Awakening Elementary School System in University
Chapter 14 - 13: An Expensive Item’s Only Flaw Is Its Price

Chapter 14: Chapter 13: An Expensive Item’s Only Flaw Is Its Price

’You understood it just like that?’

’Could the gap between people really be this huge?’

Sun Yu was a legitimate graduate student in applied mathematics, after all. He had spent a full two days poring over the proposal and related materials, only to barely grasp the general logic and limitations of the existing computational methods.

’And this "CL" on the other end, could he really have understood it all just by looking at the file and listening to me explain it once?’

’No way. I paid a fortune—2,000 yuan—for this expert. If he’s going in the wrong direction, the solution he provides will be completely off the mark. My money will have gone down the drain!’

At this thought, Sun Yu quickly spoke up, trying his best to sound tactful. "Uh, there are a few places in this proposal that are rather complex. Perhaps I should..."

Before he could finish his sentence, the voice in his headset simply ignored him and continued on.

"A typical structural stability analysis. You’ve used industry-standard finite element software for buckling analysis. The core of it is solving a generalized eigenvalue problem: ([K]-λ[KG]){φ}= 0."

"CL’s" voice was unhurried.

Sun Yu, who had been about to speak, froze with his mouth hanging open.

Because what he’d said was completely correct.

The voice continued, "Based on your description and the data in the file, the difficulty you’re facing lies with the critical buckling factor, λ. That’s the solution you’re after, and it should theoretically fall within a relatively well-defined range. I see in the file it’s specified as 4.0 ± 0.3."

"But the results from your software’s repeated calculations sometimes drop as low as 3.2 and at other times inexplicably jump to 4.8. The results are diverging and failing to converge stably within the theoretically expected interval."

Sun Yu was utterly dumbfounded.

’He realized he was the one being a total clown.’

’I probably used up all the luck of the first half of my life to stumble upon a genius of "CL’s" caliber. And what did I do? First, I tested his actual skills with some undergraduate-level math problem, and then I even doubted if he could understand my project after a single explanation.’

Sun Yu raised a hand and gave himself a light slap across the face.

Then, like a thief, he quickly moved his mouse and hit the mute button.

It was true that "CL’s" voice sounded young, completely free of the weariness that came from years in the professional world.

But to Sun Yu’s ears now, that calmness was almost terrifying.

He sounded completely indifferent to the material he was explaining, as if he were mechanically reading from a script.

"...So, the essence of the problem is clear. Your existing solver exhibits numerical instability when dealing with eigenvalue clustering or ill-conditioned matrix problems. What you need from me is a brand-new algorithm that has better numerical stability and can accurately solve for the minimum eigenvalue you’re looking for."

As soon as he finished speaking, Sun Yu saw that "CL’s" gray avatar in the meeting window suddenly lit up.

He had turned on his camera.

A stir of curiosity went through Sun Yu, and his eyes darted to the small video window.

However, the video didn’t show the face of a genius as he had imagined.

The camera was angled down, aimed directly at a clean sheet of blank paper laid out on a desk.

A moment later, a hand holding a pen entered the frame.

"CL" explained to Sun Yu as he wrote.

"I’ll design a hybrid approach. It has several steps."

"First, for your generalized eigenvalue problem [K]x = λ[KG]x, we need to construct an effective preconditioner [M] to improve the spectral properties of the coefficient matrix."

The tip of the pen flew across the paper.

"Here, I suggest using a preconditioning technique based on incomplete Cholesky factorization or Algebraic Multigrid (AMG), applied directly to your stiffness matrix [K]. Because according to your data, the [K] matrix is usually a well-conditioned positive-definite matrix..."

"Second, we use a solver algorithm that is more robust to ill-conditioned problems and eigenvalue clustering."

"I recommend using the Jacobi-Davidson method. Compared to what you’re using now, or the more traditional Lanczos or subspace iteration methods, it is less dependent on the initial guess vector..."

"Third, to further improve the accuracy and efficiency of solving for the minimum eigenvalue, we need to incorporate a spectral transformation technique."

"Within the Jacobi-Davidson framework, apply the Shift-and-Invert transformation. Choose a shift σ close to your expected minimum eigenvalue λ_cr—in your case, that means choosing σ = 4.0. This transforms the original problem into a standard eigenvalue problem that is easier to handle..."

"Finally, adaptive adjustment. To address the mesh sensitivity issue you mentioned, the algorithm’s key internal parameters, such as convergence tolerance and subspace dimension, should be adaptively adjusted based on estimates of the matrix’s local condition number, rather than using fixed values."

"At the same time, during the algorithm’s execution, you need to monitor the rate of change of the residual norm and the eigenvalue estimate in real-time. As soon as convergence stagnation is detected, the preconditioner must be updated dynamically..."

By the time he reached this point, the sheet of A4 paper was already covered in a dense scrawl of variables and formulas.

Sun Yu’s eyes were glued to the screen, but his brain had long since stopped processing.

Suddenly, Sun Yu jolted.

’Who am I? Where am I?’

’He felt like he’d seen or heard many of the terms and concepts the guy was talking about back in grad school, but when they were all put together, it was just gibberish to him.’

But, none of that mattered anymore.

He was now certain: this "CL" before him possessed a level of mathematical skill so high that it was completely beyond his comprehension. He could only look on in worship.

That was enough!

’All I need to do is latch onto this giant, and I’ll be set!’

Just then, the video window went dark. "CL" had turned off his camera.

He continued, "The core algorithm steps and key formulas are all on that sheet of paper. What you need to do now is take the time to fully understand it, then write out these steps as pseudocode. After that, find a programmer at your company to help you implement it as a script, and finally, integrate that script into your finite element software."

Sun Yu’s mind reeled. ’What? Me? Go up against Tang Monk and his disciples?’

Sun Yu quickly said, "No no no no, please, could you write up the pseudocode for me? As you can see, our time isn’t up yet."

Afraid of being rejected, he quickly added, "Look, just look at the time! Our consultation started at around ten, and it’s not even eleven yet. The three-hour package... we’ve still got plenty of time left!"

Sun Yu now felt that the only drawback to expensive things was their price. Otherwise, they were all upside.

A short silence followed from the other end of the headset.

"Alright. Give me a moment."

A few minutes later, a file-sharing notification popped up in the meeting interface.

"CL" had shared two images.

Sun Yu downloaded them to take a look. One was the sheet of scratch paper with the algorithm steps and key formulas from before, and the other was the pseudocode "CL" had just written.

Sun Yu felt a heavy weight lift from his shoulders.

He glanced at the time. It was already 11:30.

He quickly said over the voice channel, "Okay, sir, I’ll take it from here and get this organized. It’s getting late, you should rest."

The young-sounding voice on the other end replied, "Alright, I’ll be logging off now. I’ve logged the time; we still have 83 minutes left. For your next consultation, contact me three days ahead of time to make an appointment."

Then, Sun Yu watched as "CL" left the meeting.