Questions tagged [nonlinear-equations]
Solution of nonlinear systems of equations. The equations might be algebraic or differential equations.
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When is Newton-Krylov not an appropriate solver?
Recently I have been comparing different non-linear solvers from scipy and was particularly impressed with the Newton-Krylov example in the Scipy Cookbook in which they solve a second order ...
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Is it possible to solve nonlinear PDEs without using Newton-Raphson iteration?
I am trying to understand some results and would appreciate some general comments on tackling nonlinear problems.
Fisher's equation (a nonlinear reaction-diffusion PDE),
$$
u_t = du_{xx} + \beta u ...
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15
votes
1
answer
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Comparing Jacobi and Gauss-Seidel methods for nonlinear iterations
It is well known that for certain linear systems Jacobi and Gauss-Seidel iterative methods have the same convergence behavior, e.g. Stein-Rosenberg Theorem. I am wondering if similar results exist for ...
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votes
3
answers
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Solution of quartic equation
Is there a open C-implementation for the solution of quartic equations:
$$ax⁴+bx³+cx²+dx+e=0$$
I am thinking of an implementation of Ferrari's solution. On Wikipedia I read that the solution is ...
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votes
2
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Numerical method for equation solving that works on stochastically computed functions
There are many well known numerical methods for solving equations of the type
$$ f(x) = 0, \quad x \in \mathbb{R}^n,$$
e.g. bisection method, Newton's method, etc.
In my application $f(x)$ is ...
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10
votes
2
answers
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Is the shooting method the only general numerical method for solving nonlinear boundary value ODEs?
During my wandering in Mathematica.se, I gradually noticed that a certain kind of differential equation solving problem is "troubling" us all the time, that is, the boundary value problem (BVP) of ...
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10
votes
3
answers
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Methods of solving non-linear advection-diffusion systems beyond Newton-Raphson?
I'm working on a project where I have two adv-diff coupled domains through their respective source terms (one domain adds mass, the other subtracts mass). For brevity, I'm modeling them in steady ...
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10
votes
1
answer
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Solving a difficult system of equations numerically
I have a system of $n$ non-linear equations that I want to solve numerically:
$$\mathbf{f}(\mathbf{x})=\mathbf{a}$$
$$\mathbf{f}=(f_1,\dots,f_n)\quad\mathbf{x}=(x_1,\dots,x_n)$$
This system has a ...
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9
votes
3
answers
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Basin of attraction for Newton's method
Newton's method for solving nonlinear equations is known to converge quadratically when the starting guess is "sufficiently close" to the solution.
What is "sufficiently close"?
Is there literature ...
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What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?
I am reading a paper [1] where they solve the following non-linear equation
\begin{equation}
u_t + u_x + uu_x - u_{xxt} = 0
\end{equation}
using finite difference methods. They also analyse the ...
- 265
9
votes
0
answers
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Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system
Problem
Solving a non-linear system of equations.
The number of variables is the same as the number of equations.
When I fix a set of variables (say $\vec{y}$) and keep another set free (say $\vec{...
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votes
0
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views
Review of modern homotopy methods and practical techniques
I'm hoping someone can recommend recent literature concerning homotopy methods for solving systems of nonlinear equations. Already by the time of Layne Watson's 1986 paper there were a lot of methods,...
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8
votes
5
answers
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Iterative solution to a nonlinear equation
I appologize in advance if this question is silly.
I need to compute the root of
\begin{equation}
u -f(u) =0
\end{equation}
Where $u$ is a real vector and $f(u)$ is a real-vector valued function.
...
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8
votes
1
answer
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Nonlinear wave equation - Finite element or finite difference
I would like to know the which is more advantageous when it comes to solving nonlinear hyperbolic equations, Finite Element or Finite difference methods? Which method will be better in capturing ...
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votes
2
answers
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Solve non-linear set of three equations using scipy
I need to solve a non-linear set of three equations using scipy.
However, I do not have any clue on which algorithm is suitable for my problem from a mathematical point of view (stability, convergence ...
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8
votes
1
answer
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Newton iteration applied to nonlinear PDE
I'm having difficulty understanding how to apply Newton iteration to nonlinear PDEs and then use a fully implicit scheme to time step. For example, I want to solve Burgers equation
$$u_{t} + u u_{x} -...
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8
votes
1
answer
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F(x) = 0 vs. ||F(x)||^2->min
In many areas of application, one needs to solve a nonlinear system of equations
$$
F(x) = 0.
$$
Sometimes, the formulation
$$
\|F(x)\|^2 \to\min
$$
is used. Clearly, every solution $\hat{x}$ of $F(x)=...
- 3,118
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votes
4
answers
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views
How to find more than one root of a polynomial?
