Questions tagged [nonlinear-equations]

Solution of nonlinear systems of equations. The equations might be algebraic or differential equations.

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9
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192 views

Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system

Problem Solving a non-linear system of equations. The number of variables is the same as the number of equations. When I fix a set of variables (say $\vec{y}$) and keep another set free (say $\vec{...
9
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0answers
119 views

Review of modern homotopy methods and practical techniques

I'm hoping someone can recommend recent literature concerning homotopy methods for solving systems of nonlinear equations. Already by the time of Layne Watson's 1986 paper there were a lot of methods,...
6
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569 views

Implementing a Hill-Type Muscle Model

I'm interested in implementing the muscle model used in Geijtenbeek and Wang et al's work. Both papers link to the paper by Geyer and Herr, which describes this model: However, the paper on this ...
5
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69 views

2nd-order TVD criteria for flux-limiter

Consider a nonlinear hyperbolic conservation equation: $$ \partial_{t}u = -\partial_{x}f(u) $$ The spatial derivative of $f(u)$ may approximated after a spatial discretization by $x_{j}=j\Delta x$ $$ \...
5
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701 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} a_{kj}^...
5
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188 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 \mathbf{u}\...
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52 views

Continuous limit and nonlinear functional analysis

I have a kind of general question about approximation schemes in nonlinear functional analysis. Given a nonlinear map $\Phi$ from an open set (in an infinite dimensional Banach space) of functions to ...
4
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110 views

Integrators for Nonlinear/Stiff PDE

It was suggested I ask this question in this section. Anyway: I have a particular nonlinear PDE of the form $$ u_t(x,t)=iu_{xx}(x,t)+f(x,u(x,t)) \tag{1} $$ Where f is some nonlinear function. With ...
4
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254 views

Nonlinear least squares and regularization

Consider the nonlinear least-squares minimization of a vector of $n$ residuals $\mathbf{f}$ in $p$ parameters $\mathbf{x}$: $$ \min_{\mathbf{x}} || \mathbf{f}(\mathbf{x}) ||^2 $$ This can be done with ...
4
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185 views

Poisson equation in frequency domain

I need some help in numerically solving the nonlinear Poisson's equation for electrons in frequency domain. The steady-state equation is: \begin{equation} \nabla.(\epsilon\nabla\varphi) = q\left(n_i\...
4
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149 views

numerical analysis of a partial integro-differential equation

I have to numerically solve a nonlinear partial integro-differential equation. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-1/2}^{1/2} \frac{\pi\cos u}{\sin\pi u-\sin\pi x} \frac{\...
3
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90 views

Is the matrix exponential and the Jordan canonical form actually useful for solving differential equations?

All of my yearlong graduate-level Linear Algebra course notes from my professor—an algebraist/representation theorist—shows his love for the exponential map $e^A$ and the Jordan canonical form—and one ...
3
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63 views

Conjugate Gradient for nonlinear equation system

Is it possible to apply adaptions of the conjugate gradient algorithm i.e. Fletcher-Reeves, Polak-Ribere or others to systems of nonlinear equations? How should the equation system be adjusted so one ...
3
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69 views

Computation of Troullier-Martins pseudowavefunctions

The computation of Troullier-Martins pseudowavefunctions has been described in [1]. The pseudowavefunction $R^{\textrm{PP}}_l$ is defined by $$ R^{\textrm{PP}}_l(r) = \left\{ \begin{array}{ll} R^{\...
3
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77 views

Large-scale optimization of nonlinear equations

I'm looking to find a computationally efficient solution to a large system of nonlinear equations. I'm trying to maximize the following function: $$ f(\vec{x}) = \sum_i^N C_i (x_i-A_i)x_i^{\epsilon_{...
3
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33 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 ...
3
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85 views

PETSc SNES for user defined state

How to use PETSc SNES (scalable nonlinear equation solver), when the solution is not a vector but a user defined state? I am implementing a non-linear mechanics problem (geometrically exact shell 5-...
3
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266 views

How to avoid the Broyden's jacobian approximation becoming poorer with the number of iterations?

