Questions tagged [jacobian]
Questions related to the calculation or use of the Jacobian matrix or its determinant. Not to be confused with the Jacobi iterative method for solving systems of linear equations. For those, use [iterative-method] instead.
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Step size constraint in Euler backward
I am dealing with an assignment in MATLAB. It has to do with 'self-driving' cars which are driving in-front/behind eachother. Assuming M cars on a single-lane road, each car adjusts its speed based on ...
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Number of function calls and jacobian calls in scipy.root
Just as an exercise, I am numerically solving the following system of equations:
$$
\begin{equation}
\begin{cases}
x^2 + y^2 = 32 \\
3x + 7y = 15
\end{cases}
\end{equation}
$$
...
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PETSc non-linear solvers (SNES): specifying single Eval & Jacobian function
The PETSc documentation example of a non-linear solver call has the user provide separate functions for the Jacobian and function evaluations:
...
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Spot redundant equations within nonlinear system of equations
Is there a general procedure to detect if in a system of m-nonlinear equations (also non polynomial) of n-unknowns some of the equations are redundant? Can the rank of the Jacobian matrix tell me ...
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How to classify ODE equilibria that are stable but slowly changing in value with time?
I'm numerically solving a system of coupled ODEs where time is the independent variable. At each time, I can solve for the equilibrium values of my state variables where their respective derivatives ...
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Computing Jacobian in WENO scheme for advection in a porous media
I am trying to implement an advection equation for a coupled system of a two-phase flow in a porous media using a WENO scheme [1].
My equation is of the form:
\begin{align}
\frac{\partial (\phi(x,t) C(...
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Calculating Lyapunov exponent (LE) for pendulum using ellipsoid growth - code yields negative LEs
I was redirected here from physics stack exchange where hopefully my question is more appropriate. Per my advisor, I have read the textbook Chaos, an introduction to dynamical systems by Alligood, ...
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Solving stiff ODEs: Dealing with Jacobian terms which take too long to compute with finite differences
I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
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Why can bad jacobians sometimes works better for implicit ODE method?
I'm solving a system of stiff ODEs describing atmospheric chemistry and transport. I am using CVODE BDF from Sundials Computing. I have two ways to approximate the jacobian:
Allow CVODE to ...
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How do I calculate the Jacobian of this function?
I have two vectors $r$ and $m$. Both vectors are $N\times1$.
A function is calculated as -
$F(1:N) = \phi r + (r^3 + rm^2)$
$F(N+1:2N) = \phi m + (m^3+mr^2)$
...
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Calculating the Jacobian for a function containing a derivative
I have the equation
$F(t) = \phi u + \frac{1}{2}\frac{d^2u}{dt^2} + u^3$
and broadly speaking, my task is to calculate the $\phi$ and $u(t)$ such that $F(t) = 0$.
I am testing out a new algorithm to ...
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Specifying ode solver options to speed up compute time
I'm specifying the 'JPattern', sparsity_pattern in the ode options to speed up the compute time of my actual system. I am sharing a sample code below to show how I ...
- 411
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Jacobian matrix cutoff in ODE solver
I am studying an implementation of a 3rd semi-implicit Runge Kutta method (siRK3) from the book by Villadsen & Michelsen (1978), Solution of differential equation models by polynomial ...
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Solving differential equation by specifying jacobian pattern
This is a follow up to my previous question posted here
I'm trying to construct the sparsity pattern of the jacobian matrix to speed up the computation of a large system of odes. The following is the ...
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solving differential equations with jacobian pattern
I'm trying to compare the simulation time for solving a system of differential equations with and without jacobian pattern for a toy model using ode15s in MATLAB.
...
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Method of Lines: How to simplify Jacobian with periodic BCs?
Consider the advection equation
$$\frac{\partial u}{\partial t}+c(x)\frac{\partial u}{\partial x}=0.$$
With periodic boundary condtitions in $x$ with period $L$, i.e. $u(x,t)=u(x+L,t)$ and initial ...
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How to determine the Jacobian Ratio for triangle element?
I am trying to implement an algorithm to find the Jacobian ratio for each triangle in mesh as a part of mesh quality check.
