A Robust Technique for Lumped Parameter Inverse Boundary Value Problems

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A Robust Technique for Lumped Parameter Inverse Boundary Value Problems. P. Venkataraman. Todays Presentation. Introduction to Lumped Parameter Inverse BVP The Solution Procedure The Example Conclusion. Lumped Parameter Inverse BVP. An Example :.

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### A Robust Technique for Lumped Parameter Inverse Boundary Value Problems

P. Venkataraman

Todays Presentation

Introduction to Lumped Parameter Inverse BVP

The Solution Procedure

The Example

Conclusion

Lumped Parameter Inverse BVP

An Example :

Fluid flow in a long vertical channel with fluid injection

Forward Problem:

Given:

and

Find:

Inverse Problem:

and

Given:

Find:

Lumped Parameter Inverse BVP

Forward Problem:

Given:

and

Find:

• 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 Problem:

and

Given:

Find:

• 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

Lumped Parameter Inverse BVP

The solution of inverse BVP in this paper is robust:

is natural

based on derivative information

procedure can be adapted fro the forward problem too

does not require regularization

does not require dimensional control

does not require partial information

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

The Solution Procedure

Step 1:

Data Smoothing using a recursive Bezier filter

Bezier 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 procedure

Obtain 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 procedure

Obtain 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 trajectory

Trajectory is compared to underlying smooth trajectory

The Solution Procedure

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

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.

The Example – Solution to Forward Problem

Solution to forward problem using Bezier function –

order 20

The Example – Inverse Problem with Smooth Data

The Bezier function technique with 18th order functions

IG : Initial Guess

Opt1 : First Optimization

Opt2 : Second Optimization

ES : Expected Solution

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

The Example – Inverse Problem with Non Smooth Data

10% perturbation – 31 points

Variable y2

Variable y1

The Example – Inverse Problem with Non Smooth Data

10% perturbation – 31 points

Variable y3

The Example – Inverse Problem with Non Smooth Data

20% perturbation – 31 points

Variable y2

Variable y1

The Example – Inverse Problem with Non Smooth Data

20% perturbation – 31 points

Variable y3

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