I have to solve many times a nonlinear system of the form $$f(x) = b^{(n)}$$ inside a loop. The function $f$ is expensive to compute and I do not have its jacobian, so I have tried the good Broyden's ...
3
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185 views

Solving a nonlinear poisson equation via variational minimization

I am kind of new in finite elements and I am solving simple "Poisson nonlinear" problem. $- \nabla ((1 + u^2) \nabla u) = f$ $u = 0 \ \text{on} \ \Omega $ I am using Newton solver, where I have ...
3
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102 views

Applications of algorithm for solving systems of equations with uncertainty

We have been developing algorithms for detecting "robust" zeros of multidimensional functions $f: X\to\Bbb R^n$ where $X$ is an $m$-dimensional domain in $\Bbb R^m$. More precisely, for a given $f$, ...
3
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187 views

Integration of nonlinear PIDE via spectral methods

At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction $\...
2
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68 views

How does the error work for the Strang Splitting?

We know in Strang splitting that the splitting error in the steady state solution is proportional to $h^2$. I want ask 2 things: If this error in the steady state solution is the global error? If we ...
2
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67 views

Linearising Nonlinear Coupled Partial Differential Equations - Alfvénic Diffusion

I am trying to solve the following coupled partial differential equations with a finite difference scheme: $$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$ $$\partial_tW+v\partial_zW-\...
2
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59 views

Multilevel minimization - boundary conditions

I am interested in minimizing $$min_{x \in R^{n^l}} f^l(x),$$ where $f^l(x)$ is nonlinear objective function arising from discretization of PDE. I would like to use nonlinear multilevel minimization ...
2
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0answers
228 views

Understanding MP-PIC implementation in OpenFOAM

The multiphase particle-in-cell (MP-PIC) method is characterized by mapping particle properties from the Lagrangian coordinates to the Eulerian grid. However, the implementation of this method in ...
2
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0answers
55 views

References to solve system of differential equations which describe the evolution of sandpile surface using the finite element method

I want to solve the following nonlinear system in 1D \begin{cases} \dot{R} + v \frac{\partial R }{\partial x} - \frac{\partial }{\partial x}\left( D \frac{\partial R }{\partial x} \right) -\Gamma =...
2
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82 views

How to solve $y(x) y'''(x)=f(x)$

I have a PDE of the form $\partial_t y(x,t)+\partial_x(y(x) y'''(x)-f(x))=0$, where $f(x)=\cos(x)$. Suppose a stable equilibrium exists, and I want to find the steady-state solution $y(x) y'''(x)=f(x)...
2
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0answers
81 views

Nonlinear system with diagonal nonlinearity

Consider a nonlinear system of the form $\boldsymbol{f}(\boldsymbol{x}) = \boldsymbol{0}_{\mathbb{R}^n}$ for $\boldsymbol{x} \in \mathbb{R}^n$, where the function $\boldsymbol{f}$ is given by \begin{...
2
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127 views

Neumann boundary conditions in the Maccormack scheme

I am trying to solve the viscous Burger equation $$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \mu \frac{\partial^2 u}{\partial x^2} $$ with Neumann boundary conditions. I am ...
2
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286 views

Convergence of a very large non-linear least squares optimization

(note: I also posted this question on stackoverflow before finding this community here, which seems a better place for it) I'm trying to solve the following problem: I have a lot (~80000) surface ...
2
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0answers
54 views

Nonlinear Sylvester-Like Equation

Maybe you can point me to some results already developed for this. I have to solve for $X$ the following "Sylvester-like" equation: $$ AX - XB = F(X)$$ where $A\in\mathbb{R}^{a\times n}$, $B\in\...
2
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0answers
128 views

Precision not improving by decreasing step-size in nonlinear Schrödinger

I tried to simulate soliton propagation by solving the nonlinear Schrödinger equation using the split-step Fourier method. The following is an example of the Matlab code copied from a textbook. ...
2
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0answers
107 views

Backing out a function of parameters from system of nonlinear equations

I have a system of equations that cannot be solved for in closed form: $F_1(x_1,x_2,\beta)=0 ~\&~ F_2(x_1,x_2,\beta)=0 $ I want to solve for functions $x_1=x_1(\beta) ~\&~ x_2=x_2(\beta)$ ...
2
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0answers
52 views

