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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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1 vote
1 answer
125 views

How to refine $h$ and $\Delta t$ for convergence tests on evolution PDE

Setting I am solving for $u(x,y,t)$ the wave equation $\partial_{tt} u - \partial_{xx} u = f$ on $(x,y)\in\Omega=[0,1]\times \mathbb{R}$ by splitting it into an equivalent first order system: $$\...
0 votes
0 answers
108 views

Is there a fast matrix-free inverse power iteration?

Problem: I want to solve the eigenvalue problem $$x=Ax$$ to the eigenvalue $1$ for a large matrix (roughly $N^3\times N^3$ and $N$ ranges from 10 to 100) where $A$ is stochastic (i.e. all entries are ...
0 votes
0 answers
50 views

Calculating a 2D Ewald sum for a multipolar expansion

I am attempting to calculate the potential of a particle at the center of an infinite two-dimensional lattice as per the following reference: Reference: Lambin, PH & Senet, P. Ewald Summation of ...
2 votes
1 answer
287 views

Why do we use modified pressure in incompressible multiphase solvers with gravity?

The context of my question is two-phase incompressible solvers such as interFoam in OpenFOAM, but I have seen this trick used ...
3 votes
1 answer
185 views

Stability of Euler forward method

I am trying to solve a linear system of ODEs of the form: $$ \frac{du}{dt} = A u, \quad u(0)=k$$ where $A$ is a 2x2 matrix and $u(t)$ is a 2x1 column vector. I want to solve this numerically, using ...
3 votes
1 answer
150 views

Role of rotation's pivot point in optimization?

In this paper, the authors describe how to use locally rigid transformations (sampled on nodes in space) to deform mesh vertices. In the paper, rotations are relative to the pivot point, which ...
1 vote
2 answers
77 views

Why is the definition of convergence different for root finding algorithms as compared to sequences?

The definition of convergence for root finding algorithms is given in a few sources as: A sequence ${x^k}$ generated by a numerical method is said to converge to the root $\alpha$ with order $p\geq 1$ ...
4 votes
2 answers
1k views

Convergence problem for Poisson equation with periodic BC

I have written Poisson solvers using two different methods: A classic Jacobi scheme and one using the multigrid solver Hypre. I made up a couple of test cases ensuring the validity of those solvers. ...
0 votes
0 answers
64 views

Faster convergence for minimizing least squares of forward modelling problems

This specific question was raised from optimizing parameters of column experiments in the hydrogeological context. I want to optimize a parameter of interest (in this case $D$), based on experimental ...
3 votes
1 answer
236 views

Convergence-test for ODE approximates wrong limit

I am trying to numerically solve a differential equation but I am having trouble getting the convergence test to run properly. The problem is as follows: Consider an ODE $$y'(t) \enspace = \enspace f(...
1 vote
1 answer
402 views

L2 norm optimization problem

I have an optimization problem where i need to find an image x, that is very close to x' such that: monitor(x') is valid but monitor(x) is invalid. (output is valid when the neural network output is ...
6 votes
1 answer
300 views

Why FEM for incompressible materials is ill-posed?

I am an engineer who is trying to get a deeper understanding of FEM. I have been using the Zienkiewicz texts as my bible. It touches on the issue of incompressibility but I need a more intuitive way ...
0 votes
0 answers
77 views

Convergence rate for not smooth solution with classical $P^1$ Lagrangian FEM

I'm using classical $P^1$ finite elements to solve $- \Delta u = f$ with Dirichlet BC in a 2D domain $\Omega$. I know from theory that the solution is not in $H^1$ for my particular choice of $f$, so ...
1 vote
0 answers
140 views

Convergence of Evolutionary Algorithms

When it comes to Evolutionary Algorithms (e.g. Genetic Algorithm), I have often heard people make the following broad statement: "Evolutionary Algorithms Do Not Converge." I was curious ...
5 votes
3 answers
183 views

Why in scientific papers convergence of finite difference and finite volume schemes is tested using multiple norms ($l_1$, $l_2$ and $l_{\infty}$)?

