4 Inverse Problem of Liquid Crystal Director Profile
4.1 Introduction to the Inverse Problem
In Chapter 3, a model extraction technique has been implemented by comparing the model simulation result with the measured data to retrieve the model parameters.
However, in the case of LC, the direct retrieval of a LC director profile from the measured data is difficult to check its accuracy and reliability. Recently, C. J. P.
Newton in Hewlett-Packard Laboratories investigated the possibility of inverse problem to retrieve the LC director profile with singular value decomposition scheme [17]. Although it is inspiring, the work is limited to the stability and converging properties of inverse problem technique developed. No practical applications on the LC director profile retrieval from experimentally measured data were reported. In this chapter, we go one further step by deriving the necessary equations and detailed steps to successfully achieve the target of the LC director profile retrieval from the measured data.
4.1 Introduction to the Inverse Problem
The relation between the inverse problem and the forward problem can be understood as follows: In Eq. 4.1, a model of a physical system is described by a matrix and a state vector depicts the system status. Then the forward problem is that we can predict the system response with Eq. 4.1
A x
b
= .
Ax b (4.1)
The inverse problem can then be easily understood as follows: Based on the measured system response vector and the model matrix , we can retrieve the parameters of the physical system .
In the past, inverse problems are usually treated as a data fitting procedure with forward problem. This can be done by varying the state vector to obtain the best fit. However, some problems are not suitable for data fitting because either it may be difficult to fit the response data to a model or the best fit could yield a spurious solution. Hadamard introduced some useful criteria to categorize the problems: [18]
Criterion 1: For all admissible data, a solution exists.
Criterion 2: For all admissible data, the solution is unique.
Criterion 3: The solution depends continuously on the data.
A problem that violates any of the three criteria is called ill-posed. The third criterion is actually the stability condition, which requires that a small perturbation to the input
does not produce a large change in the output. When a problem is ill-posed, it is not easy to determine the true solution objectively. We will start from a simple mathematical problem to illustrate the difficulties of an inverse problem.
Assuming that the model matrix A and the input vector x are:
four-digit accuracy,
1 1 1
Therefore, the first two Hadamard criteria are satisfied. To analyze the stability criterion, we add noise to with a noise level around 0.1 percent of . We then introduce the following parameters to reveal the instability of a problem with S:
x x
2
We apply Eq. 4.5 to calculate S by adding 10,000 different random noise to and present the distribution of in Figure 4-1(a). We can find that most of the values fall between 0 and 1.1, which indicates that a small the perturbation to does not generate a large variation in b. Thus a stable solution can be obtained for this problem.
x
S
For the inverse problem, we repeat the calculation by adding small noise to and estimate the variation of and the resulting instability parameter S is plotted in Figure 4-1(b). By comparing Figure 4-1(b) to Figure 4-1(a), it is clear to find that the instability parameter S of the inverse problem is 10 thousand times larger than that of the forward problem. Therefore, the inverse problem fails to satisfy Hadamard’s third criterion of stability and is ill-posed in nature.
b
(a) (b)
Figure 4-1. (a) The Stability distribution of . (b) The Stability distribution of .
x b
By looking further into Ax = b, in a general case withA R∈ M N× , x R∈ N, and
∈ M
b R . To find a solution x, we can encounter the following three situations:
(1) If M =N, we encounter a linear system of variables with equations. If A is nonsingular with , the system possesses a unique solution. On the
other hand if is singular (i.e.,
N N
( )
det A ≠0
A det
( )
A =0), the system has infinitely many solutions.(2) If M <N, we have a linear system with less equations than variables, which is called under-determined. This problem can be reduced to the situation (1) by expanding matrix A and matrix b with N−M rows of zero, respectively.
(3) If M >N , we have a linear system with more equations than variables, which is called over-determined and ill-posed. However, we can find the least-squares solution
of Ax = b by finding a vector x in Rn that minimizes Ax b− .
Let us return to our simple problem. Since the problem is ill-posed, any matrix inversion algorithm will fail to find the desired solution . Therefore, specialized technique must be invoked to solve the inverse problem. One of the successful approaches is the so-called regularization [19]. We will briefly describe the idea of the regularization scheme, known as Tikhonov regularization. As noted above, we can
find the least-squares solution of Ax = b by finding a vector x in Rn that minimizes x
Ax b [20]. The key issue is how to choose the sensible solution from the space of − 2
reasonable solutions. An idea is to reduce the size of the solution space by invoking
additional constraints. We can implement this idea by adding an additional term to Ax b with a carefully selected regularization parameter − 2 λ
{
2}
min λ .
= +
x x Ax - b Ix2 (4.6)
A graphical tool, which is termed as the L-curve, can be used to help us choosing the regularization parameter. The L-curve graphical technique plots the λIx 2 on the y-axis and the Ax - b 2 on the x-axis by varyingλ . Figure 4-2 exhibits the L-curve
for our illustrative example, which explains the name of the L-curve to be due to the shape of the plot. The optimal value of λ is at the corner of the curve. By using this
value, we can find the solution x=
[
1.0163 0.9188 1.0481 1.0429]
T, which is very close to the known state vector x given in Eq. 4.3.λ Ix
2Ax - b 2
Figure 4-2. The L-curve is a curve in a log-log scale with λIx 2 on the y-axis
and Ax - b 2 on the x-axis by varying λ from 10−5 to 1. The optimal value of λ is be chosen is at the corner of the curve labeled with the red circle.