All Questions
453 questions
0
votes
0
answers
72
views
Developing a poisson equation solver for arbitrary input geometry and boundary conditions
I need to develop a Poisson equation solver that can take a input geometry and boundary conditions, produces a FEM mesh, generates the matrices and solve it. Problem is I want a solver that can use ...
- 101
0
votes
0
answers
60
views
Solving linear-stabilty problem numerically
Consider following coupled equations
$$\sum_nC_{n,(f)}(\theta, \phi , \psi) \partial_\theta^n f + \sum_nC_{n,(g)}(\theta, \phi, \psi) \partial_\theta^n g = \partial_t f (\theta, \phi) $$
$$\sum_nD_{n,(...
1
vote
4
answers
170
views
Solve Large Scale Underdetermined Linear Equation with per Element Equality Constraint
I have the following system on $\boldsymbol{x}$:
$$ \boldsymbol{A} \boldsymbol{x} = \boldsymbol{0}, \quad \text{subject to} \; {x}_{i} = {v}_{i} \; \forall i \in \mathcal{V} $$
Where $\boldsymbol{A} \...
- 573
0
votes
2
answers
90
views
Diagonal and Upper-Triangular pre-conditioning for Jacobi
I am interested in analyzing convergence of the Jacobi method to solve the linear system $Ax=b$,
$$\begin{pmatrix}
2 & 4 \\
1 & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2
\end{pmatrix}
=
\...
- 131
0
votes
0
answers
128
views
Heat Equation for fast source with FiPy
I'm trying to solve the following differential equation with FiPy, basically laser irradiation on a surface
$$
\rho_{s}C_{p,s}\frac{\partial T}{\partial t} = k_{s}\frac{\partial^{2}T}{\partial x^{2}} +...
- 11
2
votes
1
answer
140
views
Efficient Solver for Solving a Large Linear System Sequentially of a Positive Definite Matrix
In my case, I am solving $AX=B$ repeatedly, but the solution usually doesn't change much. So it'd probably be faster than me when I start from the previous solution and iterative, rather than solving ...
- 149
2
votes
1
answer
77
views
When does linear system have linearly growing singular values?
Suppose $W$ is a large matrix where $i$th smallest singular value grows as $O(i)$. What kind of matrix can $W$ be?
For instance, this appears to hold for random matrix with IID entries and for lower-...
- 2,799
3
votes
0
answers
53
views
Datasets for inverse heat transfer problems
I was wondering if there is an available, real-life known inverse heat transfer problem dataset to benchmark oneselfs algorithm, as in MNIST for deep learning. Talking about... (well in this case I ...
- 191
1
vote
1
answer
256
views
Coupled Partial Differential Equations
I'm trying to solve the following system of coupled differential equations, the two-temperature model for $e$ = electrons and $l$ = lattice.
$$
\rho_{e}C_{p,e}\frac{\partial T_{e}}{\partial t} = k_{e}\...
- 11
1
vote
0
answers
50
views
How can I apply a mixed boundary condition to a multi-material heat transfer problem using Crank-Nicolson?
I am working on a mixed material model for a melting material and need to enforce both a Dirichlet and Neumann type condition at the interface. Subject to an external surface heat flux at the top of ...
- 11
0
votes
0
answers
38
views
Thermo Hydraulic Mechanical modeling of energy wall slab in Comsol multiphysics
I am currently working on a complex simulation project involving an energy wall slab, and I need assistance in accurately modeling and validating it using COMSOL Multiphysics. Here are the details of ...
1
vote
0
answers
54
views
Particular linear systems: sparse matrix + column
I am trying to understand a limitation in a routine in the interval arithmetic software Intlab. From matrices starting from a given size (in my particular problems),...
- 1,077
4
votes
1
answer
389
views
Matrix Diagonalization and Computational Requirements
I have some questions about diagonalizing matrices. My interest lies in computing all eigenvalues of a given matrix. To avoid wasting time and improve my research efficiency, I want to understand the ...