In peer reviewed numerical papers, the order of accuracy of finite difference and finite volume for PDEs is computed in multiple norms, usually $l_1$, $l_2$ and $l_{\infty}$, and other times $l_{\...
2 votes
0 answers
84 views

"Unspoken Tradeoff" in Convergence Rates for "Quasi-Newton vs. Gradient Descent"

Is there an "Unspoken Tradeoff" in Convergence Rates for "Quasi-Newton Methods vs. Gradient Descent"? As a quick summary: Gradient Descent based algorithms try to find the minimum ...
1 vote
0 answers
50 views

Solution fails to converge with different collocation point selection

I'm trying to learn about collocation. As an example, I am using Numerical Recipes, 20.7.12, but changing the basis. To wit, I'm trying to solve $$y'' + y' - 2y + 2 = 0, -1 \le x \le 1, y(-1) = y(1) = ...
3 votes
1 answer
161 views

A name for a numerical phenomena when using numerical methods

I have a nonlinear solver for equation $$g= c_1f(x_1,y_1)+c_2f(x_2,y_2)$$ Note that $c_1$ is much bigger than $c_2$. After using Levenberg–Marquardt algorithm, it seemed to only optimize $x_1$ and $...
1 vote
1 answer
121 views

interface value on the error equation

https://www.jstor.org/stable/pdf/2157482.pdf, here I have a problem in last equation of (2.6) in section (2.1). When they are considering error equation on the interface $\Gamma$ they get $e_v^{(n)} = ...
4 votes
0 answers
95 views

Computationnal Mechanics : Three Bodies contact problem with ALM

I am facing an issue with convergence of a contact problem consisting of three successive bars, as presented below. A force is applied on the right hand of the first bar so there is contact between ...
6 votes
2 answers
153 views

Integrating exponential of second degree polynomials

I'm looking to compute the value of the following integral, for small values of $|a|$. $$u_n(a,b)=\frac{1}{2}\int_{-1}^1 x^ne^{ax^2+bx}\mathrm{d}x$$ In this equation, $a,b \in \mathbb{R}$ and $n \in \...
0 votes
0 answers
141 views

Getting euclidean distance between vector A and C without anyway of retrieving them when their distances with a common vector B is known

Motivation: My plan is to get the overall euclidean distance matrix for all the vectors in N number of dataset. Each dataset is basically an array of n-dimensional points. For e.g: A dataset can be ...
2 votes
1 answer
121 views

Discontinuous Galerkin order of convergence on arbirary refined mesh: step-12 deal.ii tutorial

I'm learning DG methods and in order to practice a little bit I'm using the deal.ii library. In particular, I'm looking at step-12, where they solve $$\operatorname{div}(\beta u) = 0$$ $$u = g_D \text{...
9 votes
3 answers
989 views

Accuracy issues with Arpack in Julia for eigenvalues of smallest magnitude

Following the documentation of Julia's Arpack package (Cf. https://julialinearalgebra.github.io/Arpack.jl/stable/eigs/) I have computed some largest and smallest magnitude eigenvalues of sparse ...
6 votes
1 answer
308 views

What causes periodic humps in residual plots?

When using many iterative methods, whether for solving linear systems, looking for steady-state convergence in CFD, etc., the semilog plot of the residual often shows "humps" as the residual decays. ...
4 votes
3 answers
2k views

Which absolute and/or relative stopping criteria do use for Newton's method?

I saw many stopping criteria for Newton's method all around Web and books. Some are defined from the residuals: of either current iteration only: $$ \|f(\mathbf{x}^{(k)})\| \leq \epsilon $$ (https://...
1 vote
0 answers
255 views

Confused on how Method of Manufactured solutions works?

I am new to computational science and I am trying to wrap my head around how MMS works. I am solving the time independent Helmholtz equation as a simple test of the technique so my starting equation ...
2 votes
2 answers
188 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. ...
1 vote
1 answer
322 views

FEM solution for Poisson is not exact at nodes

Let's consider the usual Poisson problem for FEM $$-u''(x)=1 \quad x \in [0,1]$$ with homogeneous Dirichlet boundary conditions. The solution is $u(x)=-\frac{1}{2}x(x-1)$ I know that the FEM solution (...
1 vote
1 answer
143 views

Step3 in deal.II - Convergence of the mean

I'm trying to understand the Convergence of the mean part of the Step-3 tutorial in deal.II. The authors say that $\frac{1}{|\Omega|}\int_{\Omega} u_h(x)dx$ converges with $\mathcal{O}(h^2)$, but I ...
2 votes
1 answer
511 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} ...
0 votes
1 answer
107 views