This program finds the first root of the function f, defined in the code. There are 5 roots of this function. (x=1,2,3,4,5) I wish to find all of the roots in this program and print them to the screen....
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votes
3
answers
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Convergence of fixed point iterations of a non-linear matrix system
I'm working on modeling two phase immiscible flow in a porous medium. When I setup the system of equations, I obtain a non-linear system of equations that can be expressed in the form:
$A(x)x=b$
...
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votes
2
answers
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For a non-linear PDEs should the source term be discretised at $u_j$ or averaged over $(u_{j+1} + u_{j-1})/2$?
The non-linear Poisson equation in one-dimension,
$$
0 = \frac{\partial^2u}{\partial x^2} - f(u)
$$
can be discretised as to give,
$$
u_{j-1} -2u_{j} + u_{j+1} = h^2 f(u_j)
$$
where $h$ is the ...
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7
votes
3
answers
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Strong coupling of a non-linear multiphysic problem: failure with Newton Raphson method
I am trying to solve a multiphysic problem using finite elements and a Newton Raphson solution scheme. I have two non-linear subsystems that are coupled bi-directionally.
The first subsystem includes ...
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7
votes
1
answer
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Understanding the Eisenstat-Walker method for choosing the tolerance of a linear solver when solving a non-linear PDE
We are working on the solution of large non-linear PDE (say the Navier-Stokes equation) which we solve using Newton's method with an analytical formulation of the jacobian. For very large systems, we ...
- 1,147
7
votes
1
answer
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views
What would be a good approach to solving this large data non-linear least squares optimisation
Introduction to Problem
I'm using a Truncated Signed Distance Function to perform 3D reconstruction from depth images.
Essentially I have a large voxel grid where each voxel contains the signed ...
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7
votes
1
answer
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views
Eikonal Equation solver with different grid densities
The Fast Marching Method, Fast Iterative Method, and Fast Sweeping Method are three ways of solving the Eikonal Equation on a discrete grid, essentially just a wavefront spreading out from initial ...
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votes
1
answer
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views
Numerical method for nonlinear system of algebraic equations of special type
I have a nonlinear system of algebraic equations of special kind:
$$
\begin{array}{rcl}
x_{i}+y_{i}+z_{0,1}+c_{i,1}z_{1,1} & = & d_{i,1}, \\
x_{i}^2 + y_{i}^2 + z_{0,2} + c_{i,1} z_{...
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votes
3
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Solving a linear equation system with pure Neumann condition
I am trying to solve a linear equation system $\textbf{A}\textbf{x}=\textbf{b}$, e.g. a Poisson equation discretized in strong form, using biCGstab method.
Since there are only natural Neumann ...
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6
votes
2
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What are the numerical methods for huge polynomial systems?
Let a system of $n$ polynomial equations of degree $d$ with $m$ variables.
I'm interested in a sparse system with $d = 3$, $n \sim 2000000$, $m \sim 50000$ and integer coefficients.
What techniques ...
6
votes
1
answer
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views
Newton-Raphson method for nonlinear partial differential equations
For the numerical solution of Reynolds equations (a non-linear partial differential equation), the Newton-Raphson method is generally proposed.
After getting algebraic equations from a finite ...
P. S.
6
votes
4
answers
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Large-scale nonlinear optimization problem
I want to solve a nonlinear optimization problem of the following form
\begin{equation}
\min\left(\sum_i d^{x_i}c_{i}\right)\\
0 \leq x_{i} \leq a\\
\sum_{i} x_{i} \leq b
\end{equation}
$a$, $b$, $...
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votes
4
answers
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parameters estimation
I have to estimate a parameter (K), but I don't know how I can do it. I think by a regression model (minimum least square?), but I'm not sure. The system is:
...
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6
votes
1
answer
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Algorithm for solving system of quadratic equations and linear equations
Let $x \in R^N$. From a Spectral Chebyshev collocation method, I have a system of quadratic and linear equations. Denote them,
$$
x^T Q_i x + L_i^T x = 0
$$
and
$$
A x = 0
$$
Furthermore, I know ...
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6
votes
1
answer
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Numerical methods for boundary-value ODEs with a jump condition
I want to solve a non linear system of equations of a particular kind. I find it hard to formulate clearly so I directly give a simple example.
$
f''=A(f,g)\\
g''=B(f,g)
$
with the boundary ...
- 247
6
votes
1
answer
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views
Non-linear optimization using approximate gradient
I'm working with non-linear optimization for imaging, such as MRI and CT.
Our problem is of the form $\|Ax-b \|_2^2+\lambda \|Wx\|_1$. $A$ is never formed explicitly, so we're limited to approaches ...
