On the many different ways to turn non-linear systems of equations into linear ones.
Questions tagged [linearization]
382 questions
15
votes
1 answer
What is the 'linearization' of a PDE?
Specifically I am looking at the proof of Lemma 4.1 on page 9 here, where the graphical form of curve shortening flow is given, and then its 'linearization'. I am struggling to find any online resources that explain what this means, and what the…
jl2
- 1,455
12
votes
3 answers
Set a variable the max of two other variables
Suppose I have two real variables, $A$ and $B$, and another one $C$. I want to store the max between $A$ and $B$ in $C$ for a problem I am modeling. I can neither use a max function nor multiply variables. What can I do?
Magic code
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12
votes
2 answers
Lyapunov stability question from Arnold's trivium
V.I. Arnold put the following question in his Mathematical Trivium:
Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization?
It puzzled me for a while, since my experience doesn't include such a…
H. M. Šiljak
- 448
11
votes
0 answers
Noether's theorem in the critical heat equation
I am watching a serie of lectures on "Blow up solution for the energy critical heat equation" from Monica Musso on YouTube and at some point she states a result I do not understand. Let met recall the setting.
We are studying the following…
Falcon
- 4,433
8
votes
0 answers
Linearization of scalar curvature: $DR|_g(h)=-\Delta_g(\mathrm{tr}_g h)+\mathrm{div}_g(\mathrm{div}_g h)-\langle\mathrm{Ric}_g,h\rangle_g$
I'm working on an exercise from Geometric Relativity by Dan A. Lee, but things didn't go well:
Following Lee's hint, I was trying to use Exercise 1.12 and view $\color{red}{g}$ as the background metric ($\bar g$ in Exercise 1.12). In Exercise…
Boar
- 357
8
votes
2 answers
How can the least-norm problem in the $1$-norm be reduced to a linear program?
Problem Statement
Show how the $L_1$-sparse reconstruction problem:
$$\min_{x}{\left\lVert x\right\rVert}_1 \quad \text{subject to} \; y=Ax$$
can be reduced to a linear program of form similar to:
$$\min_{u}{b^Tu} \quad \text{subject to} \; Gu=h,…
p.koch
- 536
6
votes
2 answers
Converting sum of infinity norm and L1 norm to linear programming
So I'm trying to convert this minimization problem,
min $\parallel Ax-y \parallel_{\infty}$ + $\parallel x \parallel_{1}$ where $A$ is $m$ by $n$, $y$ is $m$ by $1$ and $x$ is $n$ by $1$.
into a linear program. My attempted solution is to rewrite it…
Eagle1992
- 529
6
votes
4 answers
What is the derivative of $\ddot x = x + A \dot x^2$ with respect to $A$?
Find the derivative of the solution of the equation $\ddot x = x + A \dot x^2$, with initial conditions $x(0) = 1, \dot x(0) = 0$, with respect to the parameter $A$ for $A = 0$.
-- Vladimir Arnold
This is a great ODE question posed by Vladimir…
SRobertJames
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6
votes
1 answer
Proper linearization of ODEs of the form $\dot{x}(t) + f(x(t)) + \sigma(t) = 0$?
For a scalar ODE of the form $$\dot{x}(t) + f\left(x(t)\right) = 0 \label{1}\tag{1}$$ where $f \colon \mathbb R \to \mathbb R$ is some smooth function admitting a unique root $x^*$ such that $f(x^*) = 0$. The linearization of \eqref{1} is…
Fei Cao
- 2,988
6
votes
3 answers
How to write boolean expressions as linear equations
I want to convert a set of boolean expressions to linear equations. In some cases, this is easy. For example, suppose $a, b, c$ $\in$ {0,1}. Then if the boolean expression is: $a$ $\ne$ b, I could use the linear equation $a + b = 1$.
To give a more…
user66360
6
votes
1 answer
What is the significance of the linearization of a non-linear PDE?
This may be too general a question so please let me know if I need to make it more specific.
I am a first year graduate student in PDEs, and as such have not had much exposure to non-linear PDEs. I am starting to look at research papers, in many of…
jl2
- 1,455
5
votes
1 answer
Linearization of Gradient Flow
As someone who has only "theoretical" knowledge in Riemannian geometry, I have a hard time trying to wrap my head around how to actually compute the so called "linearization" of a gradient flow on a manifold.
The setting is: We have a symplectic…
Symplectic Witch
- 3,106
5
votes
0 answers
Example for Carleman Linearization resulting in a linear system
The Carleman linearization came to my attention due to this article. I tried to understand this method but so far i wasn't succesful i tried to understand page 39 of this presentation however the example didn't make sense for me.
Could someone…
worldsmithhelper
- 272
5
votes
2 answers
Newtons method for finding reciprocal
Define a function 1 which is $f_1(x)=a-1/x$
and function 2 which is $f_2(x)=1-ax $
If I set both to zero I am looking for when $x=1/a$ as the root using Newtons method.
When I do this I get two different answers however and they should surely both…
user129299
- 69
5
votes
1 answer
Understanding a proof about Riemannian metrics in three dimensions always being diagonalizable
I've recently been working through Deturck's and Yang's Existence of elastic deformations with prescribed principal strains. First and formost, I'm interested in it's proof that Riemannian metrics can in three dimensions always be diagonalized, that…
moran
- 3,147