12 votes
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Why do we solve non-linearity in hyperbolic PDEs that way?

The good thing about the conservative form is that this comprises multiple models, such as Shallow Water Equations, Euler Equations or traffic models. An essential feature of hyperbolic equations is ...
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  • 483
11 votes
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Is the shooting method the only general numerical method for solving nonlinear boundary value ODEs?

Is the shooting method the only general numerical method for solving BVP of nonlinear ODE(s)? No. Most other methods consist of three parts: Discretization. This may be done with finite ...
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8 votes
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How to solve the problem without using symbolic computation

You can solve this numerically in Python without symbolic computation. from __future__ import print_function, division import numpy as np from numpy import exp from scipy.integrate import quad from ...
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8 votes

Limitations with dynamical systems vs. PDEs?

PDEs are a form of dynamical system where there is another continuous variable. Usually this is space, so you're looking at how things over time and space instead of just over time. Here's an ...
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8 votes

Jacobians with automatic differentiation

Julia has a whole ecosystem for generating sparsity patterns and doing sparse automatic differentiation in a way that mixes with scientific computing and machine learning (or scientific machine ...
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7 votes
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Solving a system of nonlinear equations with an ODE solver is faster than with the Newton method?

This is somehow unexpected, but my recent experience with solving a system of nonlinear equations is that treating them as the right hand side of a system of ordinary equations and then evolve the ...
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7 votes

Strong coupling of a non-linear multiphysic problem: failure with Newton Raphson method

The issues you're running into now are not a failing of Newton-Raphson, but a question of coupling. You're doing iterated sequential coupling -- solving each equation sequentially and then iterating ...
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7 votes
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Solve non-linear set of three equations using scipy

Since your problem is small, you're probably best off trying fsolve or root. Both of these are interfaces to MINPACK and call <...
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7 votes
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Newton iteration applied to nonlinear PDE

It's a bit easier to see if you write your equation in the a semi-discretised system of the form $u^{\prime}(t) = F(u(t))$ and with the application of the $\theta$-method and approximating $u^{\prime}(...
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  • 5,259
7 votes
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Understanding the Eisenstat-Walker method for choosing the tolerance of a linear solver when solving a non-linear PDE

$s_k$ is the "approximate Newton" search direction. So in essence, when they say Choose $s_k$ such that $\|F(x_k)+F'(x_k)s_k\| \le \eta_k \|F(x_k)\|$ they are saying: Solve the Newton system $F'(...
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6 votes
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Implementation of nonlinear term in FEM

You should rather think of what you will need in the following step, which is probably the numerical time integration of the semi-discrete equations. If you are going to use a (semi) explicit time ...
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  • 3,398
6 votes
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Is "tangent stiffness matrix" the same as "stiffness matrix"?

$K u$ equals the internal forces only in the linear case. The tangent stiffness matrix, $K$, in a nonlinear problem is normally used in a Newton-Raphson algorithm to calculate updates to the ...
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  • 5,734
6 votes

Solving a linear equation system with pure Neumann condition

Pure Neumann problem is unique up to a constant. My two favourite solution strategies: Modifying the equation $-\Delta u = f$ to $-\Delta u + \varepsilon u = f$ for some small $\varepsilon>0$. If ...
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  • 1,832
6 votes
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treating "almost linear" nonlinear least-squares problems

If you change variables to optimize for the residual of the linear part, then the Hessian will be a low-rank update to the identity. Then L-BFGS would work very well. Specifically, your problem takes ...
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  • 3,003
6 votes
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GMRES vs Newton-GMRES for Solving nonlinear PDE's

The reason is that GMRES can only be used for solving linear equations, i.e. equations of the form $Ax=b$, where $A$ is some matrix and $x,b$ are vectors. What GMRES does, essentially, is it ...
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  • 96
6 votes

What are systematic ways of approximating a non-smooth (non-continuously differentiable) system dynamic to be n-smooth?

Two systematic ways of smoothing a function $h$ would be: 1. Join the piecewise smooth parts of your function using Hermite interpolation so that the derivatives are matched to your satisfaction. 2. ...
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5 votes

Is the shooting method the only general numerical method for solving nonlinear boundary value ODEs?

No it is not. There is also multiple shooting collocation finite differences fixed point iterations and probably some more.
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  • 3,398
5 votes

What are the numerical methods for huge polynomial systems?

Certified homotopy continuation methods are used both for finding roots and for proving that they indeed exist (inside a certain interval). A quick web search turned out this paper: Reliable ...
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5 votes

Suggestions for open source C++ library for medium scale non-linear solver

What do people suggest for the linear system of this type? I know about trilinos, petsc, and sundials, but don't know the other alternatives or have exposure to them. Generally speaking PETSc, ...
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5 votes
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Suggestions for open source C++ library for medium scale non-linear solver

I am a PETSc developer so take my suggestion with a grain of salt, but I would use PETSc because the problem sizes are large enough that execution overhead should be minimal, you can trivially switch ...
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  • 25.2k
5 votes
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Convergence issues for a non-linear system

A common strategy is to employ a damping strategy, i.e., to compute $\vec{w}^{\ast} = F \left( \vec{w}^{(n)} \right)$ and then set $\vec{w}^{(n+1)} = \alpha \vec{w}^{(\ast)} + (1-\alpha)\vec{w}^{(n)}$ ...
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5 votes
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Solving a nonlinear equation with random variable

Since $\phi$ is a scalar between $0$ and $1$, the easiest method for finding a root is bisection. If you cannot calculate the expectation of the nonlinear function $$f_\phi(\theta) = \left(\phi r_z +(...
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5 votes
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Bessel EVP and fzero

As you can see in this plot of $\log|f(\lambda)|$, $$ f(\lambda) = J_{\lambda-1}(1) - 2J_\lambda(1) -J_{\lambda+1}(1), $$ the roots $\lambda_k$ are really regular, and are approximately equal to $-k$...
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  • 11.4k
5 votes

solving for unknown inside an expectation

Since you are assuming $\eta$ is normal what i would do is try to compute the expectation as fast as possible for every $\theta$. So I would compute the expectation using any numerical integration I ...
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5 votes

Eikonal Equation solver with different grid densities

This may not be your definitive solution, but what you need in any case is to change the discretization method that replace the continuous Eikonal equation with some discrete numerical scheme on the ...
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5 votes
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2d Euler manufactured solutions

This 2004 paper by Roy et. al in Int. J. Numer. Meth. Fluids 2004; 44:599–620 (DOI: 10.1002/d.660) should contain exactly what you're looking for: ftp://ftp.demec.ufpr.br/CFD/bibliografia/...
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  • 2,096
5 votes
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What is the preferred method for evolving the Nonlinear Schrödinger Equation?

You won't find universal agreement among experts on a single best way to solve this equation. On the other hand, since it is a PDE in one spatial dimension there are many approaches that will give ...
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5 votes
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Pseudo Code for non linear power function fit needed

Chapter 10 of Numerical Analysis (Richard L. Burden, J. Douglas Faires) gives good readable pseudo code for Newtons method. The start parameters are taken from the solution of the linear problem of ...
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5 votes
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What would be a good approach to solving this large data non-linear least squares optimisation

I don't know much about tracking implicit surfaces, so I'm just going to start with the optimization problem and go from there. The optimization problem is, at the core, nonlinear least squares, and ...
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  • 3,003
5 votes
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Stopping criteria in iterative methods for solving nonlinear equations

It can happen in principle that the condition on $f$ will never be satisfied, if, for example, $|f'(x_*)|u>\mathit{tolfun}$, where $u$ is the unit roundoff. When I tried your example with $K=10^8,...
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