Questions tagged [iterative-method]

A method which produces a sequence of numerical approximations which converges (provided technical conditions are satisfied) to the solution of a problem, generally through repeated applications of some procedure. Examples include Newton's method for root finding, and Jacobi iteration for matrix-vector solves.

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516 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 ...
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152 views

Implementation of Lanczos method that returns tridiagonal matrix

The Lanczos method can be used to obtain extremal eigenpairs of sparse symmetric or hermitian matrices. I know there are several implementations of the Lanczos method (as well as Arnoldi, Davidson, ...
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91 views

Trying to understand splitting-based iterative method for 2D Laplace problem

I am trying to understand the theory behind a splitting based iterative method which uses the incomplete Cholesky factorization. Before giving the specific details, let me first give the problem ...
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78 views

Improving convergence of Jacobi iteration to Schur form

I'm using SIMD processor arrays to compute the eigen-decomposition for large numbers of small (up to $32\times 32$) matrices. For assorted technical reasons, Jacobi iteration maps well to the SIMD ...
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459 views

Solving a PDE implicitly by iteration in python

Connected to this question here on Computational Science, I've posted a follow-up question on how to solve a PDE using an implicit scheme like Crank-Nicholson in general in this question on SO. But I ...
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185 views

Preconditioning technique for large sparse non-hermitian matrix

I am attempting to solve a computational acoustics problem that involves solving an underlying sparse matrix. The size of the problem varies with grid size (3D) and fill-in's obviously make direct ...
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182 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. ...
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779 views

Can Gauss-Seidel/SOR (preconditioned?) be applied to a non-diagonally dominant matrix?

After applying finite difference method to a Laplace/Poisson problem always arises a diagonal dominant system of equations that can be solved with Gauss-Seidel or SOR methods. If the original PDE does ...
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203 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 $x_*$ ...
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344 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 ...
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1answer
242 views

Robust smoothers for geometric multigrid

I'm searching for robust smoothers for geometric multigrids. By robust I mean: Effective for high order approximations (say spectral element, spectral Discontinuous Galerkin), Parallel (suitable for ...
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15 views

solving linear system whose symmetrized matrix is positive definite

Are there iterative methods for the solution of nonsymmetric linear systems $Ax=b$ that can take (theoretical or practical) advantage from knowing that $A+A^T$ is positive definite? These matrices are ...
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240 views

Nonlinear least squares and regularization

Consider the nonlinear least-squares minimization of a vector of $n$ residuals $\mathbf{f}$ in $p$ parameters $\mathbf{x}$: $$ \min_{\mathbf{x}} || \mathbf{f}(\mathbf{x}) ||^2 $$ This can be done with ...
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503 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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46 views

Python routine to calculate shape resonances of H2

I am currently doing a project in which my aim is to write a program that can be used to calculate single and multi-channel shape resonances. So I'm looking at bound states and quasi-bound states. ...
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44 views

Subspaces for Iterative methods

In the original paper of Conjugate Gradients, the authors mention that if we pick the canonical basis $\{e_1,e_2,\ldots,e_n\}$, to obtain A-orthonormal vectors, we end up with the Gaussian elimination ...
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197 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 ...
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85 views

PETSc SNES for user defined state

How to use PETSc SNES (scalable nonlinear equation solver), when the solution is not a vector but a user defined state? I am implementing a non-linear mechanics problem (geometrically exact shell 5-...
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262 views

How to avoid the Broyden's jacobian approximation becoming poorer with the number of iterations?

I have to solve many times a nonlinear system of the form $$f(x) = b^{(n)}$$ inside a loop. The function $f$ is expensive to compute and I do not have its jacobian, so I have tried the good Broyden's ...
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1answer
77 views

Project to nearest point on convex polyhedron

I have a point $y \in \mathbb{R}^d$ and a convex polyhedron $\mathcal{P}$ given as the intersection of half-spaces: $$\mathcal{P} = \{x \in \mathbb{R}^d \mid a_1 \cdot x \le b_1, \dots, a_n \cdot x \...
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108 views

Residual of Poisson equation with periodic boundaries

I am trying to write a multigrid solver for Poisson's equation, $-\Delta u=f$, on the unit square, $\Omega=(0,1)^2$ with periodic boundaries. My primary source has been Multigrid by Trottenberg, ...
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93 views

How to reproduce the numerical examples in Prof. Saad's Book about Krylov subspace methods?

After reading Prof. Saad' Book, "Iterative methods for Sparse Linear Systems, 2nd version", I want to do the numerical examples about the Krylov subspace methods not only to reproduce the results in ...
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100 views

Small residual but wrong results

When I use BiCGStab to solve a linear matrix system, I use the relative residual to exit the iteration and output the results. For calculating the relative residual I divide the norm of vector $r$ ...
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209 views

Understanding MP-PIC implementation in OpenFOAM

The multiphase particle-in-cell (MP-PIC) method is characterized by mapping particle properties from the Lagrangian coordinates to the Eulerian grid. However, the implementation of this method in ...
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162 views

Richardson's Iteration, Gradient Method and Spectral Radius

Richardson's iteration introduce a scalar $\alpha$ to the update formula: $$ \textbf{x}^{(k+1)} = \textbf{x}^{(k)} + \alpha \textbf{r}^{(k)} $$ And compute $\alpha$ by minimizing the spectral radius:...
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69 views

Simultaneous update to barycenters

Suppose a tiling is given in 2D (an embedding of a planar triangulated graph), with all faces convex. Now suppose one moves each point, one by one, to the barycenter of its neighbors. I think that ...
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70 views

How can I numerically solve a saddle point problem with repeated constraints?