2 point BVP solver: how to compute errors

Background I am working with chapter 2 in LeVeque's book: https://faculty.washington.edu/rjl/fdmbook/ I have build my own solver in Python to solve the 2 point BVP: $$ \epsilon u''+u(u'-1) =0 , \\ u(0)...
1 vote
0 answers
44 views

Convergence of the Roothaan-Hall equations

Suppose that we are given a time-independent quantum mechanical system whose wavefunction depends on three space coordinates. Let $F$ be the Fock operator of the system. Suppose also that we have a ...
15 votes
1 answer
4k views

Comparing Jacobi and Gauss-Seidel methods for nonlinear iterations

It is well known that for certain linear systems Jacobi and Gauss-Seidel iterative methods have the same convergence behavior, e.g. Stein-Rosenberg Theorem. I am wondering if similar results exist for ...
3 votes
4 answers
770 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 ...
1 vote
1 answer
82 views

Rate of convergence - Stochastic Euler Method

The absolute error criterion of the pathwise approximation of an Ito process $X$ by an Euler approximation $Y$ is: $$ \epsilon=E\left(\left|X_{T}-Y(T)\right|\right) $$ We shall say that a time-...
2 votes
2 answers
211 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+\...
0 votes
1 answer
419 views

Finite elements with CFL condition - How to obtain correct order of convergence

I have discretized a PDE with continuous finite element method in spatial variable and with implicit Euler or Crank-Nicolson in temporal variable. In both cases, I have error estimates in $L_2$ norm ...
0 votes
0 answers
40 views

Devising Convergent Numerical Scheme for PDE

I'm currently looking at the PDE \begin{align*} u_t + \left[x(1-y) - (1-x)\right]u_x - (1-y) u_y + (z-xy) u_z = (z-xy) u_{xy} - (1-x)u& \\ \end{align*} with \begin{align*} u(x,y,z,0) = 1& \\ ...
1 vote
1 answer
169 views

Finite Difference for Advection Equation With Source

I'm trying to find a convergent finite difference scheme for the PDE \begin{equation} \begin{split} u_t + (x-1) u_x &= (x-1)u, \hspace{.5cm} x \in [0,1] \\ u(x,0) &= 1 \\ u(1,t) &= 1. \\ \...
1 vote
1 answer
70 views

Where could error terms that blow up in SWE come from?

I have been working on a solver for shallow water equations with reflective boundary conditions. I have found that it diverges very fast. As a workaround I noticed yesterday that if I smooth the ...
4 votes
1 answer
428 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 (...
1 vote
0 answers
72 views

Multigrid Reduction In Time Convergence

I am trying to solve a 2D dynamic linear elasticity model parallel in time using Xbraid. The spatial domain is [0,1]x[0,1] and time domain [0,1]. For time integration I am using a backward Euler ...
3 votes
1 answer
101 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 ...
1 vote
0 answers
101 views

Convergence of Conjugate Gradient Algorithm

I am trying to solve a linear elasticity model using finite element discretization in a rectangle domain [0,1]x[0,1]. For the solution of the the linear system $Ku=F$ I am using the CG algorithm. ...
3 votes
1 answer
435 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 ...
1 vote
1 answer
95 views

Issue solving nonlinear equation containing a quotient

I have a coupled set of PDEs that need to be solved as part of a larger problem. I am currently approaching this by computing spatial derivatives with finite differences and using PETSc's nonlinear ...
5 votes
1 answer
156 views

Accurate computation of Gauss-Kuzmin entropy

The Gauss-Kuzmin distribution gives the probability of an integer appearing as a partial denominator in the continued fraction of a real number $x$ as $$ P(a_k = k) = -\log_2\left(1 - \frac{1}{(k+1)^2}...
1 vote
0 answers
52 views

Effect of reducing flux consistency order at boundary on convergence order

Consider the 1D nonstationary convection-diffusion PDE $$ \begin{alignat}{2} \partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\ f(t) &= \left.\left( a ...
2 votes
1 answer
2k views

what is non-asymptotic convergence?

I guess convergence in general means it is in asymptotic sense but what does non-asymptotic convergence mean?. Can someone please explain with an example?

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