The military-industrial scientific research system of the academic master.

Marcel Schmidt would never have imagined that many years later, when artificial intelligence research even began to occupy seats for the Nobel Prize and the Fields Medal, his remark "neural networks are also mathematics" just for the sake of argument would become a hit and be regarded as a guiding principle by researchers in the field.

Even Schmidt himself became one of the objects of worship in certain ceremonies to appease the server's machine spirit.

But these are all things for later.

In any case, at this point in time at the end of 2004, the fact that Advances in Mathematics chose to publish a paper related to global optimization and data models was big news that could cause a slight earthquake in the industry.

Fortunately, one of the authors of the paper is the winner of the last Fields Medal, and this is his first documented collaborative paper.

Therefore, the focus of the industry quickly shifted from the content of the article itself to the research direction that Chang Haonan was concerned about.

When bigwigs publish articles in top journals, they can themselves set the trend.

Chasing hot topics is a feature that most researchers, both in China and abroad, have to experience...

Delft University of Technology, The Netherlands.

Professor Joseph Blatt was standing in front of a white light interferometer, flipping through a series of test results displayed on the computer screen with a complicated expression.

Not far away, his student Qin Shaofeng was concentrating on checking a test report that had just been spit out by the equipment in front of him.

"Professor, the results have been compared..."

A few minutes later, Qin Shaofeng took off his glasses and put them aside, then spoke in a somewhat lost tone.

However, before he could finish his sentence, Blatt interrupted him:

"It should not match the previous two tests?"

Qin Shaofeng was stunned at first after the result was directly guessed, then nodded:

"The same sample was tested, and the same 1D measuring instrument was used. However, the difference between the peak-to-valley values ​​in the three tests was more than 0.3mm, and the root mean square value was also around mm, which completely failed to meet Party A's requirements..."

As a higher education institution located in a small lowland country and in an unknown city, the Delft University of Technology is not only far less famous than giants like Harvard, MIT, and Oxford, but is even far inferior to the University of Amsterdam next door.

However, it has considerable strength in optics and optical engineering.

Joseph Blatt is one of the most outstanding scholars in this field.

He worked for Philips and ASML in the 70s and 80s, and moved to academia in 1988. He subsequently made a series of outstanding contributions in the fields of optical imaging and optical storage technology.

He is also one of the founders and current president of the European Optical Society.

The project he is currently following up is to develop a new non-contact three-dimensional coordinate measurement technology commissioned by Taylor Hobson to detect aspheric lenses with a peak-to-valley (PV) value better than 0.1mm.

This accuracy is not exaggerated for contact measurement technology based on atomic force analysis, but because the probe needs to contact the surface of the workpiece, it is not suitable for some particularly fragile optical components.

In comparison, non-contact measurement using interference method and geometric light method is obviously much more friendly.

Of course, it is much more difficult to achieve the same accuracy as contact measurement.

Faced with so many failures in the past few months that he couldn't remember how many times, Qin Shaofeng realized that there was not much point in continuing to conduct repetitive experiments.

What is needed now is a change of thinking:

"Have you considered recalibrating the public observation points of the equipment, so that the position offset error can be reduced?"

He suggested first.

The errors in the surface detection process mainly come from three aspects: motion axis error, probe error and position offset error.

There is not much difference between the first two methods for contact and non-contact methods, so the accuracy problem of the non-contact method mainly appears in the position offset error.

Since there is no probe that can directly contact the surface and directly determine the measurement reference origin, the error data captured by the non-contact device is essentially equivalent to the superposition of surface error and position offset error.

Only by separating the latter through algorithms can we obtain relatively accurate results.

So Qin Shaofeng’s idea is actually correct.

But recalibrating the observation points...

It is basically equivalent to "try restarting" after a computer malfunction.

This is an attempt with a bit of metaphysical color after I couldn't find the specific reason.

Although it does work wonders sometimes, it is a bit too unpresentable for researchers in optical engineering.

Therefore, Blatter rejected it almost immediately:

"It is not meaningful."

