A Robust Technique for Lumped Parameter Inverse Boundary Value Problems

1. A Robust Technique for Lumped Parameter Inverse Boundary Value Problems P. Venkataraman

2. Todays Presentation Introduction to Lumped Parameter Inverse BVP The Solution Procedure The Example Conclusion

3. 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:

4. 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

5. 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

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

7. 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

8. 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

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.

10. The Example – Solution to Forward Problem Solution to forward problem using Bezier function – order 20

11. 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

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

13. The Example – Inverse Problem with Non Smooth Data 10% perturbation – 31 points Variable y2 Variable y1

14. The Example – Inverse Problem with Non Smooth Data 10% perturbation – 31 points Variable y3

15. The Example – Inverse Problem with Non Smooth Data 20% perturbation – 31 points Variable y2 Variable y1

16. The Example – Inverse Problem with Non Smooth Data 20% perturbation – 31 points Variable y3

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