3 1 3

1 1 2 3

1 1 2 3

0;

1 0;

0;

(0) 0; (0) 0; (0) 0; (0) 0;

(1) 1; (1) 0; (1) 0; (1) 1;

e

y R y y y RA

y Ry y

y P y y

y y y y

y y y y

P. Venkataraman




10

The Example – Solution to Forward Problem

Solution to forward problem using Bezier function – order 20

P. Venkataraman




11

The Example – Inverse Problem with Smooth Data

The Bezier function technique with 18th order functions

IG : Initial GuessOpt1 : First OptimizationOpt2 : Second OptimizationES : Expected Solution

P. Venkataraman




12

The Example – Perturbation in Boundary Conditions

Most solution to inverse BVP do not consider change in BC

This work accommodates changes in BC as it works with a clipped region

P. Venkataraman




13

The Example – Inverse Problem with Non Smooth Data

10% perturbation – 31 points

Variable y1Variable y2

P. Venkataraman




14



Variable y3

P. Venkataraman




15



Variable y1Variable y2

P. Venkataraman




16



Variable y3

P. Venkataraman




17

Conclusions

This paper presents a robust method for inverse lumped parameter BVP

The method is based on describing the data using Bezier functions

The method involves three sequential applications of unconstrained optimization

The method does not require regularization

The method does not require dimensional control

The method does not require additional information on the nature of the problem or solution

The method accommodates the perturbation of the boundary conditions

a robust technique for lumped parameter inverse boundary value problems

Documents

robust technique

lumped parametersthe

chicago il

posed solution

boundary conditionserror

solution procedurestep

parametersquality of

smooth data