Questions tagged [convergence]

Questions related to whether the sequence of iterates generated by an iterative method has one or more limit points, and if those limit points have the correct properties.

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1answer
187 views

Convergence rate Jacobi/Gauss-Seidel with mesh resolution

In the book A Multigrid Tutorial - Briggs, Henson. McCormick in the beginning of Chapter 3, it is mentioned that ...because the convergence factor behaves as 1-$O(h^{2})$, the coarse grid ...
4
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1answer
776 views

CFL condition for variable coefficients

I understand that the constant in the Courant-Friedrichs-Lewy condition is defined as $\mathrm{CFL} = \frac{u \Delta t}{\Delta x}$, where $u$ is the principal coefficient. I came across this post: ...
4
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1answer
642 views

Is it possible to ensure global convergence of a fixed point iteration?

Suppose I have a fixed point iteration of the form $$x_{n+1}=f(x_n).$$ Suppose further that after some initial testing, I found that it does not converge to an a priori known fixed point $x^*$. I ...
4
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1answer
206 views

convergence of unconstrained convex optimization

I encounter an optimization problem. The simplified version is like following: Denote function $F(x):\mathbf{R}^n\rightarrow\mathbf{R}$, where $F(x)$ is a smooth lower bounded convex function (i.e. $...
4
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1answer
416 views

Finite element error for second order ODE at nodes equal to zero

I coded a finite element method with linear basis elements for the problem $$-u'' = f(x), x\in[0,1], u(0) = u(1) = 0$$ The nodes are uniformly spaced and I will denote them as $x_i$. I initially ...
4
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1answer
2k 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 ...
4
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1answer
434 views

Are self-convergence tests reliable?

I'm developing a solver for solving linear hyperbolic equations of first order with respect to time and spatial derivatives. The formal order of accuracy of the solver must be 5 because I use 5th-...
4
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1answer
259 views

Multigrid stops converging when more grid levels are used

I'm having a problem with multigrid code I wrote. If I solve Laplace's equation in 2D and use more than 5 grid levels, the V-cycles stop converging after a few cycles (see below, convergence factor > ...
4
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1answer
125 views

When does L-BFGS outperform GD?

In practice, L-BFGS is frequently held comparably to other inexact QN methods, and it provides a middle ground of sorts between Hestenes–Stiefel CG and BFGS as memory goes from zero to infinity (...
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113 views

Preconditioned residual converges, but true residual doesn't

I'm using Albany w/ Trilinos to solve an elasticity problem with thermal expansion mismatch. I'm using block GMRES with MueLu preconditioning. It works for problem size of several million dofs, but ...
4
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529 views

Convergence rate of Picard iterations

Given a first order ODE $y'(x)=f(x,y)$ with the initial condition $y(x_0)=x_0$ such that it satisfies Picard thoerem of existence and uniquness, one can compute the solution by Picard iterations : $$ ...
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229 views

Nonlinear conjugate gradient restart threshold 1/10

Nocedal and Wright on Conjugate Gradient Methods, p. 123, describe a restart strategy ... whenever two consecutive gradients are far from orthogonal $\qquad {{| \nabla f_k^T \ \nabla f_{k-1} |} \...
3
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4answers
299 views

How does gmres method iteration behave for this non-diagonalizable matrix?

Recently, I have been studied my lessons about gmres iteration, probably the most popular iteration method for general large sparse linear system of equations Ax=b. And the convergence is obtained ...
3
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4answers
2k views

What is the meaning of “preasymptotic” and “superconvergent”?

Precisely the title of the question. I have encountered these terms in two areas: conjugate gradient method, and adaptive finite elements.
3
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2answers
7k views

What are some reasons that Conjugate Gradient iteration does not converge?

I would greatly appreciate it if you could share some reasons the Conjugate Gradient iteration for Ax = b does not converge? My matrix A is symmetric positive definite. Thank you so much! Edit with ...
3
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2answers
631 views

How many generations does it typically take for a differential evolution method to reach a global optimum?

