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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54 views

MInimizing cost function using iterative search for a minimum method

I want to estimated the parameters $\ \hat{\theta} $ of a model using an iterative search for the minimum of a cost function. The cost function is defined as follows: $$ V_N(\hat{\theta}) = \frac{1}{...
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How to implement the Hessenberg QR Algorithm?

For context, I'm creating a linear algebra library from scratch for learning purposes in C. Right now I'm working on calculating eigenvalues but my implementation of the QR Algorithm is diverging. ...
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Solving a nonlinear problem with a very small components with finite element method

In solving nonlinear hyperelastic solid mechanics problems, to converge to the correct solution we need to do step-by-step loading which makes the deformation at each step very small (for my ...
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What online optimisation algorithm can be used for a noisy cost function?

I am trying to optimise a function, but the function can be noisy and give varying results for the same parameters. Furthermore, it needs to be online, as the data from each new iteration happens ...
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1answer
72 views

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 ...
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1answer
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How to solve for f(A)x=b without GMRES?

How to solve for $f(A)x=b$? For GMRES, an answer is given in this book chapter: http://link.springer.com/chapter/10.1007%2F978-3-642-58333-9_2. Ungated version: https://www.researchgate.net/profile/...
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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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1answer
58 views

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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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 $...
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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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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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Why MATLAB chooses the Householder in its built-in function gmres.m?

Recently, I have studied how to construct an orthonormal basis for Krylov subspace to solve $Ax=b$, where $A\in \mathbb{R}^{n\times n}$ is nonsingular. As we know, there are usually 4 ways to ...
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How to understand the choice of Krylov subspace orthonormal basis?

This semester, I study the Krylov subspace iterative methods (about Ax=b) using the book H. A. Van der Vorst. Iterative Krylov Methods for Large Linear Systems, volume 13. Cambridge University Press, ...
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138 views

What's wrong with the **PCG and MINRES** in matlab?

Last week, I have learned the details of the robust iterative methods of PCG, MINRES, GMRES, which will converges to the exact solution $x^*$ of nonsingular system within $N$ steps for $A\in \mathbb{R}...
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When should I write a matrix-vector function to handle the sparse matrix vector multiplication?

This semster, I have been studying the iterative methods for large sparse matrix system. But I have some questions. For large sparse matrix, we must use an economic storage to store them. The most ...
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Does the k-th approximate solution of a stationary iteration belong to the k-th Krylov subspace?

For an stationary iteration method solving $Ax=b$ as follows: $$ Mx_k = Nx_{k-1}+b, $$ I have known that when $M = I$, i.e., the Richardson iteration, the k-th solution $x_k = x_{k-1}+r_{k-1}$ is in ...
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What is the standard, extrapolation, and modified version of Richardson iteration method?

I have been studying the iterative methods recently. For classical iterative methods solving $Ax=b$, I have seen that the most simplest iteration method is the so-called "Richardson iteration". But I ...
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114 views

Why Krylov subspace iterative methods are faster than classical iteration?

This semester, I have been studying the most popular iterative methods, i.e., Krylov subspace iteration methods. For a large sparse system linear $$ Ax=b, $$ where $A$ is nonsingular, I know that ...
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Why do not we choose the error solution norm as an iterative method's criterion?

For solving linear system $$ Ax=b, $$ using iterative mehods, we often use the terminate criterion as follows: $$ \frac{\|r_k\|}{\|r_0\|}=\frac{\|b-Ax_k\|}{\|b-Ax_0\|}<eps. $$where $x_0$ is the ...
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Why does the initial guess for linear system usually choose by zero vector?

For solving linear system $$ Ax=b, $$ using iterative mehods, we often use the terminate criterion as follows: $$ \frac{\|r_k\|}{\|r_0\|}=\frac{\|b-Ax_k\|}{\|b-Ax_0\|}<eps. $$where $x_0$ is the ...
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How to implement Geometric Multigrid in non-rectangular grids?

It is quite easy to implement multigrid on a rectangular grid but what about an non-rectangular?How to coarse a non-rectangular grid and apply multigrid(assume an easy non-rectangular grid capital ...
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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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Are there any other better methods than block diagnoal and block upper triangular precondtioner for saddle point problems?

For stokes problems, $$ -\Delta \vec{u} + \nabla p =\vec{f},\qquad \nabla . \vec{u} = 0; $$ with appropriate boundary conditions which guarantee there is a unique solution. Using FDM or FEM, ...
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1answer
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What is the **contraction factor and convergence factor** of a iteration method?

For any iteration method from A=M-N, e.g., $$ Mx_{k+1}=Nx_{k}+b,\quad k=0,1,... $$ we know that it converges iff $\rho(M^{-1}N)<1$. And when it converges, there exists a concept called asymtotic ...
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131 views

How suitable is multigrid method for time-dependent PDEs?

