Switch Search Direction Algorithm versus GN and LM Parameter Estimation Algorithms in Modeling Inverse Heat Transfer Problem

The SSDPE method is a robust parameter estimation technique which minimizes the error. Modeling inverse heat transfer is estimating values of parameters of a mathematical model from measured data. A governing differential equation, its boundary and initial conditions are used to derive the solution, and then the solution is coupled with an efficient parameter estimation algorithm to search for the unknown thermal parameters. The SSDPE technique was developed to couple with a Green’s function solution of a biological system to estimate three unknown parameters [1]. A Green’s function solution is combined from superimposing all finite effects caused by the applied forcing function; such as, step-, ramp-, or pulsed-function. The common nonlinear least squares techniques are hard or impossible to couple with this kind of nonlinear discontinued solution. The preciseness of a parameter estimation technique can be measured from its ability to minimize the random noise effect on the estimated results. Therefore, the SSDPE is compared with two known methods. The three techniques are investigated with different levels of random noise added on top of the simulated measured data. This paper validates the ability of the SSDPE to estimate parameters by testing the method versus the Gauss-Newton, and the Levenberg Marquardt estimation techniques.


INTRODUCTION
Parameter estimation algorithms are very important tools to estimate unknown parameters from experimental measurements. Parameter estimation techniques are built to connect experimental and computational research work. Inverse heat transfer is about estimating parameters of the studied thermal system using thermal measurement. The properties of a studied thermal system are commonly the aim for many computational and experimental researches, which requires parameter estimation. Engineers, economists, mathematicians, physicians, etc. need parameter estimation techniques to predict the behavior of their studied systems. Economists use these techniques to predict future of their markets, etc. Parameter estimation techniques help physicians to predict the spread of diseases or to detect burn depth.
Two of the best parameter estimation methods are used in this paper to compare the ability of our SSDPE algorithm to predict the correct parameters from a mathematical model.
In my previous published papers [1][2][3][4], we coupled the Bioheat mathematical model {Green's function solution} with the SSDPE Algorithm to predict blood perfusion, core temperature, and thermal resistance (Burn depth). The reason for developing the new SSDPE algorithm was to overcome the difficulty associated with calculating the Jacobian matrix for the Green's function solution. In this paper, we demonstrate that the SSDPE algorithm works efficiently to search for two optimal parameters by comparing the functionality of our algorithm versus the Gause-Newton and Levenberg-Marquardt algorithms. A set of simulated temperature measurements were created to evaluate the ability of the three parameter estimation algorithms to detect the two parameters {β 1 , β 2 } and whether their results are close. Five different noise levels of simulated measurements were used to analyze the three parameter estimation algorithms.

BACKGROUND ON PARAMETER ESTIMATION
The searched parameters are β = [β 1 , β 2 , …,β 3 ] which fit the analytical solutions T m (β) = T(x s , t m , β) and the measured data (from a sensor) The optimal searched value of ( ) ∑ ( ) is where, ,

GAUSE-NEWTON & LEVENBERG-MARQUARDT ALGORITHMS
The objective is to minimize the sum of squares of the residual function S(β). The two common algorithms for solving nonlinear least-squares problems are the Gause-Newton (GN) Method and the Levenberg-Marquardt Algorithm (LM) [5][6][7][8][9][10][11][12]. The two methods are built from the Newton's method Eq. 2: Where H is Hessian (matrix of all 2 nd derivatives) and G is gradient (vector of all 1 st derivatives) [ ] For only one term from the Hessian matrix: Eq. 6 is reduced to: The sum of squares of the residual function is defined as: The G gradient is expressed as: Also the Hessian matrix is expressed as: Where J(β) is the Jacobian and f (β) is: When r i = (Y i -T i ) → 0 that makes the value of f(β) is small, compared to the product of the Jacobian, the Hessian matrix can be approximated as: This gives the Gauss-Newton algorithm One limitation that can happen with this algorithm is that the simplified Hessian matrix might not be invertible. To overcome this problem a modified Hessian matrix can be used: where I is the identity matrix and μ is a value such that makes Hm(β) positive definite and therefore can be invertible.
This last change in the Hessian matrix corresponds to the Levenberg-Marquardt algorithm:

SWITCH SEARCH DIRECTION ALGORITHM-SSDPE
The best fit of the analytical model is when the value of S is the minimum value. To minimize S the parameters are varied one at a time over a range of values to iteratively arrive at the minimum. First the parameter β 1 is held constant while varying the 2 nd parameter "β 2 ".
The best value of "β 2 " is then used while varying the 1 st parameter "β 1 ". This is repeated over finer ranges of values until sufficient resolution has been achieved. An example of the search process is demonstrated in Fig. 7.

SYSTEM MODEL
The SSDPE algorithm was developed and coded in Matlab and we have demonstrated its ability to estimate two parameters of the thermal system. The two unknown parameters are consequently needed to be determined from the simulated experimental measurements. We modeled a simulated temperature probe which provides eleven measured points to estimate the base temperature and the thermal convection coefficient of the fluid around the probe.
Then a comparison of the estimated values is made using the three parameter estimation algorithms. The probe is modeled as a cylindrical fin, which conducts energy and transfers it by convection through its surface area to the ambient environment at temperature T ∞ .

Figure (3) Physical representation of the mathematical formulation
The effective thermal conductivity of the temperature probe is taken as k = 100 W/m-C o and the dimensions of the duct are indicated in figure (3) The solution was formulated from the governing equation, boundary conditions and the initial condition of the thermal system: The first parameter to be estimated is the base temperature of the fin (probe), β 1 = θ b . The second parameter is, β 2 = √hp/kA c from which we will get the convection coefficient. The model is taken as steady state.
Over this eleven measurement points the analytical model results are matched with the experimental data. The measured temperature is the input for the model. The fit between the model and data is quantified using the average root of the squared residual values where the residual is

Figure (4) eleven clean and noisy simulated temperature measurements
An example of one set of iterations for β 1 and β 2 is shown in Fig. 5. For the top part of the two curves, β 2 is kept constant while β 1 is varied. Then the process goes to switching the search direction by varying β 2 and keeping β 1 constant. The minimum value of each curve is then used for the next set of iterations.   Table (1).

RESULTS
The results from figure-(6 & 7) and table (3) indicate the ability of the SSDPE Algorithm to search for the optimal estimated parameters, since the method gave the same results  As a test of the parameter estimation, simulated sensor data was generated with different levels of random noise added to the temperature signals. Table 3    The corresponding input temperature curve is shown in the top portion of Fig. 8 for the ± 0.7 C o noise case. The matching analytical temperature is also shown for comparison. The two temperature curves align very well. The analytical curve noise comes from the noise on the simulated measured temperature signal that is the input driver for it.

CONCLUSION
In conclusion, the estimated parameters β 1 and β 2 using the SSDPE method are clearly very close to those estimated by the GN & LM methods which prove the stability of the SSDPE method. The sensitivity of the SSDPE to predict parameters from noisy data was also demonstrated for wide range of possible level of noise which is commonly associated with the sitting of the probe, wiring data acquisition, environment, etc. The results support clearly the ability of the model to predict the searched parameters from different level of noisy measurements. The SSDPE has significant use with some complicated numerical or analytical solutions which have higher degree of complexity. Based on these encouraging results, further study will be done with the SSDPE to improve its ability to guess the predetermined domain of each parameter {β i,min , β i,max }.