A test suite of large systems of nonlinear equations

I am looking for a kind of modern test set of large nonlinear problems. The only option I managed to find so far is rather dated: http://folk.uib.no/ssu029/Pdf_file/Testproblems/testprobRheinboldt03....
2
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0answers
352 views

Finite difference scheme for solving nonlinear least-squares problem

I am dealing with following problem: $$ \min_{u,\gamma}\Bigg\{ \frac{1}{1000} \iint_{S_2} {\gamma (x,y)^2 dxdy} + \iint_{S_2} {[u(x,y) - u_0 (x,y)]^2 dxdy} + \iint_{S_2} {[\Delta u(x,y) - \gamma (...
2
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0answers
139 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
66 views

Solving a nonlinear equation with a Markov process and RVs

Assume that we have the following equation and the following assumption. The scope is to solve for some particular variables expressed later. Update $$E_{t}\left[ b(A_{t+1})^{1-\gamma} *R_{t+1}^{-\...
2
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0answers
54 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 ...
2
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0answers
75 views

Estimating eigenvalues from time-dependent non-linear operator

I have a very sparse non-linear system $N(u) = 0$ that can be solved as a time-dependent ODE, $\frac{du}{dt} = N(u)$, and explicitly integrated until $\frac{du}{dt} = N(u) = 0$, e.g. by forward euler, ...
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30 views

Approximation in the derivation of the Arc Length method

I am studying the proof of the Arc Length method in section 2.2 of this thesis. In equation (2.2) the author introduces the supplementary conditions $$ (\Delta {\bf u} + \delta {\bf u})^T \cdot (\...
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0answers
101 views

Solving coupled PDEs with self-consistency condition

I am figuring out how to attack a problem (the Usadel equations of superconductivity) in which I need to solve a set of nonlinear PDEs for the fields $\{G_i (r)\}$ $$ U(G_i(r), \nabla G_i(r), \Delta(r)...
1
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57 views

Stochastic differential equation system (SDE) : overflow encountered in double-scalars

I'm trying to integrate the following SDE system from Dekker et al. To do that I'm using the Euler-Maruyama method which is basically a forward-Euler scheme plus a Gaussian noise term where the mean ...
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0answers
51 views

Solving Laplace equation with constraint on boundary

I have found the following PDE problem in a paper: Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium ...
1
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0answers
98 views

Optimize speed for calculating the approximate inverse of a large matrix

I am searching for a faster method to calculate an approximate inverse of a large matrix (up to 32000x32000) resulting from a discrete non-linear system of partial differential equations. I'm using C++...
1
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0answers
31 views

Numerically estimate the Jacobian from a scalar time-series

I'm trying to numerically estimate the Jacobian from a time-series. Following the paper, https://journals.aps.org/pra/abstract/10.1103/PhysRevA.43.2787. Consider that I have a scalar time series $x = (...
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0answers
75 views

RK4-method starts oscillating above certain input parameters

I am trying to solve an equation of the following type $$\partial_zE(z)=-c_0J$$ with $$J=c_1\beta E^3(z)$$ using the boost::odeint-framework and a fixed time stepper, with $c_0$, $c_1$ and $\beta$ ...
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0answers
70 views

Bifurcation points on homotopy path by numerical continuation?

I am trying to implement an algorithm that finds (possibly) all solutions of a system of nonlinear polynomial equations $$F(X) = 0$$ I thought about using the (convex) homotopy $$H(X,T) = TF(X) + (1-T)...
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0answers
90 views

Linearize non-linear PDE with BCs to hyperbolic problem: How does linearization affect BCs?

I am working with the Shallow Water equations that is a system of non-linear PDEs that simulate water waves propagation on some domain, in my cases the $x$-axis. I have here a GIF showing the results (...
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0answers
123 views

Incorporating radiation boundary condition at the edge in finite difference

I am trying to solve the 2-d heat equation on a rectangle using finite difference method. I am confused as to how to incorporate non linear radiation boundary condition at the edge. $-k\frac{\partial ...
1
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0answers
56 views

Oscillations when solving parabolic heat equation with FTCS

I'm wondering if someone could help me out, or point me in a direction of how I can understand the following oscillations that occur when I solve the Porous Medium Equation $$u_t = u_{xx}^{m+1}$$ ...