The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
0answers
33 views

How to measure the computational costs of solving CSPs and optimization problems?

I would like to define an abstract cost (machine and implementation independent measure) of the work needed to solve CSPs as a function of constraint and Jacobian evaluation counts. I am trying to ...
0
votes
2answers
112 views

Systems of nonlinear equations

Consider the nonlinear system of equations $$ (1) \quad qk^2a_1^2E^2+wna_0a_1AE+pnka_0^2a_1E+rn^2a_0^2A^2-rn^2a_0^3A^2+qk^2a_0a_1ABE-qk^2na_0^2E^2=0, $$ $$ (2) \quad ...
1
vote
1answer
69 views

System of nonlinear equations in MATLAB

I've got some problems solving (numerically) this system of equations. \begin{array}{l} 40 \cdot \cos (2t) + 105 \cdot \cos ({\theta _3}) - 75 \cdot \cos ({\theta _4}) - 91.924 \cdot \cos ({337.62}) ...
0
votes
0answers
38 views

Solving a non linear equation and iterating for various values in python

problem:Unable to solve a unknown for multiple known values in a non-linear equation ...
5
votes
2answers
118 views

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 ...
5
votes
0answers
58 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 ...
7
votes
2answers
111 views

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 ...
3
votes
1answer
75 views

System of quadratic algebraic equations

I have this problem $H_i(x_1,x_2,\dots, x_N) = a_{ijk} x_j x_k + b_{ij} x_j + c_i = 0 \quad 1\leq i \leq N$ And I need to show that applying Newton-Raphson can fail to find even one real solution ...
2
votes
1answer
63 views

Methods to solve this equation on finite fields?

Is there any analytical (exact, closed-form solution) or numerical method to solve an equation such as $p(x) = r^x$ where $p(x)$ is a polynomial whose coefficients are drawn from a finite field, ...
1
vote
2answers
82 views

finding wave function for anharmonic oscillator

I'd like to find the normalized ground state wavefunction for the anharmonic oscillator (Duffing) whose potential for which there is no analytic solution; an oscillator with a quartic potential, in ...
3
votes
2answers
162 views

Slow convergence of Newton's method for finite elements

The application is a simple non-linear advection diffusion problem (steady state) using DGFEM. My error at each iteration is given by $$ e_{n+1} = ...
10
votes
1answer
254 views

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 ...
3
votes
0answers
202 views

Finding roots of systems of equations with a Jacobian that is singular everywhere

Given $a\in\mathbb{R}^{mn\times n}$, find a $C\in\mathbb{R}^{n}$, $x\in\mathbb{R}^{m\times n}$ such that $$ 0 = f_{k}(\boldsymbol{C}, \boldsymbol{x}):=\sum_{i=1}^{m} C_{i} \left(\prod_{j=1}^{n} ...
2
votes
2answers
176 views

How can i solve this first order system of differential equations

Edit: I am trying to solve $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu\frac{\partial^{2}u}{\partial x^{2}}\\ u(0,t)=0\nonumber \\ u(1,t)=0\nonumber \\ u(x,0)=\sin(\pi x)\\ ...
7
votes
1answer
110 views

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 ...
3
votes
1answer
89 views

Switch branch in bifurcation

I have a system of nonlinear equations $F(x,a) = 0$ and I know that at a specific point $a_c$ a bifurcation occurs, thus the Jacobian becomes singular. How can I switch branches and start following a ...
3
votes
0answers
92 views

Time-stepping for coupled nonlinear PDEs

What are good references for time-stepping of the coupled incompressible Navier-Stokes-heat equation (Boussinesq flow), $$ \begin{cases} \rho\left(\dot{\mathbf{u}} + \mathbf{u}\cdot\nabla ...
2
votes
0answers
37 views

Solving nonlinear wave equation in a dispersive infinite waveguide

I would like to solve a three-dimensional nonlinear wave equation in an infinite cylindrical waveguide numerically. Since the waveguide is dispersive, shocks are less likely to form. Both the ...
4
votes
1answer
228 views

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. ...
9
votes
2answers
452 views

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 ...
9
votes
2answers
129 views

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 ...
3
votes
1answer
115 views

Should the Jacobian of a system of PDEs be calculated from the main equations of the discretised equation?

I am solving a coupled system of non-linear PDEs in 1D. Something like, $$ u_t = F_1(u,v,w) \\ v_t = F_2(u,v,w) \\ w_t = F_3(u,v,w) $$ where each variable is a function of $x$ (the spatial ...
0
votes
0answers
119 views

Phase space reconstruction

I am looking for a toolbox for phase space reconstruction of univariate and multivariate time series using any method like the False nearest neighbourhood or the autocorrelation method for determining ...
3
votes
1answer
528 views

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 ...
2
votes
2answers
150 views

Nanoseconds vs. picoseconds in numerical quantum problems with Matlab ODEs

Hello there and thanks for taking a look at this problem. This problem is related to my previous question and I will therefore use a similar introduction from, Choice of step size using ODEs in ...
3
votes
1answer
208 views

Fenics: time-independent Sine-Gordon equation

Is there a code for the equation $$ \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2} = \sin(u) $$ or for the sine gordon equation in two dimensions because I want to change some ...
0
votes
1answer
976 views

How do I extrapolate data from a NON-LINEAR (logarithmic) standard curve in Excel?

I have made a standard curve. The X-axis is logarithmic. The y-axis is linear. I have added a logarithmic trendline (y = -1.546ln(x) + 39.254; R² = 0.9906). How can I re-arrange the equation to ...
2
votes
2answers
82 views

Problems where SPD linear system arises

I know some of the places where SPD linar systems arises such as elliptic PDEs and normal equations. Can I have a more comprehensive list of scientific applications which require solving SPD linear ...
5
votes
4answers
1k 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 ...
3
votes
2answers
428 views

C++ alternatives for simulating dynamic systems

I'm looking for alternatives to Matlab/Simulink and Dymola for simulating a non-linear dynamic system. I know it's possible to implement the time-domain behavior without a lot of code and a good ...
2
votes
1answer
234 views

Non-linear root finding when the Jacobian is almost singular

I'm trying to solve a system non linear-equations: $$ \frac{\partial K(\mathbf{\lambda})}{\partial \lambda_i} - c_i = 0 $$ for $i = 1, \dots, 15$, using Newton's method: $$ \lambda^{k + 1} = \lambda^k ...
3
votes
1answer
195 views

Stability of numerical schemes for non-linear equations with a Jacobian with negative eigenvalues

Let us assume I have an A-stable numerical scheme. I believe that given any linear equation $y' = Ay$, it means that the numerical scheme applied to this equation is stable (and therefore convergent ...
6
votes
4answers
568 views

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: ...
3
votes
2answers
115 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 ...
6
votes
1answer
245 views

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 ...
8
votes
1answer
274 views

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 ...
5
votes
2answers
149 views

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 ...
6
votes
0answers
264 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} ...
4
votes
1answer
1k 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 ...