For differential evolution methods in optimization, how many generations does it typically take to reach a global optimum? How do we know if the values are never going to converge?
3
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1answer
487 views

Scaling/nondimensionalization for numerical optimization

I have a numerical optimization problem that I am trying to scale appropriately, in order to allow for the solver to achieve faster and more accurate results. I found a paper here that had a short ...
3
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3answers
301 views

$H^1$-convergence rate of finite element method for Poisson equation, depending on element order

I wanted to verify my FEM-program by applying the method of manufactured solutions, while solving the Poisson equation in two dimensions using the continuous Galerkin method $$-\nabla^2u=f$$ with $$u=...
3
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2answers
234 views

Asymptotic error of forward Euler

I'm trying to understand the asymptotic error behaviour of forward Euler (finite difference method), as timesteps are decreased (refined), so I feel trust in the method of manufactured solutions (MMS)....
3
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1answer
2k views

Gauss-Seidel iterations node spacing

I am working on an assignment where I am determining the temperature distribution of a chip on a substrate. When I decrease the nodal spacing the results change drastically. The smaller the nodal ...
3
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1answer
305 views

steady state solution from parabolic problem vs solution of elliptic problem

My question is related, but not a duplicate of Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde Suppose I solve the parabolic PDE: $u_t = \Delta u + f(x,t)...
3
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1answer
149 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 ...
3
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1answer
161 views

Conjugate Gradient for singular 2D poisson finite element with Neumann Boundary Conditions

Heavily edited question after I realised partly what the problem was I have programmed a simple 2D square finite element solution to the Poisson equation $-\Delta u = f$ The source function ...
3
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2answers
1k views

Numerical Sensitivity in Density of States of Tight-binding model

I'm working with the tight-binding model, and I'm trying to learn the basics of how to compute the Density of States (DOS) $N(E)$ numerically. The DOS is given by $$N(E) = \frac{1}{N}\sum_k \delta(...
3
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1answer
222 views

Numerical Lax-Wendroff scheme order of convergence on Burgers equation

I was suggested to move that question here. The question to be as follows. Statement of the problem Is it possible to achieve the second order of convergence (OOC) of Lax-Wendroff (LxW) scheme ...
3
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2answers
158 views

In numerical methods, eg, finite differencing approaches, does there exist convergent schemes that are not both consistent and stable?

In a book that our course is following this semester, the theorem given is only in one direction: if the scheme is both consistent and stable, then the scheme is convergent. However, since this ...
3
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1answer
266 views

Measure the convergence rate of a discretization of a wave equation

I'm currently trying to approximate the following type of wave equation (in weak formulation): Let $\Omega \subset \mathbb{R}^d$ ($d=2$) be some polygonal domain. We seek a function $u \in L^2\left(0,...
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2answers
128 views

How does constraint resolution affect the stability/accuracy of numerical integration?

I understand some basic analysis techniques (local truncation error, global error, zero-stable, absolute stable, etc.) of numerical integration. But I find it hard to apply these techniques in ...
3
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1answer
81 views

Numerical integration in time for finite elements

I am trying to solve $M\ddot{u}=-Ku+F_\text{ext}$ for a 2D linear elastic model with $M$ be the mass matrix,$K$ the stiffness matrix and $F_\text{ext}$ the external load vector coming from a uniformly ...
3
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1answer
77 views

Choosing an appropriate time step for a discrete & continuous dynamics simulation

I have inherited of a flight dynamics simulation in C++ which represents a small drone with it's autopilot, actuator dynamics and a solid state IMU. Hence, it is composed of a few models, some ...
3
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1answer
108 views

Determine stability of an algorithm?

This is related to a question I answered on Stack Overflow regarding calculating the square root of a number. I was thinking about it and realized that the formula is just the first in a family of ...
3
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0answers
63 views

Why GA convergence curves continue as two parallel lines?

I'm working on a optimization problem and using GA algorithm (in MATLAB, ga function). As you know MATLAB plots GA result with two curves, one for the best values and other to show the mean values ...
3
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0answers
206 views

Unstable convergence of a Poisson equation

What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ...
3
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0answers
116 views

Finite difference method for coupled PDEs: optimizing performance (time step, iterations per step)

I'm solving coupled PDEs using finite difference method: Incompressible Navier-Stokes and the divergence-free induction equation (Maxwell's equations) with non-uniform electrical conductivity. The ...
3
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0answers
2k views

BFGS Fails to Converge

The model I'm working on is a multinomial logit choice model. It's a very specific dataset so other existing MNLogit libraries don't fit with my data. So basically, it's a very complex function ...
3
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0answers
40 views

