What I am not certain about is how linearization helps in the context of a non-linear problem. ... would the linearization get the solution closer to a global optimum if it exists or would the problem just compute faster such that the problem can be made larger in scale.
If a portion of the equation has a linear section this can be partitioned from the non-linear sections and solved using simpler equations, this will be faster; especially if the area is large or is involved in a complex analysis or optimization.
If the partitioned section can only be approximated using linear equations this will still be faster, but with a loss of accuracy.
It is possible to approximate, optimize, and for a final pass use exact equations. In a complex system this can result in a huge speedup without loss of accuracy, but possibly at the cost of missing the most optimal solution due to missing the true global optimum and relying on the approximations of local optimums to hopefully arrive at a global optimum. That is called the Hartman–Grobman theorem.
Where a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior that is called bifurcation theory or a crisis.
The Wikipedia explanation of a non-linear system is:
"Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.
Linearization of a non-linear equation allows the use of linear equations to estimate a point of a non-linear function, the further from that point the greater the likelihood of error. An iteration of linear approximations applied to a non-linear functions necessitates choosing a step size, too small a step size will be slow but more accurate while larger steps will be faster but may lead to instabilities.
For example a well-posed ordinary differential equation can be processed using a finite difference method and solved using matrix differential equations. A matrix of small simple equations is easier, and faster, to solve than a matrix of polynomials.
For much easier to follow examples see:
"What Is the Difference between Linear and Nonlinear Equations in Regression Analysis?"
Compare a linear versus a nonlinear regression model of the relationship between density and electron mobility. The nonlinear equation is so long it that it doesn't fit on the graph:
$$\Tiny{\begin{align}
\text{Linear equation (left side)} & \qquad \qquad \qquad \qquad \\
\text{Mobility } = & \text{ } 1243 + 412.3 \times \text{Density Ln} - 94.29 \times \text{Density Ln}^2 - 32.90 \times \text{Density Ln}^3 \qquad \quad \\
& \\
\text{Non-linear equation (right side)} & \\
\text{Mobility } = & (1288.14 + 1491.08 \times \text{Density Ln} + 583.238 \times \text{Density Ln}^2 + 75.4167 \times \text{Density Ln}^3) / (1 + 0.966295 \times \text{Density Ln} + 0.397973 \times \text{Density Ln}^2 + 0.0497273 \times \text{Density Ln}^3) \quad \\
\end{align}}$$
On the left:
"The fitted line plot shows that the raw data follow a nice tight function and the R-squared is 98.5%, which looks pretty good. However, look closer and the regression line systematically over and under-predicts the data at different points in the curve. When you check the residuals plots (which you always do, right?), you see patterns in the Residuals versus Fits plot, rather than the randomness that you want to see. This indicates a bad fit, but it’s the best that linear regression can do.".
On the right:
"The fitted line plot shows that the regression line follows the data almost exactly -- there are no systematic deviations. It’s impossible to calculate R-squared for nonlinear regression, but the S value (roughly speaking, the average absolute distance from the data points to the regression line) improves from 72.4 (linear) to just 13.7 for nonlinear regression. You want a lower S value because it means the data points are closer to the fit line. What's more, the Residual versus Fits plot shows the randomness that you want to see. It’s a good fit!".
Beginner level references:
"Linear or Nonlinear Regression? That Is the Question."
"How to Predict with Minitab: Using BMI to Predict the Body Fat Percentage, Part 1"
"How to Predict with Minitab: Using BMI to Predict the Body Fat Percentage, Part 2"
"What Is the Difference between Linear and Nonlinear Equations in Regression Analysis?"
