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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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Why not use the preconditioned residual as termination criterion for preconditioned CG?

I have a Poisson equation with wildly varying material parameters (1 .. 1000), wildly varying element sizes (5 nm .. 100 um) and some quite anisotropic (tetrahedral) elements (100 nm x 100 um). I use (...
Thomas Klimpel's user avatar
8 votes
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586 views

DIIS method to accelerate SCF convergence for stretched geometries

I am implementing from scratch an Hartree-Fock calculation in the STO-3G basis set to perform Born-Oppenheimer molecular dynamics. I have a Restricted Hartree-Fock procedure that can reproduce very ...
user avatar
8 votes
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171 views

Accelerated convergence for Sparse NMF

In the Non-Negative Matrix factorization (NMF), you basically compute an approximation of a given matrix $V \in \mathbb{R}_{+}^{n \times m}$ into matrices $W$ and $H$ such that $W \in \mathbb{R}_{+}^{...
Gilles's user avatar
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Are there any benefits of computable analysis to numerical algorithms

Computers can work only with computable numbers, while most of the algorithms are based on analysis of real numbers (real analysis). When I heard of the existence of computable analysis I ...
Milind R's user avatar
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Conjugate residual/gradient convergence checking in practice

Let's say we want to solve $Ax=b$ ($A$ symmetric positive /semi/definite) with the conjugate residual/gradient method. $A$ comes from FEM where the mesh is being refined. The exact solution is $x_*$ ...
Christian's user avatar
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Understanding the meaning of Computational Order of Convergence

I am a postgraduate student with interest in numerical methods for solving nonlinear systems of equations. I have read some papers that discussed about 'computational order of convergence' for some ...
Hassan's user avatar
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Non-convergance when calculating temperature/heat flows through a section of rock

I am attempting to calculate temperature of section of rock in the earth as a function of vertical position in the rock and time. Along with it I am calculating the heat flow through the rock as a ...
skybluecodeflier's user avatar
4 votes
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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 ...
BlueTacZac's user avatar
4 votes
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184 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 ...
J. Ni's user avatar
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765 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 : $$ ...
Amir Sagiv's user avatar
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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} |} \...
denis's user avatar
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3 votes
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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 ...
Davood's user avatar
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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 ...
e-ae's user avatar
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3 votes
0 answers
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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 ...
Charles's user avatar
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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 ...
Titanic's user avatar
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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 ...
Misery's user avatar
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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) \...
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2 votes
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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 ...
stats_noob's user avatar
2 votes
0 answers
37 views

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{...
Mike D's user avatar
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Difference between contraction rate and convergence rate

I am slightly confused about the concept of contraction rate. For me it sounds equivalent to a convergence rate. Could somebody clear up the difference for me? I have those two definitions of ...
hcl734's user avatar
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2 votes
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How to determine the order of convergence of the Euler-Maruyama method?

This question is originally posted in Quant.StackExchange but has been unanswered for some time so I ask in here. To make this simple let us consider the Geometric Brownian Motions (GBM). My ...
k.dkhk's user avatar
  • 255
2 votes
0 answers
412 views

Numeric integration over Dirac delta

I'm trying to solve the following integral numerically. $$H(y) = \int dx \, f(x) \, \delta(g(x,y)).$$ For this I chose a representation of the delta-function and employ convergence with respect to $\...
J. Goe's user avatar
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Does applying the Newton-Raphson iteration for matrix reciprocal refine a matrix inverse from LU/GE?

This is a follow-up to this answer. Suppose you have a possibly very ill-conditioned matrix $A$, and you compute its inverse with LU/GE to get $X_{\text{lu}}\approx A^{-1}$. The Newton-Raphson ...
Kirill's user avatar
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2 votes
0 answers
132 views

Strange convergence behavior of WENO5 for Hamilton--Jacobi equations

I have the following question. I have a code function that computes right-biased and left-biased approximations of the derivative of a function using WENO5 for Hamilton--Jacobi equations as described ...
Dmitry Kabanov's user avatar
2 votes
0 answers
81 views

Dynamic Successive Over/under Relaxation (SOR) with several variables

I am solving a partial differential algebraic equation (PDAE) system which has the following dependent variables: $f=f(X,T)$ and $g=g(T)$, along with a few others My current method for coupling is ...
james506's user avatar
2 votes
0 answers
1k 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 ...
user1640255's user avatar
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 ...
stats_noob's user avatar
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) = ...
user14717's user avatar
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1 vote
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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 ...
TheCodeNovice's user avatar
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 ...
tohoyn's user avatar
  • 331
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 ...
spyros's user avatar
  • 481
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. ...
spyros's user avatar
  • 481
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 ...
cos_theta's user avatar
  • 451
1 vote
0 answers
57 views

