10.3 The physics of springs  (Page 11/11)

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$e=Fc+ϵ,$

where ϵ is the error in the measurements. Note that if A T is not invertible we can replace ${A}^{-T}$ with ${A}^{+T}$ , where ${A}^{+T}$ is the pseudo inverse of A T , so that $F=diag\left({A}^{+T}f\right)$ .

Maximum likelihood estimate

With our problem now rewritten in a statistically more usable form, we can easily apply a method involving the maximum likelihood estimate, described by Calvetti and Somersalo [link] , pages 35-37. Applying this method to our rewritten problem yields the equation

$\left[{F}^{T},{\Gamma }^{-1},F\right]c={F}^{T}{\Gamma }^{-1}\overline{e},$

where

$\overline{e}=\frac{1}{N}\sum _{j=1}^{N}{e}_{j},$

and Γ is the covariance matrix of the random variable e . This method assumes that the error is normally distributed, which is a reasonable assumption (see "Distributions" ). Now with this formulation we can solve our problem using statistical knowledge of the problem.

Problems with maximum likelihood estimate approach

The covariance matrix, Γ , comes from the distribution of ${e}_{j}=A{x}_{j}$ . Then, ${\Gamma }_{\left(e\right)}=A{\Gamma }_{\left(x\right)}{A}^{T}.$ In our problem where $A\in {\mathbb{R}}^{16×14}$ and ${\Gamma }_{\left(x\right)}\in {\mathbb{R}}^{14×14}$ , ${\Gamma }_{\left(e\right)}$ is a singular matrix. In fact in any system where $A\in {\mathbb{R}}^{m×n}$ where $m>n$ , ${\Gamma }_{\left(e\right)}$ is singular. This presents a problem when solving the system

$\left[{F}^{T},{\Gamma }_{\left(e\right)}^{-1},F\right]c={F}^{T}{\Gamma }_{\left(e\right)}^{-1}\overline{e}.$

We can avoid the issue of invertibility of ${\Gamma }_{\left(e\right)}$ by instead of solving the problem $e=Fc$ , we solve the problem $x={F}^{\text{'}}c$ where

${F}^{\text{'}}={A}^{+}F,\phantom{\rule{1.em}{0ex}}{A}^{+}=\phantom{\rule{4.pt}{0ex}}\text{pseudoinverse}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}A.$

Our problem then is

$\left[{F}^{\text{'}T},{\Gamma }_{\left(x\right)}^{-1},{F}^{\text{'}}\right]c={F}^{\text{'}T}{\Gamma }_{\left(x\right)}^{-1}\overline{x},$

where

$\overline{x}=\frac{1}{N}\sum _{j=1}^{N}{x}_{j}.$

However we see that even if F is nonsingular ${F}^{\text{'}}={A}^{+}F$ is singular, and our problem is not avoided.

Finding optimal force vector

Setting up our problem with the equation $e=Fc$ (see "Rewriting the Problem" ) helps us see an important realization. Because F is diagonal,

$e=\left[\begin{array}{c}{e}_{1}\\ {e}_{2}\\ ⋮\\ {e}_{n-1}\\ {e}_{n}\end{array}\right]=\left[\begin{array}{c}{F}_{1,1}{c}_{1}\\ {F}_{2,2}{c}_{2}\\ ⋮\\ {F}_{n-1,n-1}{c}_{n-1}\\ {F}_{n,n}{c}_{n}\end{array}\right].$

Thus assuming ${F}_{i,i}\ne 0$ ,

${c}_{i}=\frac{{e}_{i}}{{F}_{i,i}}\phantom{\rule{1.em}{0ex}}\text{or,}\phantom{\rule{1.em}{0ex}}{k}_{i}=\frac{1}{{c}_{i}}=\frac{{F}_{i,i}}{{e}_{i}}.$

