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.
201
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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 ...
- 31
3
votes
1
answer
149
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 ...
- 31
1
vote
2
answers
66
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 $p\geq 1$ ...
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1
vote
0
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123
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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 ...
0
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0
answers
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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 ...
- 23
3
votes
1
answer
230
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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(...
- 185
6
votes
1
answer
216
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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 ...
- 201
0
votes
0
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73
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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 ...
- 281
1
vote
0
answers
122
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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 ...
- 129
5
votes
3
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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_{\...
- 435
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"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 ...
- 129
1
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0
answers
50
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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) = ...
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0
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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 ...
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6
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2
answers
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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 \...
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2
votes
1
answer
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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
$$\operatorname{div}(\beta u) = 0$$
$$u = g_D \text{...
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0
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Getting euclidean distance between vector A and C without anyway of retrieving them when their distances with a common vector B is known
Motivation:
My plan is to get the overall euclidean distance matrix for all the vectors in N number of dataset. Each dataset is basically an array of n-dimensional points. For e.g: A dataset can be ...
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1
vote
0
answers
148
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Confused on how Method of Manufactured solutions works?
I am new to computational science and I am trying to wrap my head around how MMS works. I am solving the time independent Helmholtz equation as a simple test of the technique
so my starting equation ...
- 201
2
votes
2
answers
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Manufactured solution for $-\operatorname{div}(a(x) \nabla{u}) = f$ when $\alpha(x)$ is discontinuous
I'm studying the dealii tutorial number 4,5 and I understand the workflow. I've also been able to find the EOC by using manufactured solution where $f$ is a smooth r.h.s. and $\alpha(x)$ smooth too.
...
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1
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1
answer
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Step3 in deal.II - Convergence of the mean
I'm trying to understand the Convergence of the mean part of the Step-3 tutorial in deal.II. The authors say that $\frac{1}{|\Omega|}\int_{\Omega} u_h(x)dx$ converges with $\mathcal{O}(h^2)$, but I ...
- 281
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votes
1
answer
95
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2 point BVP solver: how to compute errors
Background
I am working with chapter 2 in LeVeque's book: https://faculty.washington.edu/rjl/fdmbook/
I have build my own solver in Python to solve the 2 point BVP:
$$
\epsilon u''+u(u'-1) =0 , \\
u(0)...
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1
vote
0
answers
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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 ...
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2
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1
answer
480
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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}
...
1
vote
1
answer
81
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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-...
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votes
3
answers
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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://...
- 43
1
vote
1
answer
281
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 (...
- 281
1
vote
1
answer
287
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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 ...
- 11
0
votes
1
answer
377
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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 ...
- 145
9
votes
3
answers
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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 ...
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votes
0
answers
39
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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& \\
...
- 141
1
vote
1
answer
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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. \\
\...
- 141
1
vote
1
answer
70
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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 ...
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4
votes
1
answer
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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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1
vote
0
answers
54
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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 ...
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3
votes
1
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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 ...
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1
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0
answers
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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. ...
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3
votes
1
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336
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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 ...
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1
vote
1
answer
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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)} = ...
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1
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1
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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 ...
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votes
2
answers
199
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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
$$
\frac{1}{1 + \frac{1^2}{1 + \frac{2^2}{1 + \frac{3^2}{1+\...
- 2,145
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votes
1
answer
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Accurate computation of Gauss-Kuzmin entropy
The Gauss-Kuzmin distribution gives the probability of an integer appearing as a partial denominator in the continued fraction of a real number $x$ as
$$
P(a_k = k) = -\log_2\left(1 - \frac{1}{(k+1)^2}...
- 2,145
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vote
0
answers
50
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Effect of reducing flux consistency order at boundary on convergence order
Consider the 1D nonstationary convection-diffusion PDE
$$
\begin{alignat}{2}
\partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\
f(t) &= \left.\left( a ...
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votes
1
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what is non-asymptotic convergence?
I guess convergence in general means it is in asymptotic sense but what does non-asymptotic convergence mean?. Can someone please explain with an example?
2
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0
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Convergent Finite Difference Scheme for Parabolic Equation
Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$
where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$.
In Strikwerda's book, the ADI scehme \begin{align*}
\left( 1 - \frac{...
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votes
1
answer
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Non-conforming bi-linear finite element
The four-noded bi-linear rectangle element, which sometimes goes under the name Melosh element, is non-conforming unless the element sides are aligned.
Out of curiosity I have implemented this element ...
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2
votes
1
answer
322
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Best way to check if SOR solution has converged for 2d matrix
I have written a SOR algorithm to solve the Laplace equation on a 2d grid. The outside of the grid is fixed at 0 and the central square is fixed at 10.
I can obtain the fully converged solution for ...
- 95
3
votes
1
answer
118
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Determine stability of an algorithm?
This is related to a question I answered on Stack Overflow regarding calculating the square root of a number. I was thinking about it and realized that the formula is just the first in a family of ...
- 151
2
votes
4
answers
312
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Testing the time dependent Schrodinger Equation with an analytical solution?
I am numerically solving the Schrodinger Equation in 1D first and in higher dimension later,
but I want to know the convergence rate of my numerical solver in different grid size and numerical methods....
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1
vote
1
answer
129
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Unexpectedly Slow Convergence Implicit Euler
I'm solving the coupled ODE
$$ \left[\begin{array}{c}x^\prime(z)\\p_x^\prime(z)\end{array}\right] = C(z)\cdot\left[\begin{array}{c}x(z)\\p_x(z)\end{array}\right] = \left[\begin{array}{cc}0& A(z)\\...
2
votes
1
answer
146
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How the number of pre-smoothing and post-smoothing steps affect the asymtotic convergence rate of geometrical Multigrid?
Does the convergence rate of multigrid depend on the total number of smoothing steps or on the number of pre and post smoothing steps seperately?
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3
votes
1
answer
161
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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 $...
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