Let's say that I have vertices of the triangle: $P_1(x_1, y_1, z_1)$, $P_2(...
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Does the k-th approximate solution of a stationary iteration belong to the k-th Krylov subspace?
For an stationary iteration method solving $Ax=b$ as follows:
$$
Mx_k = Nx_{k-1}+b,
$$
I have known that when $M = I$, i.e., the Richardson iteration, the k-th solution $x_k = x_{k-1}+r_{k-1}$ is in ...
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Calculating the jacobian of norm and square root terms in the Finite Element Method
In the code that my group is writing (Lethe) we use a stabilized approach to solve the Navier-Stokes equation. The GLS stabilized method we use has a stabilisation term which contains a stabilization ...
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On using Ritz Method to solve a Mindlin–Reissner plate
I am trying to replicate the method given in the this paper. I have written a Matlab program which determines the displacement field of Mindlin–Reissner plate theory using Ritz method. The limitation ...
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Numerically Approximating the Jacobian and Comparing the Eigenvalues With Analytical Form
I am trying to study the stability of numerical discretization schemes using the Jacobian matrix of the residues with respect to the vector of conserved variables.
For a simple diffusion equation ...
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Numerical Stability of a Generalized Spatial Discretization Scheme
After reading the matrix stability chapter (10) of Hirsch [1], I decided to dive in the reference list of the chapter. One of the papers [2], which is cited as reference shows an very interesting ...
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Detecting blocks in non-linear system of equations
When solving systems of non-linear equations using Newton's method, it is often observed that the system has an independent sub-system, e.g. :
$$ f(x,y) = 0 $$
$$ g(x,y) = 0 $$
$$ h(x,y,z) = 0 $$
If ...
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Debugging Newton-method used in a CG-approach
I am currently proof-checking my program, which is intended to use Newton's method for solving nonlinear equations, using a continuous galerkin approach. Thus, as first step I checked it using a time-...
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Implementation of the Jacobian-free Newton method
In my calculation (of a simple heat equation, for testing) using the Newton method, I tried to replace the full Jacobian matrix with an approximation vector, i.e. replacing $J$ in
$$J(u)\delta u=-F(u)...
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Shooting method implementation
I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved.
$$
\begin{aligned}
\dot x_1(t)&=x_2(t)\\
\dot x_2(t)&=p_2(t)−\sqrt 2 ...
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What is the fastest method to invert millions of matrices?
My project involves large simulation and estimation. For each simulation I need to solve 600,000 systems of nonlinear equations. Currently I am using Newton's method to find the solutions. That ...
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How to calculate/derive analytic FEM Newton Jacobian
I trying to wrap my head of derivation of the analytic FEM Jacobian
for the Newton method. Say we have a nonlinear Poisson problem of the
(weak) form
$$ \int a(u)\nabla\ u\cdot \nabla v = \int f v $$
...
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How to verify solution to pre-conditioned linear systems solver?
I am solving Ax=b. A has a very large condition number (> O(10^10))
I am using the conjugate gradients method with point jacobi pre-conditioning. I obtained a solution 'x' that "looks" reasonable. ...
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Newton - Raphson method : maxima of function in 2 variables
I am computing the maximum of a function (with two-variables) using Newton-Raphson method. The function is : $e^{-(x \ - x_0)^2 - (y \ - y_0)^2}$, whose maxima exists at $(x_0,y_0)$.
The Jacobian ...
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Confusion about determining the jacobian in a rootfinding algorithm
I have written some Python code to determine the numerical roots of the following non-linear equation:
$$f_m=\tan\lambda_m - \frac{\lambda_m}{1+a}$$
where $\lambda_m\gt0$ and $a\geq0$. The code is:
<...
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FEM on tet10 element: negetive determinant at the Gauss point
I am trying to implement a fem code on tet10 elements. I follow the lecture notes for tet10 implementation given in
http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch10.d/AFEM.Ch10.pdf
...
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FEM/FVM/FD for structural modeling and stability issues due to large structural constants?