This formulation helps us see another important realization. Namely the importance of ${F}_{i,i}\ne 0$ . Recall that $F=\text{diag}\left({A}^{-T}f\right)$ . We have control over the f we choose, so it would be wise to choose an f so that ${A}^{-T}f$ has no zero elements. This happens only if every row of ${A}^{-T}$ is not orthogonal to f . We would like to choose an f that had as many zero elements as possible, but also is not orthogonal to any of the rows of ${A}^{-T}$ . We can formulate our search for an optimal f into the optimization problem,

$\text{min}\phantom{\rule{1.em}{0ex}}{f}^{T}f+\gamma \frac{1}{\delta +\text{min}\left(\left|{A}^{-T},f\right|\right)},$

where γ and δ are adjustable parameters to get the best result. Because of the unsmooth nature of this minimization problem, this is best solved using MATLAB's fminsearch function.

In some networks we may have limitations on the f that we choose. For example in our original network

${f}_{1}={f}_{6}={f}_{7}={f}_{8}={f}_{10}={f}_{13}=0$

for every f that we choose. In the case that ${f}_{j}=0$ , we can rewrite our optimization problem to

$\text{min}\phantom{\rule{1.em}{0ex}}{f}^{\text{'}T}{f}^{\text{'}}+\gamma \frac{1}{\delta +\text{min}\left(\left|{A}^{\text{'}},{f}^{\text{'}}\right|\right)},$

where

${A}^{\text{'}}=\left[\begin{array}{cccccc}{A}_{1}^{-T}& \cdots & {A}_{j-1}^{-T}& {A}_{j+1}^{-T}& \cdots & {A}_{n}^{-T}\end{array}\right],\phantom{\rule{1.em}{0ex}}{A}_{i}^{-T}=\text{the}\phantom{\rule{4.pt}{0ex}}i\text{th}\phantom{\rule{4.pt}{0ex}}\text{column}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}{A}^{-T},$

and

${f}^{\text{'}}=\left[\begin{array}{c}{f}_{1}\\ ⋮\\ {f}_{j-1}\\ {f}_{j+1}\\ ⋮\\ {f}_{n}\end{array}\right].$

Using these methods, we found optimal f 's using MATLAB's fminsearch function on simulated data from small simple spring networks. In every network the A matrix was square and invertible. We tested networks having 2, 4, 6, and 8 springs. In each case our initial force vector was a perfectly reasonable guess that gave incomplete solutions for the spring constants. However when we inputted that initial force vector into fminsearch , the output was a force vector that made physical sense and gave accurate solutions for the spring constants.

Further research

There are two main areas of further research that directly follow the research presented in this paper. One is applying the maximum likelihood estimate approach to spring systems having a singular A matrix. We saw in "Problems with Maximum Likelihood Estimate Approach" the problems we ran into when we applied the statistical technique to a system with a singular A matrix. But this problem is expected due to the underdetermined nature of the problem (see "An Inverse Problem" ). We dealt with that problem by stacking originally, so further research could be done in using the statistical approach and stacking.

The other area for further research is in optimizing the f vector (see "Finding Optimal Force Vector" ). Further work could be done on tweaking the minimization problem so that the optimal f can be found for larger networks and networks with more restrictions on f . Also, if our system is underdetermined then even an optimal f won't give us a complete solution without stacking. So more research could be done in trying to find a set of optimal f 's that, when stacked, give the most accurate and complete solutions.

Acknowledgements

We would like to thank Dr. Steven Cox and Dr. Mark Embree for their guidance, as well as Dr. Derek Hansen and Jeffrey Hokanson for their help and support.

This Connexions module describes work conducted as part of Rice University's VIGRE program, supported by National Science Foundationgrant DMS–0739420.

Pfieffer, P. Pfieffer Applied Probability. Connexions.org.

Cox, S. CAAM 335: Course Notes. www.caam.rice.edu.http://www.caam.rice.edu/ cox/main.pdf

Cox, S., Embree, M.,&Hokanson, J. CAAM 335: Lab Manual.

www.caam.rice.edu.http://www.caam.rice.edu/ caam335lab/labman.pdf

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