He waved his hand and then began to analyze the cause of the error to Qin Shaofeng:

"The calibration logic of this equipment is currently based on the two-dimensional model of the aspheric surface. Although some corrections have been made using the Gauss-Newton method, in essence, the center of the aspheric surface measured is still not the center of the actual surface, and the model curve obtained after two-dimensionalization is not the meridian of the aspheric surface, so simply resetting the observation point will not have any effect..."

The fact that such a detailed explanation could be given before even seeing the experimental report made the latter sigh that, after all, old people are the wisest.

But this emotion lasted less than two seconds.

Blatt paused for a moment, then continued:

"And more importantly, I tried it last night and it didn't work..."

"..."

The filter shattered instantly.

Of course, the work still has to go on.

After some thought, Qin Shaofeng came up with a new approach:

"Then if we upgrade this device with a better control computer, can we skip the two-dimensional model calculation step and directly use the three-dimensional correction algorithm to obtain the center of the circle and the coordinates of the non-spherical surface in space?"

This time, it was not rejected directly.

"The problem with the 3D correction algorithm is that it is prone to local convergence... leading to outliers with extremely outrageous errors, which is very unacceptable in industrial production..."

Blatter has clearly been thinking about this for a long time:

"In fact, if we can overcome the local convergence problem, we only need to generate three-dimensional aspheric surface data with position error and surface shape error through simulation, and then compare the generated coordinate points with the standard aspheric surface equation to obtain the error of each coordinate point. Finally, we can iteratively optimize the corresponding position error by using the minimum root mean square error principle..."

In fact, asphericity does not mean no regularity. Its standard equation is generally a quadratic surface superimposed with high-order coefficients. In three-dimensional space, there are only rotations and translations. There is no need to consider rotation along the z-axis, that is, there are only changes in 6 degrees of freedom. As for the position parameters, they can be jointly represented by two third-order matrices containing second-order partial derivatives and three error terms.

Therefore, the final problem can be summarized as: using a reasonable global optimization algorithm to optimize the objective function so that its error function value is minimized.

Qin Shaofeng's foundation is indeed solid. After listening to his mentor's ideas, he quickly caught some new ideas.

However, it is still vague:

“So the reason why you said before that the two-dimensional model still cannot achieve the desired effect after optimization is because the Gauss-Newton method is not positive definite in the process of solving the optimal solution?”

"Not quite."

Blatter shrugged helplessly:

"In fact, some people in Japan have already used the Levenberg-Marquardt method, which is positive definite when solving the Hessian matrix, for three-dimensional measurement, but the effect still does not meet expectations."

"Levenberg-Marquardt method..."

This term finally made Qin Shaofeng completely grasp the flash of inspiration, and he instantly became energetic:

"A few days ago, a classmate who majored in mathematics mentioned that a very famous scholar had just published a paper related to this algorithm, which was to improve the problem of local false optimality... and because the content was a bit out of touch with pure mathematics, it caused some controversy there..."

Blatt himself actually has a certain foundation in mathematics, but it is indeed not comparable to that of a serious mathematician.

But it was enough for him to realize the value of this paper in the field of optics.

"Is there any more detailed information? I want to take a look at this paper."

He asked with his eyes almost shining.

“I have a link to the paper…”

Qin Shaofeng took off his gloves and turned to go to the office to get his computer.

However, Blatt couldn't wait any longer and pointed to his computer next to him:

"Maybe you could just use this..."

A few minutes later, a paper of more than twenty pages was handed to Blatt.

“For a math paper, it’s really a bit too long.”

He took it half jokingly.

Qin Shaofeng's answer was quite serious:

"My classmate said that if you are only interested in the Levenberg-Marquardt method itself, you only need to read the first half..."

"understood……"

Blatt didn't even raise his head, but just waved at Qin Shaofeng, indicating that the latter could go and rest first.

However, Qin Shaofeng had just returned to his seat in the office with a long sigh of relief when the phone next to him suddenly rang.

"Qin, please help me contact Taylor Hopson Company."

"Let's say I have a new technical solution... but I haven't figured out some of the specific details yet, so I may need to have a meeting with them to discuss it..."