Research starts with PhD students

Chapter 35 Who is Zhang Shuo? Please help me tell you!
That evening, Zhang Shuo received a reply email from Friedrich:

"Hello, Mr. Zhang Shuo.

My name is Friedrich Jost, and I reviewed your paper. I'm sorry to say that I initially read it with a questionable mindset.

Because I don't believe it.

It is impossible to find a general algorithm for any nonlinear partial differential equation.

This was my opinion, and your paper changed my mind.

Among them, the most exciting part is the logic of "proving asymptotic solutions". I specially asked my old friend Maxim and sent that part to him.

You must know him, he's very famous!
Maxim told me that the part about "proving asymptotic solutions" was very complete and could form a perfect logical loop. He said that part was very interesting and he wanted to get to know you. "

The first half of the email talks about irrelevant things. The only thing that is certain is that the logic of "proving the asymptotic solution" is correct.

The second half is the main content.

“I was very interested in your paper and studied it carefully for a long time. I found that your algorithm can produce a wide range of results when it comes to nonlinear problems.

If completely nonlinear equations are involved, the results may even become meaningless.

My judgment, right?
Can your algorithm go a step further, that is, find a more precise range of solutions?"

At the end of the email, Friedrich Jost asked two questions.

One is that 'when it comes to nonlinear problems, the range of results obtained by the algorithm is very wide'. To put it bluntly, the results will become inaccurate.

Another is to ask whether the algorithm can go further.

The first question is very critical.

Partial differential equations can be divided into 'linear' and 'non-linear', and 'non-linear' is not necessarily 'completely non-linear'.

The degree of 'non-linearity' varies from equation to equation.

A linear equation is like a straight road, while a nonlinear equation is like a damaged road. As long as there is damage, it will be classified as "nonlinear".

Obviously, there are different degrees of road damage. If the road is completely damaged and its shape cannot be seen, it can be called "completely non-linear".

The problem with Zhang Shuo's algorithm is that the more nonlinear the program is, the larger the range of the calculated solutions will be.

For example, for a linear equation, the exact solution is 100, and the range of 99 to 101 can be calculated.

For a certain seriously nonlinear equation, the solution region is 99~101, but the solution may be -10000~10000, which is just boxing the solution region within the range.

Although the calculated results for completely nonlinear equations are so large that they are almost meaningless, being able to directly solve partial differential equations is already a surprising achievement.

Zhang Shuo thought about it for a while and wrote back to Friedrich, "Mr. Yost, your judgment is absolutely correct.

The study of completely nonlinear equations involves many of the world's most difficult problems. In order to ensure the accuracy of the calculation results and avoid errors, the only option is to expand the range of the results.

If you want to make the algorithm more accurate, you can modify and improve the parameter evaluation system in the second part of the method paper.

That part uses the parameters of the equation to simulate the human brain's calculations and obtain the results of substituting the numerical values.

In my thesis, what is important is the method of simulating human brain operations, rather than a more efficient algorithm.

As for the part about substitution transformation method and proving asymptotic solution, I can’t think of any further refinement of the method…”

Zhang Shuo then explained some algorithmic issues, reviewed the entire document, and sent the email after confirming that there were no problems.

……

The next morning, there was still no reply email.

Zhang Shuo went to class with Huang Kai.

He really enjoyed the feeling of attending classes with his classmates, as if he were back in his student days.

Of course, that is true.

at the same time.

In the office on the second floor of the Institute of Advanced Mathematics, a female teacher with short, neat hair stood at the door and knocked twice.

"Enter!"

A chubby middle-aged man looked up and shouted, then asked in surprise, "Tong Jie, why are you here?"

The female teacher's name is Tong Jie. She is an associate professor and master's supervisor in the School of Mathematics. She is only 33 years old.

The middle-aged man is Su Bingkang, a professor at the School of Mathematics, a concurrent researcher at the Institute of Advanced Mathematics, and also Tong Jie’s doctoral supervisor.

Tong Jie walked into the office and handed the draft book to Su Bingkang, "Teacher Su, take a look at this, a solution to the nonlinear Schrödinger equation."

Su Bingkang took the draft and read it carefully with doubts.

The analysis in the draft book contains five pages.

When he turned to the second page, his brows were already furrowed. He stared at it for a long time, and then took a pen to check the calculations.

After checking it several times, he pointed to a place on the second page and asked Tong Jie, "Is it that you don't understand this?"

"Correct!"

Tong Jie said, "This transformation is very strange. After substituting numerical values ​​for verification, I found that some are correct and some are wrong, but he finally found the exact solution."

"I checked the results and there is no problem."

Su Bingkang frowned and asked, "Who did this solution?"

Tong Jie said, "I have a student named Zhong Yijing."

"I asked her about it. She asked a PhD student. The PhD student just took a look at it during lunch and quickly solved it."