For elliptic PDEs (Poisson-type), the multigrid method is very sufficient, but how about time-dependent problems (i.e parabolic or hyperbolic PDEs)? Is it efficient to solve such problems using a ...
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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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Can a direct method like Thomas be used in a multigrid method as a smoother?

As far as I know, multigrid uses stationary iterative methods as smoothers (i.e GS), but can we use a direct method also? For example, in case we have a tridiagonal system (for example 1D heat ...
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1answer
92 views

Solve linear system with Newton-Raphson method

Is it possible to solve a linear matrix system $A x = b$ using the Newton-Raphson method? If yes, how can this be done? More special, how is the derivative build?
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Low rank update of QR of inverse

I am in a situation where as part of a sort of inverse power method scheme, I want to very often perform the following step: Apply a symmetric rank one update $uu^\top$ to my inverse matrix $A^{-1}$ ...
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153 views

Does iterative method work for singular consistent linear system Ax=b?

Recently, I have been studied iterative methods for large sparse linear system Ax=b, where A is nonsingular, so there is a unique solution x. And the stopping criterion is usually chosen with norm(b-...
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Divergence issues when using intrinsic cohesive elements approach

When I model the strain localisation of a microscopic sample (or say RVE ) with cohesive elements approach, the convergence performance looks very terrible. I have to use extremely time increments (...
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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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Evaluation of slope at iteration ith - Newton-Raphson method

I'd like to know how Ansys computes the slope (=stiffness matrix) at point x1 in figure. I'm studying the way in which Ansys uses the Newton-Raphson method when there are nonlinearities. In the slide ...
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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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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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1answer
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The proper way to assess the error of Jacobi iteration (for 2D Poisson equation)?

Motivation: I'm using 2D regular grid (it's actually a quadtree but I can still treat it as a finite difference thing if I weight-average the solution over smaller scale cells for the purpose of ...
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Why iterative method: AMG preconditioned PCG is slower than Matlab direct method 'A\b'?

Recently, I have met a question that a saying goes that for large linear system: iterative methods are required because of memory problem of direct methods. But when I implement some experiments ...
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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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1answer
373 views

What's the terminology for this alternative minimization algorithm?

Say the model is $F(x_1)G(x_2)Z(x_3) = y \in \mathbb{R}^N$, with $F,G,Z$ explicitly known, we are given observation of $y$ as $y_b \in \mathbb{R}^N$ to find the value of $x_1$, $x_2$, $x_3$ for each ...
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Iterative linear solver for “ugly” saddle point system

I am a graduate student majoring scientific computing. The numeric model I made caused a very ugly-looking saddle-point linear system. It is not symmetric at all and I will attach the sparsity pattern ...
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1answer
88 views

Iterative solution of ill-conditioned matrix systems

I want to solve a matrix system of the form $Ax=b$ where $A$ is ill-conditioned. The matrix system comes from a structural simulation problem which was discretized using finite elements. I do not have ...
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1answer
137 views

How to set an initial guess for the iterative solver in Comsol?

How to set the initial guess for the iterative solver GMRES or FGMRES for linear problems (Helmholtz equation of RF module) in Comsol?
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1answer
143 views

Complex differentiation of linear solvers

I have a linear system $$Ax=b$$ which I'm solving approximately, and I need to take the frechet derivative of x with respect to z. Were I solving the problem exactly (either analytically or to machine ...
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1answer
280 views

Why my parallel code using MPI is much slower than the serial one?

I know that is not the first time someone asks this question but I'm really confused.I'm new to MPI, and I tried to implement the Jacobi solver for a linear system $Ax=b$. I want to compare the time ...
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1answer
105 views

Optimality of block-Jacobi preconditioner

For a dense $N \times N$ matrix $A$, is the block-Jacobi preconditioner comprising the inverse of the diagonal blocks of $A$ the optimal block-diagonal preconditioner? Could there exist another matrix ...
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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 ...
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A preconditioner for self-consistent iteration

I tried to derive a preconditioner for self-consistent iteration similar to section IX in arXiv:0804.2583. For simplicity, consider here only one orbital (one or two electrons) systems. Suppose that ...
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1answer
144 views

In iterative methods, are matrix decompositions considered useful for implementation?

When we study an iterative method from textbooks, for example, see the Gauss-Seidel Method, the given matrix is decomposed with suitable splittings. In the example, $A = L+U$. So we can proceed with ...
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1answer
65 views

Blowup of error in Conjugate Gradient method with periodic Dirichlet Poisson matrix

My problem is that the L2-Norm of the residual for the periodic Poisson matrix $P$ is initially decreasing but starts to blow up after a certain number of iterations. The blowup happens earlier the ...
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1answer
101 views

Re-using LU factorization within iterative (?) setup for a sum of two matrices

So, I would love to make at least some use of my preexisting data, no matter how small, and just out of ideas. Maybe I am just a prisoner of a Kahneman-like theatre-ticket paradox, and don't know ...

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