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votes
1
answer
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Nonlinear dynamics: algorithm suggest
I've just started a thesis on nonlinear dynamics which entails numerical analysis of the Duffing oscillator (DO). It's basically just a second order ODE, or equivalently a set of ODEs.
Say, after ...
- 305
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votes
0
answers
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Implementing a Hill-Type Muscle Model
I'm interested in implementing the muscle model used in Geijtenbeek and Wang et al's work.
Both papers link to the paper by Geyer and Herr, which describes this model:
However, the paper on this ...
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5
votes
1
answer
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Why do we solve non-linearity in hyperbolic PDEs that way?
I am used to solve parabolic or elliptic non-linear PDE and the common methods to tackle non-linearity are Picard's iteration and Newton's method. I am a bit confused by the way things are done with ...
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votes
2
answers
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Implementation of nonlinear term in FEM
Although there are similar questions, I am also struggling with the implementation of the following term in "my own code" by Finite Element Method, namely, $\nabla \phi \cdot \nabla \phi$. $\phi$ is ...
5
votes
2
answers
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How to do upwinding in finite volume schemes for nonlinear equations?
In finite difference theory, you learn, that you have to use upwinding for equations with high convection, like Burgers' equation. What does the finite volume equivalent look like? What if the ...
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Confusion regarding the Adam-Moulton and Backwards Differentiation Formula (BDF) of the VODE solver
I am exploring the Method of Lines as a way of time stepping semi-discretised PDEs with ODE time-integration solvers. For an excellent introduction to this technique see the scholarpedia.org article. ...
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votes
1
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What are systematic ways of approximating a non-smooth (non-continuously differentiable) system dynamic to be n-smooth?
I have a system dynamic that is non-smooth because it has several signum and absolute value functions in it (three-tank level control). I can obviously choose different sigmoid functions to ...
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votes
2
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Convergence issues for a non-linear system
I have a nasty system of coupled integral equations, which I managed to discretize and recast a non-linear system, i.e. something like:
$$ \vec{w} = F \left( \vec{w} \right) \hspace{32pt} w \in \...
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votes
1
answer
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views
Nonlinear least-squares solvers vs. generic minimization
A nonlinear least-squares problem with $F:\mathbb{R}^m\to\mathbb{R}^n$,
$$
F(x) \to \min_x \quad (\text{in the least-squares sense})
$$
really means minimizing
$$
\frac{1}{2} \|F(x)\|^2 \to \min_x.
$$
...
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votes
1
answer
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Can redundant variables be beneficial for root-finding convergence
Suppose I have $n$ generally nonlinear equations for $n$ variables, like e.g. for $n=2$ the system $F(x,y)=0$
$$
\begin{aligned}
x^2+2y-4&=0\\
\sqrt{8}x+y^2-5&=0
\end{aligned}
$$
By ...
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votes
1
answer
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Quantifying the degree of nonlinearity in a heat transfer problem
I’m working with a heat equation of the form.
$$\frac{d(\rho(T)c_p(T)T)}{dt}-\nabla\cdot(k(T)\nabla T)=f$$ with temperature dependent density $\rho(T)$, specific heat $c_p(T)$, and thermal ...
- 12k
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votes
1
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Evolving nonlinear Schrodinger equation with higher-order algorithms?
First I will give the relevant information for my question, and then I'll ask the question.
$\large{\textrm{Background}}$
For evolving the nonlinear Schrodinger equation (NLS), one typically uses [a ...
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votes
1
answer
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Numerical solution of non-linear diffusion equation via finite-difference with the Crank-Nicolson method
I want to numerically solve the non-linear diffusion equation:
$$
\frac{\partial}{\partial t} T(x,t)= \frac{\partial}{\partial x}\left(T^{5/2} \frac{\partial T}{\partial x} \right)
$$
I want to use ...
- 51
5
votes
1
answer
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Newton's method goes to zero determinant Jacobian
I am using the Newton's method to solve $3\times3$ systems.
For some particular cases, it turns out that at a given iteration, the Jacobian matrix cannot be inverted and that its determinant is very ...
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votes
1
answer
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PETSc or Trilinos for GPU?
I have to choose between the PETSc and Trilinos libraries for a project that involves the solution of non-linear systems of equations.
From their web sites they both mention support for GPUs, ...
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votes
2
answers
216
views
Solving a nonlinear algebraic system that includes a linear term
I am trying to solve a particular system of non linear equations written as $F(x) = 0$ in an efficient way.
More specifically, $$F(x) = (I - \gamma A)x - g(x) + C$$ where $\gamma$ is a scalarconstant,...
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votes
1
answer
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Analysis of nonlinear finite element methods
I have been doing a lot of reading on the development of finite element methods and their analysis using, e.g., functional analysis. I am clear on the formulation of the weak form of a PDE and ...
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