"Doctoral student? What's his name?" "Zhang Shuo!"

"Zhang Shuo?" Su Bingkang thought the name sounded familiar and asked immediately, "Is he the doctoral student who got the project from the Institute of High Energy Physics?"

"It seems to be him, with the same name, but I'm not sure."

Su Bingkang nodded, he stood up and called Tong Jie, "Come with me to the first floor."

The two of them went downstairs together.

There was a large office on the first floor, where several people were talking, including Qi Zhixiang, who was very authoritative in the Supreme Court.

Su Bingkang went in and shouted, "Come and see this!"

He placed the draft on Qi Zhixiang's desk and explained, "A manual solution of a nonlinear Schrödinger equation, finding a set of exact solutions."

"But there is one step that I don't understand. Please help me!"

People in the office immediately became interested.

Su Bingkang is a regular researcher of the High Court. Most of the doctoral supervisors and professors in the School of Mathematics are only on secondment to the High Court, and those who can serve as researchers are all of very high level.

In addition, Su Bingkang focuses on the research of partial differential equations.

He directly said that he couldn't understand the solution process of the Schrödinger equation, so it must be very interesting.

Several people studied it together and soon found the key point. They also discussed it together, "I checked it, there is a problem with this conversion!"

“Some values ​​were substituted correctly, some were wrong, but the key point is that the exact solution obtained was correct!”

“How can we find an exact solution if we have made the wrong transformation?”

"The question is, how did it get turned around?"

“I don’t understand!”

"Who did this solution? And they found an exact solution. This kind of equation usually doesn't have an exact solution, right?"

They discussed it over and over but could not come to any conclusion.

As a professional mathematician, when I find that I can't understand a small problem in a field, I feel as uncomfortable as being scratched by a cat.

Su Bingkang talked about the solution completed by Zhang Shuo.

Qi Zhixiang made a straightforward decision, "Go ask him! Go now, this problem must be solved, otherwise I won't be able to sleep!"

"It's not a good idea to ask a doctoral student, right?" a professor hesitated.

"What are you afraid of? Confucius was not ashamed to ask questions from people of lower status. Zhang Shuo is not an ordinary student. He is the one who solved the problem. What's wrong with us asking him?"

Qi Zhixiang said indifferently.

A few people simply went to look for Zhang Shuo together, and after asking around, they found the classroom door.

……

Zhang Shuo is taking a class called "Numerical Solutions to Inverse Problems" and the lecturer is Professor Lin Zhitao.

An inverse problem is a research question involving a mathematical model that can be used to solve a class of problems.

These questions themselves are not analyzable.

The basic idea of ​​the mathematical solution to the inverse problem is to first establish a model to describe its possible behavior, and then use numerical techniques to solve the model.

Lin Zhitao's lecture style is passionate. He stood on the stage and talked non-stop. "For example, the dynamic calculation of gas and liquid on the fluid flow surface is a very complicated problem."

"What we need to do is to build a model and use numerical methods to solve this problem."

"This allows prediction of the dynamic behavior of fluids (gases and liquids) on flow surfaces..."

As he was speaking, he noticed a few people standing at the door.

Qi Zhixiang, Su Bingkang?

Two researchers from the High Court came to see him. Were they inviting him to join the High Court? Or were they asking him about some professional research issues?
The former is impossible.

If it’s the latter, maybe you can join the math program at the Supreme Court?
It’s definitely more funding than doing your own project!
Lin Zhitao suddenly lost interest in continuing his lecture. He looked at the time and simply stopped, "That's all for today."

"In the next class, we will talk about numerical methods to solve classic examples of the dynamic behavior of gases and liquids on fluids. It's best for everyone to prepare in advance."

"get out of class is over!"

As soon as Lin Zhitao announced the end of get out of class, the people at the door came in impatiently, and directly ignored him, and asked the students, "Who is Zhang Shuo?"

"Is Zhang Shuo here?"

"Who is Zhang Shuo..."

Everyone immediately looked at Zhang Shuo who was sitting at the side of the second row.

Zhang Shuo just felt confused. He pointed at himself and asked in confusion, "Looking for me? I'm Zhang Shuo..."

Qi Zhixiang walked over quickly, opened the draft book and handed it over, and asked, "Student Zhang Shuo, can you tell us the transformation method of this step?"

"Why is this so? I made the wrong transformation, but got the exact solution?"

Zhang Shuo took the draft and looked at it carefully, then asked subconsciously, "Isn't this the question that the second-year graduate student asked in the cafeteria?"

"Yes, Zhong Yijing, she is my student."

Tong Jie walked to the front and said, "In this solution process, how did the transformation on the second page get turned over?"

Zhang Shuo took another look at the draft, a little confused. Isn’t it just the transformation of the target solution of the nonlinear Schrödinger equation?
These professors don’t understand?
(End of this chapter)