# Questions tagged [linear-system]

For questions about solving linear systems of equations. These typically take the form Ax=b, where A is a matrix , while x and b are vectors.

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### Interpolation and Restriction operators in Multigrid

I saw in several places that interpolation operator ($P$) and restriction operator ($P^T$) are usually transposes of each other (up to multiplication by a constant). As I understood it related to ...
170 views

### Solving huge dense square symmetric linear system

I have a linear system of the type $A x = y$ where A is a dense, square, symmetric, positive definite matrix, $x$ a vector of unknown parameters, and $y$ is a vector of observed quantity. I know that ...
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### Elementary question on numerical linear algebra

I’ve been facing a problem of solving linear systems Ax=b arising from discretized PDEs (Stokes equations in particular). Nively, it seems that solving Ax=b should not take much more time than simply ...
46 views

### Pass forward intermediate results during iterative optimization

To investigate a counter-current flow heat exchanger while considering temperature dependent physical properties (such as specific heat $c_\textrm{p,i}$, heat conductivity $\lambda_\textrm{i}$, ...
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### When do not use preconditioners for sparse linear system of equations?

I'm implementing a solver of Finite Element Method, and to solve the linear system of equations I'm using gmres from MKL of Intel. Exists the option with and without a preconditioning. In what case it ...
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1 vote
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### Solution of an underdetermined system stemming from a PDE with Neumann BC

Consider the Poisson's equation in 1D with homogeneous b.c.'s $\mathrm{d} \phi/\mathrm{d} x=0$ with the seven point Laplacian (1 -54 783 -1460 783 -54 1 / 576 on a uniform grid). The resulting system ...
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### Simplest solver for linear equation systems

Normally, this boards sees a lot of traffic about the most efficient and most powerful solvers for huge linear equation systems. But this time, I have the opposite problem: I need to implement a ...
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### Preconditioning vs. regularization

I used to be more of a numerical linear algebra and computational science person, but recently, I've crossed into stats and machine learning. For this discussion, let's focus on matrices that are not ...
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### Solve a system of coupled differential equations in Python

I have a system of two coupled differential equations, one is a third-order and the second is second-order. I am looking for a way to solve it in Python. I would be extremely grateful for any advice ...
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### Algebraic multigrid as solver and as preconditioner

My question is around the efficiency of AMG. In which case AMG can perform better,as solver or as a preconditioner(for example a Krylov space method as CG)? Assume the case of elliptic pdes.
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### Solving a linear system with block-banded matrix

It is required to solve a linear system $Ax=b$, where the matrix $A$ is symmetric, all the variables and coefficients are real. The structure of $A$ is A = \begin{pmatrix} A_{11} &...
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### Kronecker-factored least-squares?

Suppose $a_i,b_i$ are $d\times 1$ matrices, $y_i$ is scalar, and I need to find least-squares solution $w$ of the following system of $n$ equations: $$y_i=(a_i^T\otimes b_i^T)w$$ Is there a ...
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### Numerical Linear Algebra: When to use Direct methods versus iterative methods to solve a linear system - for PDEs in particular

I am reading the Chapra and Canale book on numerical methods, and was working through the chapters on solving linear systems. Now the book goes through direct methods including Gaussian Elimination, ...
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### Linearize non-linear PDE with BCs to hyperbolic problem: How does linearization affect BCs?

I am working with the Shallow Water equations that is a system of non-linear PDEs that simulate water waves propagation on some domain, in my cases the $x$-axis. I have here a GIF showing the results (...
1 vote
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### How to implement flexible gmres in matlab?

About the flexible GMRES (fgmres), we know that it is a variant of right preconditioned gmres. And the robust command gmres in matlab as follows: ...
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1 vote
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### How to compute the computational cost and storage of the Full Orthogonalization Method?

About the analysis of Full Orthogonalization Method (FOM) in Prof. Saad's book, wrote as follows: Algorithm 6.4 (FOM): \begin{array}{l} r_0=b-Ax_0,\beta=\|r_0\|_2,v_1 = r_0/\beta\\ Define \quad H_m ...
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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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### 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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1 vote
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### 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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1 vote
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### How to obtain linear tridiagonal system from PDE

I'm trying to re-solve the governing equations in hydraulic fracturing modeling as instructed step by step in a paper. After (A-9), the author stated that by substituting A-6, A-8 and A-9 into ...
79 views

### Analytic vs discrete understanding of PDE

The PDE I am working with: $$\partial_tu = \nabla \cdot (a(x)\nabla u)-\beta(x)u\\ \partial_nu=0, x \in \Omega \subset \mathbb{R}^2\\ \beta(x)>0$$ Integrate the PDE: \int_\Omega \partial_t u=\...
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### What will PDE discretization matrix look like for time and space? [duplicate]

Please note: this question is not a duplicate of this question since, while the PDE is the same, the nature of this question is different, i.e. the other question treats a different aspect of this PDE....
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### Is steady linear elasticity inherently ill-conditioned?

Compared to the transient PDE for linear elasticity, the steady equations appear to less well-conditioned. Are they inherently ill-conditioned without the transient term? The condition number for ...
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### Condition number of matrix and effects of round off errors

In my numerical linear algebra class, I learned that for some matrices, it could have an element that is a very small number that is approximately 0 (and many orders of magnitude different from all ...
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1 vote
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### Thermal stress to displacement through finite volume method

I am attempting to solve for a displacement field where I know the thermal stresses on my discretized domain, which consists of hex cells. My first question is: is easier way to solve for the ...
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### TDMA with 3rd order upwind scheme

I'm trying to implement a model I found in a paper, but there is something I do not understand. The authors say they use TDMA to solve their equations; however, they use a 3rd order upwind biased ...
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### How "sparse" should a sparse matrix be to see benefits?

I have a matrix, whose size scales as $2^N$ (assume even $N$). In each row of the matrix, only about $2^{N/2}$ of the entries are filled ($N$ can be somewhere between 10 and 40, depending on what's ...
1 vote
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### What are the names of the variables in the linear system $Ax=b$

I've been discretizing PDEs and formulating $Ax=b$ systems, and yet I don't really know what the $A$ and $b$ are in words. I occasionally call the $A$ matrix the "Jacobian matrix," but for linear ...
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I am studying the reverse mode of automatic differentiation. The reverse mode of automatic differentiation allows the efficient computation of a the derivative of a single dependent variable $y$ with ...