Solving ODE boundary problem with additional conditions

I need to solve two point bondary problem for ODE. The solution of the problem itself is very easy. The real issue is that when solving arising non linear system of equations it always converges to ...
3
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0answers
92 views

analytic or numeric integral of diverging function

I'm trying to carry out the following integral numerically $$\int_{r_\mathrm{in}}^{r_\mathrm{out}} \Sigma\left(r'\right) \frac{r'}{r} \left( \frac{1}{r-r'}\, E(L) + \frac{1}{r+r'}\, K(L) \right) \...
2
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4answers
167 views

Testing the time dependent Schrodinger Equation with an analytical solution?

I am numerically solving the Schrodinger Equation in 1D first and in higher dimension later, but I want to know the convergence rate of my numerical solver in different grid size and numerical methods....
2
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2answers
292 views

$O(h^2)$ convergence for Elliptic PDE

I am trying to solve an elliptic PDE in 2-D: $$-\nabla^{2} u = 20tanh(10x-5)(10-10tanh(10x-5)^2) = f$$ I know that the solution is $u = tanh(10x-5)$ but I am unable to get $O(h^2)$ solution with a ...
2
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1answer
3k views

How to test convergence of an algorithm for constrained optimization

I am applying an iterative method (projected newton) to an optimization problem. Theoretically, the method should converge linearly. I would greatly appreciate it if you could share how should I test ...
2
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1answer
1k views

Computing rate and order of convergence

this is a follow up question to Convergence rate vs convergence order I guess the whole confusion about how rate vs. order of convergence also came from the implementations I saw. For example, for ...
2
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2answers
925 views

How to choose the number of random points in Monte Carlo simulations?

I am struggling with convergence criteria when performing a Monte carlo simulation on a uniform distribution. Any help would be much appreciated ! Say I want to sample uniformly a 1D interval (for ...
2
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1answer
157 views

Does Lanczos have trouble with large matrix elements?

I have a large, yet very sparse, matrix that I'd like to diagonalize. Both my own Lanczos implementation and the ARPACK that's built in with scipy fail to converge properly, though. I know that my ...
2
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2answers
122 views

Normalizing error when data passes through zero

Given a time-dependent measure of error, $e(t)$, and a time-dependent function to normalize it with, $f(t)$, one would report the normalized error as $$E(t) = \left|\frac{e(t)}{f(t)}\right|.$$ ...
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2answers
2k views

Does energy decrease with basis set size in density functional theory?

Based on the variational principle, one might expect that the ground state energy of a density functional theory (DFT) calculation will decrease as the basis set size increases. (As I understand it, ...
2
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2answers
466 views

Finite volume solution of electrostatics using magnetic vector potential

I would like to solve for the electric potential and magnetic vector potential using the finite volume method (collocated grid). My equations are: $\nabla\cdot(\sigma\nabla\phi)=0$ $\nabla \cdot \...
2
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2answers
147 views

Manufactured solution for $-\operatorname{div}(a(x) \nabla{u}) = f$ when $\alpha(x)$ is discontinuous

I'm studying the dealii tutorial number 4,5 and I understand the workflow. I've also been able to find the EOC by using manufactured solution where $f$ is a smooth r.h.s. and $\alpha(x)$ smooth too. ...
2
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1answer
210 views

Solving Cahn-Hilliard equation using semi-implicit Fourier spectral methods

So, I have written both a C and python code to solve the 2D Cahn-Hilliard equation: \begin{equation} \frac{\partial c}{\partial t} = \nabla^2\left(c^3 - c - \kappa\nabla^2c\right) \end{equation} ...
2
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2answers
138 views

Accelerating convergence of a generalized continued fraction

I wish to compute $$ \frac{1}{1 + \frac{1^3}{1 + \frac{2^3}{1 + \frac{3^3}{1+\cdots} } } } $$ to high accuracy. To start, I tried computing $$ \frac{1}{1 + \frac{1^2}{1 + \frac{2^2}{1 + \frac{3^2}{1+\...
2
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1answer
123 views

Best way to check if SOR solution has converged for 2d matrix

I have written a SOR algorithm to solve the Laplace equation on a 2d grid. The outside of the grid is fixed at 0 and the central square is fixed at 10. I can obtain the fully converged solution for ...