I am interested in numerically solving the following constrained minimization problem; Find the value of $x\in \mathbb{R}^n$ that minimizes $f$ where $f\colon \mathbb{R}^n\to \mathbb{R}$ is defined ...
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153 views

Generalized eigenvalue with null space

Define $S\in\mathbb{R}^{n\times n}$ as $$S:=H+Q^\top V^{-1} Q.$$ $H,V$ are positive semidefinite. Here, $H$, $Q$, and $V$ are large, dense matrices but they are structured: I can write code for ...
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82 views

Numerical error in implementation of iterative algorithm

I am trying to implement (in Python for now) low thrust orbit propagation for spacecraft using universal variables. For a given central body with the gravitational parameter $\mu$ and an orbit with ...
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68 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 ...
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52 views

SPECT reconstrction using MLEM

In Single-Photon Emission Computerized Tomography (SPECT) parallel beam reconstruction using Maximum-Likelihood Expectation–Maximization(MLEM), is it sufficient to scan the object around 180 degree? ...
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242 views

Does Conjugate Residual really have convergence properties similar to that of Conjugate Gradient?

I have coded up a toy implementation of Conjugate Residual and have been testing it. Both wikipedia and the Saad claim that Conjugate Residual and Conjugate Gradient have similar convergence behavior....
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108 views

Modal analysis of structure with aerodynamic damping

I'm using modal decomposition to predict the steady state response of a beam structure to harmonic loading. The structure itself is very lightly damped, but we know from experiments that the ...
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71 views

What are Implications of Commutative Diagrams?

This question may be too broad. But I really want to know some concrete explanations. I often find various commutations appear here and there, which concerns the application order of two operators. ...
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29 views

Approximation in the derivation of the Arc Length method

I am studying the proof of the Arc Length method in section 2.2 of this thesis. In equation (2.2) the author introduces the supplementary conditions $$ (\Delta {\bf u} + \delta {\bf u})^T \cdot (\...
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62 views

Upper bound on condition number in linear preconditioning

I'm studying iterative methods for solving linear system, and I find the following setting in Wikipedia: Consider a matrix splitting $A = M-N$, where $A,M,N$ are all symmetric and positive definite ...
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Problem with recursive implementation of Subspace Iteration method in Numpy

I am having trouble with implementing the method of subspace iteration to find the eigenvalues and vectors of a random, symmetric matrix, A that is mxm with m = 10. ...
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273 views

Minimax optimization with an oracle

I have an optimization problem of the following form: $$\min_y\left[\max_x f(x,y)\right].$$ It is fairly straightforward to minimize $f(x,y)$ over $y$ with $x$ fixed, and similarly to maximize $f(x,...
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1answer
174 views

Calculating Error for Poisson Equation using Successive Over-Relaxation technique, Python

I am trying to solve the Poisson Equation $\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 32(x(x-1) + y(y-1))$ for a 61x61 grid using Python3 with boundary conditions being $T=...
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116 views

Is there any function to calculate condition number of sparse matrix in Eigen libraray?

The function JacobiSVD and BDCSVD can calcuate condtion number of a dense matrix via singular values. However I need to know condition number of a sparese matrix due to slow computation speed using ...
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80 views

2-norm of solution update suddenly becomes zero after a few iterations

I am trying to solve the Poisson equation in 2D for heterostructure devices. I have linearized the equation and discretized it using FDM. I am using BiCGStab to iteratively solve for the solution as ...
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63 views

1-D Conduction Steady state heat transfer using FD

I have tried to solve 1-D Conduction Steady state heat transfer problems in Matlab (see below). Here is the 1-D model: T''[x] == 0, T[0] == 100, T'[100] == 0 ...
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69 views

Bifurcation points on homotopy path by numerical continuation?

I am trying to implement an algorithm that finds (possibly) all solutions of a system of nonlinear polynomial equations $$F(X) = 0$$ I thought about using the (convex) homotopy $$H(X,T) = TF(X) + (1-T)...
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60 views

Can the standard multigrid performance be used for time-dependent PDEs?

Consider a time dependent pde(i.e u(x,t)).I know when only space-coarsening is used the standard multigrid performance can be applied but what if instead we use only time-coarsening?Can we apply the ...
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49 views

BicgStab is not able to solve while Jacobi or GaussSeidel Methods can

I am trying to solve the 2D laplace equation, $\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0; \qquad 0 \lt x \lt 1, \quad0 \lt y \lt 1$ Subjected to the boundary ...
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59 views

Kinetic preconditioning

Publication arXiv:0804.2583 describes a method for doing self-consistent iteration without having to diagonalize the Hamiltonian operator at every step. IX. PRECONDITIONING As already mentioned, ...
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34 views

Logging vs outputs in iterative optimisation

I'm coding an iterative algorithm of constrained continuous optimisation. An augmented Lagrangian algorithm (outer) calls a bound-constrained L-BFGS-B algorithm (inner), which calls a line search ...
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61 views

Fixed point iteration reduction factor

In a book for solving a nonlinear differential equation with $N+1$ points, $u_{xx} = e^{u}, u(-1)=u(1) = 0$, in $[-1,1]$ with homogeneous Dirichlet boundary conditions, the fixed point iteration is ...
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65 views

fourth order Poisson iterative solver --in Matlab

I want to calculate the stream function $\psi$ starting from a velocity field $(u,v)$ (such that $u=-\frac{\partial\psi}{\partial y}$ and $v=\frac{\partial\psi}{\partial x}$). I thus calculate the ...
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56 views

bound error for iterative method for solving linear system

$A$ is square and positive definite, and let $r_k = Ax_k - b$. Also let $M = \frac{1}{2}(A+A^T)$. I want to show that $$\frac{||r_{k+1}||_2}{||r_k||_2} \le \left(1-\frac{\lambda_\min(M)^2}{\lambda_\...