# Questions tagged [jacobian]

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29 questions
26 views

### 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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### Determination of Jacobian when there are more than 1 degrees of freedom at a node

In finite element method, the formula for Jacobian is $J =\beta^T X$ where, $\beta = [\frac{\partial N}{\partial \xi} \frac{\partial N}{\partial \eta}]$ and $X = [x_1 y_1; x_2 y_2; .. x_n y_n]$. ...
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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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### 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 ...
230 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 $$\int a(u)\nabla\ u\cdot \nabla v = \int f v$$ ...
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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: ...
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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 ...
122 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 ...
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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 ...
476 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 ...
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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 $J(\boldsymbol{x})$ ...
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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 ...