Decrease in slope during convergence analysis

I am using the method of manufactured solutions to perform the order of accuracy testing. I am using a cube for the testing. The cube is size 1m on all sides. I used 5 refinements: $dx = dy = dz = ...
user27504's user avatar
  • 321
1 vote
0 answers
98 views

BFGS convergence problem

I would like first to state that it is beyond my capability to identify whether this is a BFGS issue or a R package problem. I've been doing some mixed logit regression using the R package "mlogit". ...
Richard TR's user avatar
1 vote
0 answers
81 views

Successive iteration method for solving eigenvalue ploblem

I have a question concerning the branch of successive iteration methods (Newton, Runge-Kutta). I definitely know (or can read in Wikipedia) the implementation of these methods. But I was wondering ...
MKM's user avatar
  • 11
1 vote
0 answers
328 views

Accuracy of finite difference method for heat equation on a disk

To study an approximation for the heat equation $$\frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial\theta^2}=f(r,\theta)$$ on the ...
Daniele's user avatar
  • 11
1 vote
0 answers
147 views

Diverged HDG solution for 2D incompressible Navier-Stokes test case at SMALL time step. Why?

I wrote a hybridizable Discontinuous Galerkin code for transient Navier-Stokes flow, with thanks to Martin Kronbichler and Scott Miller's step-51 code in Deal.II. The main algorithm is from Nguyen ...
Zhenyu  Zhang's user avatar
1 vote
0 answers
62 views

Decreasing - increasing - stabilising $l_{2}$ norm

Let $\bar{x}$ denote the analytical solution of a PDE. Let $x^{(k)}$ be the solution at the $k^{th}$ iteration. The initial guess for the solution is $x^{(0)} = 0$. Let $r_{0} = ||\bar{x}-x^{(0)}||_{2}...
Gaurav Saxena's user avatar
1 vote
0 answers
84 views

Manufacturing a solution for non-smooth coefficients in elliptic problems

This question is a continuation of this answer (I can't comment) If we were going to manufacture a solution for a problem with discontinuous coefficients, I understand that the solution should have ...
balborian's user avatar
  • 601
1 vote
0 answers
111 views

Application of vector extrapolation methods to convergence to a steady state solution

I'm working on a fluid solver using dual-time stepping and everything works really well, except the convergence in pseudo-time is slow. I'd like to accelerate the convergence. I know multigrid methods ...
tpg2114's user avatar
  • 608
1 vote
0 answers
221 views

Oscillating convergence in my Resilient BackPropagation (RPROP) implementation

I have implemented in matlab a neural network that uses rprop's algorithm to update its weights. Strangely the error on the training set does not converge to a local minimum, but oscillates. Here is ...
Teo Teo's user avatar
  • 11
1 vote
0 answers
338 views

Am I using the incorrect implementation of the fast Chebyshev transform?

I was told that the fast Chebyshev transform has superior spectral convergence, but I am unable to verify its rumored convergence. I was given plots of its spectral convergence, where the signal's ...
linuxfreebird's user avatar
1 vote
0 answers
155 views

Convergence of a DASPK depending on DAE formulation

I have a system of non-linear DAE and I noticed that the system does not converge if some of the equations are not differentiated. For example, if the control volume equation is represented as this: $...
Marwan's user avatar
  • 39
1 vote
0 answers
178 views

Stationary phase approximation for an integral with infinity saddle points?

I need a hand with the numerical evaluation, in Mathematica, for this integral: $$f(t)=\int_{-\infty}^\infty Exp\{it(\omega_H-\omega_l-\omega_k) - \sum _{j\neq(l,k)} S_j [1-e^{-it\omega_j}]\}\, dt$$ ...
Gustavo Lara's user avatar
1 vote
0 answers
90 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} = 0\...
tau1777's user avatar
  • 451
1 vote
0 answers
134 views

Hatree-Fock, reasons for convergence/ non-convergence

I'm new here so please forgive me if I lack proper stack exchange etiquette. So, I was wondering if anyone here could provide insight on a problem that I am running into with with a Hartree-Fock ...
normal person's user avatar
1 vote
0 answers
4k views

Stationary solution converge but time dependent doesn't

I've coupled a COMSOL model for fluid dynamics with a very simple pde that model the transport of humidity in air. When I solve it for the stationary case, the solution converge easily, but when I ...
Lorenzo Nespoli's user avatar
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 ...
Diplodokus's user avatar
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 ...
JasonC's user avatar
  • 43