I've read that in modeling structures problems, the finite element method (FEM) is typically used. I am unfamiliar with FEM, but I am wondering, in particular, if using FEM, as opposed to finite ...
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How Jacobian matrix helps optimization faster?
I tried some python optimization functions and some of them needed Jacobian matrix prior for faster convergence. I understand Jacobians are basically transformation matrices that data from one space ...
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Sanity checking jacobians for Finite Element code
I have a legacy DG finite element code for which I would like to write some unit tests. As it is written, for 2D problems, each degree of freedom has associated with it a Jacobian matrix (for later ...
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Implementation of the Jacobi iteration to find the solution to $Ax = b$
I implemented the Jacobi iteration using Matlab based on this paper, and the code is as follows:
...
user9925
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1
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Calculate Jacobian of triangular element given coordinates of vertices and displacements?
I am trying to determine quality of my mesh elements using the Jacobian determinant as the measure. My algorithm takes vertices of nodes in triangular mesh and moves them around so as to form ...
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Optimal ordering in Jacobi SVD algorithm
In Jacobi SVD algorithm as given here every pair of columns of the matrix is orthogonalized until convergence. I want to know that how does the order of selection of the pair of columns affect the ...
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Does the limit of $\frac{\partial f}{\partial u}$ at $u=0$ exist?
For an optimization routine I needed to compute the derivative of the right-hand side $\: f_u(x_k, u_k)$ of a discrete-time system $x_{k+1} = f(x_k, u_k)$. Since $\: f_u(x_k, u_k)$ includes terms that ...
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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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Ill-conditioned Jacobian matrix from Nernst-Planck equation with Butler-Volmer reactions
The governing equations are listed here of my notes on page 4. It's a reproduction of other's paper which solves the equations with COMSOL. The problems arise when I want to solve for the consistent ...
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"Damping factor" for a set of non-linear ODEs
I have a set of four non-linear ODEs representing a negative feedback. I have done parameter variation by random sampling to study the sensitivity of steady state and other dynamic properties to ...
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Rank deficient Jacobian in discretized periodic solutions to autonomous ODE
I'm trying to numerically find periodic solutions to different systems of autonomous nonlinear ordinary differential equations. I decided to use a finite difference scheme and solve the resulting ...
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Jacobian-Free Newton-Krylov vs explicitly forming jacobian in DG
For a given discontinuous galerkin (DG) implementation for Navier-Stokes, targeting 10,000 to 1,000,000 4th order cells in 3D, I'm using PETSc's suite of linear/non-linear solvers on the back-end. It ...
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Is there a relatively simple way to extract the Jacobian from a Runge-Kutta 4/5 integrator?
I have a RKF45 numerical integrator that simulates polymerization of proteins using CUDA. It does so by tracking the populations of discrete length polymers, e.g. monomers, dimers, trimers, etc. all ...
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Levenberg-marquardt: How to calculate the jacobian with fixed parameters
So I'm working on a fitting algorithm using the levenberg-marquardt algorithm and I'm a bit stumped as to how to handle fixed parameters. Looking around at other code, like the minpack version of the ...
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Non-linear root finding with positive definite Jacobian
I am dealing with a system of non-linear equations:
$$
f(\boldsymbol{x}) = \boldsymbol{y}, \;\;\; \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^d.
$$
And I know that the Jacobian $J(\boldsymbol{x})$ ...
- 707
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680
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Poisson-Nernst-Planck equations with ill-conditioned sparse matrix
I am trying to solve Poisson-Nernst-Planck system of equations for ions diffusion problem using finite volume method. Nernst-Planck equation for mass transport and Poisson equation for electrostatic ...
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1
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How Matlab optimization works without Jacobian or Hessian
How does Matlab optimization tools works? It just gets the error function and doesn't need Jacobian (first derivatives) or Hessian (second derivatives)? How it is possible? If it is finite difference ...
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How to deal with complexity in numerical code, for example, when dealing with large Jacobian matrices?
I am solving a non-linear system of coupled equations, and have calculated the Jacobian of the discretised system. The result is really complicated, below are (only!) the first 3 columns of a $3\times ...
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