Method for finding successively better approximations to the roots (or zeroes) of a real-valued function

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45 views

Methods of solving non-linear advection-diffusion systems beyond Newton-Raphson?

I'm working on a project where I have two adv-diff coupled domains through their respective source terms (one domain adds mass, the other subtracts mass). For brevity, I'm modeling them in steady ...
3
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0answers
35 views

Quasi Newton method for block diagonal Hessian

I am having an unconstrained optimization problem, where the Hessian is positive semidefinite and block diagonal. The function is strictly convex, hence, the curvature condition ($ s^{T}_{k}y_{k} > ...
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0answers
30 views

Applying Newton-Raphson method to system of two differential equations, one time independent, one time dependent

The goal is to couple system of two nonlinear differential equations by applying appropriate space and time discretization and Newton-Raphson scheme. The equations system is $$\left[ \begin{array}{c} ...
2
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2answers
96 views

Why do Newton-Krylov iterations stagnate in this problem? [closed]

Consider this integro-differential heat equation taken from SciPy documentation page: $ \nabla^2 P = \alpha \left(\iint_\Omega \cosh(P)dx dy \right)^2 $ which was found in this question. In the ...
4
votes
1answer
47 views

Efficient and stable computation of inverse CDF

What is the most efficient and numerically stable algorithm for computing the inverse CDF $F^{-1}(y)$ of a probability function, assuming that both the PDF $f(x)$ and the CDF $F(x)$ are known ...
5
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1answer
167 views

Solving a set of linear equations with block structure and weak coupling

I have a standard set of linear equations $Ax=b$ where the Hessian matrix $A$ has the special block structure as shown: $A= \begin{pmatrix} T & U\\ U^T & V \end{pmatrix}$, $x= ...
3
votes
1answer
38 views

Implementation of Backward-Euler scheme, Newton-Raphson iteration scheme to time dependent nonlinear differential equation

I just knew how to do Newton-Raphson iteration in time-independent 1D nonlinear differential equation. Then I applied to time-dependent 1D nonlinear differential equation, and I got confused. Below ...
3
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0answers
56 views

Best way to add a positivity constraint to Newton's Method

So given an objective function $f({\bf x})$, I would like to include a positivity constraint when I perform the fixed point iteration: $${\bf x}^{(t+1)}={\bf x}^{(t)} - \text{H}_f^{-1}\nabla f({\bf ...
0
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2answers
103 views

How to implement Newton method in solving 1D PDE system? (ie. Poisson eq, continuity eq, drift-diffusion eq.)

I want to solve PDE system, which consists of Poisson equation, continuity equations for electron and hole with drift-diffusion equation numerically, by using method called Newton's method. This ...
4
votes
1answer
127 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
votes
1answer
316 views

Difference between Gauss-Newton method and quasi-Newton method for optimization

Can anybody help me? I heard that Gauss-Newton method compute an aproximation of the Hessian instead of the true Hessian, but, quasi-Newton method too, don't it? what is the differences between them? ...
1
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1answer
76 views

Solving this nonlinear system of equations

Suppose I have this set of equations: $$a = x + z\qquad (1)$$ $$b = y + \frac{z}{2}\qquad (2)$$ $$ z = k_0x\sqrt{y}\qquad (3)$$ Where $a$, $b$ $\in \mathbb{R}$ and $k_0 > 0$. The values of $a$ ...
6
votes
1answer
139 views

Newton iteration applied to nonlinear PDE

I'm having difficulty understanding how to apply Newton iteration to nonlinear PDEs and then use a fully implicit scheme to time step. For example, I want to solve Burgers equation $$u_{t} + u u_{x} ...
0
votes
1answer
83 views

Inverse Transformation of Isoparametric Quadrilaterals

I urgently need help with a problem I am having for an assignment for university. I do not expect a full solution, but I am really in need of guidance. I am required to write a FEA program in VBA, I ...
2
votes
2answers
143 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 ...
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0answers
46 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
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0answers
162 views

Linearization of two phase flow iteration for Newton Method

I am trying to implement a numerical model to solve a two-phase flow using the Newton-Raphson method. To do so, I also have to differentiate the fluid pressure $P$ with respect to the matrix velocity ...
4
votes
1answer
128 views

scaling and preconditioning for trust region Newton methods

Geometrically, scaling and preconditioning seem to address similar challenges in optimization. However, these two concepts are implemented very differently. Take trust region Newton method, as an ...
10
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1answer
181 views

Can an approximated Jacobian with finite differences cause instability in the Newton method?

I have implemented a backward-Euler solver in python 3 (using numpy). For my own convenience and as an exercise, I also wrote a small function that computes a finite difference approximation of the ...
2
votes
1answer
147 views

Hessian-free and Truncated Newton methods

In this paper on Deep Learning for Machine Learning, the approach is referred to as Hessian-free method. That is because the Hessian is never computed explicitly. Instead, the product of the Hessian ...
5
votes
1answer
163 views

Caveats of Hessian free method

Hessian free iterative optimization techniques like Newton-CG, do not explicitly compute the Hessian but instead approximate the product of the Hessian with a vector through finite difference. The ...
6
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2answers
214 views

Strong coupling of a non-linear multiphysic problem: failure with Newton Raphson method

I am trying to solve a multiphysic problem using finite elements and a Newton Raphson solution scheme. I have two non-linear subsystems that are coupled bi-directionally. The first subsystem includes ...
0
votes
1answer
156 views

Doubt regarding stopping criterion for Newton method

I am solving an unconstrained convex optimization problem, which can easily have a million variables. I am trying to get a working system with a toy problem of around 200 variables. I am noticing that ...
0
votes
1answer
117 views

Strict Feasibility in Interior Point Methods

As we know, in the interior point methods, all the iterates have to be strictly feasible. I implemented an affine scaling interior point for nonlinear objective functions. For small examples (2D), it ...
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0answers
58 views

Mapping to a computationally less expensive basis when employing Newton's method

I'm looking for advice, or references, for a change of basis to my dependent variables that leads to a less computationally expensive scheme when solving a system of coupled polynomial equations. ...
4
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1answer
122 views

Methods for Constrained Optimization Problems with Box Constraints

Consider this problem: \begin{equation} \begin{array}{ll} \text{minimize } & f(x) \\ \text{subject to } & a \leq x \leq b \end{array} \end{equation} where $a,b,x \in ...
4
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1answer
154 views

Active Elements in Projected Newton's Method?

