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.

56 questions with no upvoted or accepted answers
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8
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525 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 ...
8
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0answers
160 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}_{+}^{...
7
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0answers
81 views

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 (...
5
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0answers
131 views

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 ...
5
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0answers
215 views

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 $x_*$ ...
5
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0answers
367 views

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 ...
5
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0answers
84 views

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 ...
4
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0answers
116 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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0answers
547 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 : $$ ...
4
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0answers
236 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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0answers
64 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
213 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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2answers
919 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. ...
3
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0answers
121 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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0answers
35 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{...
2
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0answers
33 views

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 ...
2
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0answers
62 views

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 ...
2
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0answers
389 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 $\...
2
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0answers
697 views

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 ...
2
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0answers
113 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 ...
2
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0answers
69 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 ...
2
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0answers
970 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 ...
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0answers
59 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 ...
1
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0answers
52 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 ...
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0answers
68 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. ...
1
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0answers
41 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 ...
1
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0answers
52 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 = ...
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0answers
64 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". ...
1
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0answers
79 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 ...
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0answers
247 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 ...
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0answers
143 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 ...
1
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0answers
59 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}...
1
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0answers
75 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 ...
1
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0answers
103 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 ...
1
vote
0answers
202 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 ...
1
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0answers
237 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 ...
1
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0answers
145 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: $...
1
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0answers
161 views

Stationary phase approximation for an integral with infinity saddle points?

I need a hand with the numerical evaluation, in Mathematica, for this integral: $$f(t)=\int_{-\infty}^\infty Exp\{it(\omega_H-\omega_l-\omega_k) - \sum _{j\neq(l,k)} S_j [1-e^{-it\omega_j}]\}\, dt$$ ...
1
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0answers
87 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\...
1
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0answers
108 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 ...
0
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0answers
119 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 ...
0
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0answers
25 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 ...
0
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0answers
64 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 ...
0
votes
0answers
38 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& \\ ...
0
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0answers
37 views

How can the choice of coarsening factor affect Multigrid's convergence?

The linear system $Ax=b$ is coming from the discretization of an elliptic PDE. Multigrid method is used in order to solve it. Suppose $c_0$ is the coarsening factor on level 0 and $c_m$ the coarsening ...
0
votes
1answer
107 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)} = ...
0
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0answers
63 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 $c_2$. So after using Levenberg–Marquardt algorithm, I could only get $x_1$, $y_1$ and $...