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Questions tagged [jacobian]

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0
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1answer
64 views

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)$...
1
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1answer
75 views

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 ...
3
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1answer
188 views

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 ...
5
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2answers
210 views

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 $$ \int a(u)\nabla\ u\cdot \nabla v = \int f v $$ ...
0
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1answer
139 views

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. ...
1
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0answers
141 views

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 ...
1
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1answer
406 views

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: <...
1
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1answer
201 views

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 ...
1
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0answers
50 views

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 ...
0
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1answer
773 views

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 ...
2
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2answers
203 views

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 ...
1
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1answer
5k views

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: ...
1
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1answer
3k views

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 ...
1
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0answers
41 views

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 ...
3
votes
1answer
158 views

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 ...
4
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1answer
932 views

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 ...
2
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2answers
231 views

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 ...
1
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0answers
45 views

“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 ...
2
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0answers
115 views

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 ...
2
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0answers
479 views

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 ...
2
votes
1answer
441 views

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 ...
3
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2answers
794 views

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 ...
4
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2answers
138 views

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 $J(\boldsymbol{x})$ ...
4
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2answers
424 views

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 ...
2
votes
1answer
353 views

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 ...
9
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5answers
376 views

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 3x9 matrix,...