To those who are familiar with the projected Newton's method or projected gradient method... We consider a constrained optimization problem with simple bounds. Particularly, minimize f(x) subject to ...
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2answers
1k views

Solving Kepler equation for true or eccentric anomaly

Is there any reason to always solve the Kepler equation for the eccentric anomaly, $E$, instead of the more meaningful (at least to me) true anomaly, $\theta$? Solving for the eccentric anomaly ...
3
votes
3answers
288 views

Beale's function and newton iteration

I am trying to find the minimum of the so called Beale’s function given by $f(x_1,x_2) = (1.5-x_1+x_1x_2)^2 + (2.25-x_1+x_1x_2^2)^2 + (2.625-x_1+x_1x_2^3)^2$ Using Newton iteration $x^{(k+1)} = ...
4
votes
2answers
343 views

Do I need to impose boundary conditions in the Jacobian matrix?

In the framework of Finite Element Method, when the Newton method is used, we solve $J(x^k) \delta x = -f(x^k)$, and the increment $\delta x$ would not change some entries from $x^k$ related to ...
1
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1answer
324 views

Solving a system of nonlinear equations with an ODE solver is faster than with the Newton method?

This is somehow unexpected, but my recent experience with solving a system of nonlinear equations is that treating them as the right hand side of a system of ordinary equations and then evolve the ...
0
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1answer
41 views

Non linear system of equations with discretization on k-space

I want to numerically solve the following system of differential equations at the steady state: \begin{equation} \begin{aligned} \frac{\partial \rho_{11\mathbf{k}}}{\partial t} =& ...
0
votes
1answer
179 views

MATLAB Newton non-linear equation

I have the following non-linear equation: where $w0=0.25,w0=0.5,w0=0.75$. I have to prove that if $k$ is a root, then also $−k$ is a root and that there exists only one $k∈(0,1)$ root, but my ...
0
votes
1answer
76 views

Calculation of isotropic-nematic phase transition

In this paper, the theory behind the isotropic-nematic phase transition is discussed. Furthermore, an algorithm is given to calculate some properties of this phase transition. I have written a ...
1
vote
1answer
119 views

Finding null vectors of a parameter-dependent matrix

I have dense complex matrices $M(z)$ in which each element $M_{ij} = M_{ij}(z)$ depends on a complex parameter $z$. I need to find $z$ such that the matrix $M$ gets singular, i.e. I am looking for ...
2
votes
2answers
472 views

Newton-Raphson method fails!

I am trying to solve an equation like $R(x) = 0$, using Newton-Raphson method. To obtain the $x$ increment in each iteration I solve $dx = -(A)^{-1}\cdot R$ where $A = dR/dx$. But the convergence ...
3
votes
2answers
313 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 ...
3
votes
1answer
152 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 ...
3
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2answers
378 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} = ...
3
votes
1answer
135 views

Excluding roots from a system of nonlinear equations

I have a system of nonlinear equations of which I know it has a single root I am interested in, and has a continuum of roots I am not interested in. I am currently using Newton with line-searching in ...
0
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0answers
72 views

Increase convergence of non-linear equations resulting from ODEs

I am trying to solve a set of couple ODEs: $V_l(r) - r W_l(r) - f1(r) W_l' = 0\tag 1$ $r^2 h''_l(r) + f2 r h_l'(r) + f3 h_l(r) - f4 U_l(r) = 0 \tag 2$ $\kappa (U_l + h_l) + V_{l+1} + W_{l+1} = ...
1
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1answer
651 views

Newton's method in interpolation

I've seen that in Newton's method for interpolating polynomials, the coefficients can be found algorithmically using (in Python-ish): ...
4
votes
3answers
193 views

Best method to find the zero of a decreasing function numerically

I need to find the zero of a function $f(\lambda)$ which is of the form $\sum \frac{c_i^2}{(1+\lambda d_i)^2} -1 $. I tried using Newton's method, and it works sometimes, but it is higly dependent of ...
13
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2answers
1k 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 ...
3
votes
1answer
266 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 ...
4
votes
1answer
113 views

Matlab help related with the scaled newton's iteration method

I apologize if this question doesn't fit for this site. I am new for this site. I need little help with matlab code for the method mentioned in this paper for computing the inverse of the matrix. In ...
2
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0answers
579 views

Newton Iteration method convergence

I wrote a Python code which solves a second degree nonlinear differential equation using the Newton iteration method. The code converges to a 2-cycle within 50 or so iterations. The cycle only ...
3
votes
1answer
348 views

BFGS methods for constrained elasticity problems

My dear community, I am wondering why BFGS methods are not so widely used for simulating mechanical problems which heavily still relies on inverting the hessian matrix. I am essentially interested ...
2
votes
1answer
500 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 ...
4
votes
2answers
162 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 ...