I was a top student in America.

Chapter 112 Now I Understand (Please Subscribe)

Arjun's concerns are valid.

There are less than two weeks left until the interim report.

If we switch to Li Ao's candidate weights now, the entire team would need to reorganize the parameter scan and summarize the report data, which would be extremely costly to experiment.

If the final results still fail to tighten the lower bound estimate, the entire report will be greatly affected, and it may even affect the Department of Energy and Argonne National Laboratory's overall evaluation of the model.

Arjun felt much more at ease if the proposal had been put forward by the experienced Amber.

However, Li Ao was new to the area and lacked practical experience in interdisciplinary projects.

Rudy listened to Arjun's concerns but did not immediately express his opinion. Instead, he gently tapped the table with his index finger.

The office fell silent as everyone awaited their advisor's decision.

The knocking stopped after about ten seconds.

Rudy sat up straight and made a decision: "Mathematical derivation is like a microscope, helping us see the essential structure behind the numerical results. Since Leo and Amber are in charge of the boundary weights, we should give this candidate solution a chance to be validated."

"Even if it fails, the entire team will bear the risk. We should quickly open a separate validation branch and run a parameter scan with the new candidate weights."

"Yes, Professor Rudy." Arjun immediately accepted the task.

After the follow-up work was arranged, Rudi did not linger and directly announced the end of the group meeting.

After the meeting, everyone returned to their work.

The parameter scanning and results plotting modules are primarily maintained by Carter and Nicole, whose workload will increase dramatically in the coming days.

Li Ao also wanted to participate in programming, and also to test how effective the "complete concentration" state was when writing scientific research code.

However, JFI's code repository has strict access restrictions, and the team's division of labor is already clear, so he can only observe from the sidelines for the time being.

For the next two days, Li Ao went to the research institute as usual.

Although he didn't need to directly intervene in the parameter scan, he wasn't idle; he took the opportunity to learn functional analysis.

During his studies over the past few days, he has repeatedly come into contact with mathematical tools such as Hibernate spatial projection, bounded linear functionals, operator norms, weak convergence, and compactness.

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While researching online, he had seen many complaints from mathematics students about the course.

Some people joke that once a function enters the memory space, it no longer resembles a function.

If convergence is replaced with a weak meaning, the proof immediately doubles.

However, functional analysis is like strength training in the field of mathematics; it is painful to practice, but after you finish, your thinking becomes exceptionally solid.

Whether studying discrete energy spaces or dealing with the boundedness of boundary trace operators, these tools are indispensable.

By leveraging the terminology capabilities of the "Model Analysis" tool, Li Ao broke down complex concepts into four basic modules: space, norm, operator, and energy, making them much easier to understand.

That morning, Li Ao sat at his workstation with Rudin's "Functional Analysis" spread out in front of him.

Amber, who was standing next to him, was relatively free. Apart from occasionally checking on Carter and Nicole's progress, she spent the rest of her time working out her doctoral thesis on draft paper.

"If we consider the discrete energy space as a finite-dimensional Hilbert space, the boundary values ​​can be written as a linear operator."

Li Ao quickly solved a minor problem in an example question, muttered to himself, and then turned his attention to the next question.

The problem requires you to use the conclusions of bounded linear operators and HiIbert space projection to prove that the local constraints of the boundary trace operator are bounded under the discrete energy norm, and to give the corresponding operator norm estimate.

"This is a bit of a problem."

After carefully reading the question, Li Ao turned to a new page of draft paper and began to devise a proof.

He had just written down a few lines of equations when Amber's voice came from beside him: "Leo, these kinds of estimation problems can be quite difficult to figure out by yourself. I happen to have some free time now, so let me explain it to you."

Amber knew that Li Ao was taking extra classes in functional analysis, and she was actually looking forward to it, finally having the opportunity to guide him as a senior.

She wasn't able to help much with the boundary weight issue.

Two days ago, she shared some learning tips and told Li Ao to ask questions whenever he encountered difficulties, but two days have passed and she hasn't even had a chance to answer his questions once.

She even began to wonder if Li Ao was just shy and embarrassed to bother others.

Therefore, when she saw that Li Ao seemed to be stuck on a certain question today, she took the initiative to come over.

Seeing Amber's enthusiastic gestures, Li Ao felt slightly embarrassed.

Before taking the course, he read online discussions that said functional analysis was extremely difficult and worried that self-study would be inefficient. That's why he politely told Amber that he would ask her for help more often in the future.

To his surprise, once he truly delved into the subject, he discovered that much of the content corresponded to discrete energy space. He figured that breaking down the structure and then deriving the results wasn't as difficult as rumored.

Since he could solve it himself, he naturally didn't need to take up Amber's time.

However, since the other party had taken the initiative to speak, Li Ao couldn't refuse her. He quickly moved aside a little and pointed to the question on the draft paper, saying, "That's perfect. Could you please take a look at this question for me?"

"Let me see." Amber smiled confidently.

As a PhD student in the mathematics department of a top university, she naturally has unique tricks for dealing with operator estimation.

After reading the question, she immediately understood the approach: "The key point of this question is to use the boundedness of the previous operators to prove that the boundary constraints will not escape the control of the discrete energy norm. If we directly apply the global boundary operators and ignore the local projection, it will lead to an overly coarse estimate. We must combine the two."

"The core solution approach is to first perform orthogonal decomposition of the Hilbert space, and then estimate the norm of the local trace operator."

After explaining the solution, Amber was about to pick up a pen and write out the detailed proof steps.

But before she could even put pen to paper, a look of sudden realization flashed in Li Ao's eyes: "I understand."

Amber paused, then asked in surprise, "Huh? You understand that already?"

In her memory, even senior undergraduate students in the Department of Mathematics at the University of Chicago need a lot of practice when they first encounter functional analysis before they can independently handle this kind of discrete trace estimation problem.

Li Aoman had only been self-studying for less than a week; he shouldn't have mastered it so quickly.

She guessed that Li Ao was probably too embarrassed to ask her to repeat herself, and was about to test the waters when Li Ao's next sentence immediately silenced her doubts.

"However, your reminder has given me an idea for an even simpler way of writing it."

As he spoke, Li Ao's pen moved quickly across the draft paper, and a completely new local estimation route gradually appeared.

Amber followed the handwriting as she read on.

At first, she was still finding fault with it, but after seeing the end, she had to admit that this method was indeed simpler and logically sound.

For a moment, she felt as if she were nailed to the chair, unable to utter a single word. In a corner of her heart, a faint sense of regret and frustration rose up, a feeling of having taken the initiative to approach her.