- 143
0
votes
1
answer
159
views
recommended simple linear solver on gpu
I'm looking for recommendations for a simple GPU linear equation system solver that is a dropin replacement for scipy.linalg.solve. Right now, I'd rather not go the Petsc/TriLinos/Slate route. ...
- 852
2
votes
0
answers
129
views
Plasma charge conservation for a multi-Euler system - looking for quasi-linear Riemann solver that also resolves slow contact discontinuity
I am solving a multi-species plasma problem by assigning a set of ideal gas Euler equations to each species, e.g. protons and electrons.
I.e. I am solving the system
$$
\partial_tU_s + \partial_x(F_s) ...
1
vote
2
answers
121
views
How to handle non bilinear weak form?
I solved the 2D heat equation using the finite element method. It all went well first with the adiabatic case, however problems occured when I introduced cooling with the enviroment.
I modeled the ...
- 63
7
votes
3
answers
2k
views
How large is large for direct solvers?
Let us say I want to solve a large sparse linear system. It is said that iterative solvers should be better than direct solvers in this case. But how large is large? What is the exact threshold beyond ...
- 181
5
votes
2
answers
160
views
Cheap way to keep parameter matrices orthogonal during optimization?
TLDR; I can keep matrix variables approximately orthogonal by taking a single gradient step in the direction of "effective rank" of matrix at each step of iterative solver, is there a more ...
- 2,799
0
votes
1
answer
94
views
Weird runtime behavior of `scipy.linalg.solve_triangular` and `trtrs`
I want to understand the time complexity of scipy.linalg.solve_triangular, which calls trtrs from LAPACK under the hood, so I ...
- 181
4
votes
1
answer
313
views
Saddle point system
I am solving a system of the form
$$ \begin{pmatrix}
A & b^T \\
b & 0
\end{pmatrix}
\begin{pmatrix}
x \\ \ell
\end{pmatrix}
= \begin{pmatrix}
c\\
0
\end{pmatrix}
$$
Where $A$ is a symmetric ...
- 1,077
0
votes
0
answers
38
views
How conservation of momentum is ensured in (Projected) Gauss-Seidel constrain solver
I'm developing molecular dynamics where my time-step is limited by stiffness of the bonds. I trying to get inspiration from game-engines, where they solve similar problem (hard bond constrains). These ...
- 937
3
votes
0
answers
151
views
Population of the coefficient matrix of a linear system Ax=b stemming from the finite differences of an arbitrary geometry
I've been looking into solving a linear system $$Ax=b$$ where $A\in\mathbb{R}$ is the sparse coefficient matrix of size $K\times K$, $b\in\mathbb{R}$ is the right-hand side (i.e., the source term) of ...
- 83
0
votes
0
answers
67
views
Solving AU = F using linalg.cg results in 0 iterations
I am working on solving the following PDE: $$\left(\mu_{x}\frac{\partial^{2}u}{\partial x^{2}}+\mu_{y}\frac{\partial^{2}u}{\partial y^{2}}\right)=f(x,y) \tag 1$$
Which is then discretised:
$$- \mu_{x} ...
- 43
2
votes
1
answer
65
views
Solving $(I-Q)x={\bf 1}$ for sub-stochastic sparse $Q$ of dimension 5M $\times$ 5M
I have a (right) sub-stochastic CSC sparse matrix $Q$ of dimension 5 million, with 200 million nonzero entries, which is a nonzero percentage of 0.0008%, so it is indeed extremely sparse. It is not ...
- 513
2
votes
1
answer
411
views
Where am I making a mistake in solving the heat equation using the spectral method (Chebyshev's differentiation matrix)?
I would like to numerically solve the following heat equation problem:
$$ u_t = \Bigg(2{a \over l}\Bigg)^2 u_{xx} \tag 1$$
$$ x \in [ -1, 1 ] \tag 2$$
$$ u(x, 0) = 0 \tag 3$$
$$ u(1, t) = A \sin \Bigg(...
3
votes
1
answer
118
views
Solving for $X$ in $\sum_{a,b} b a^T b^T X a = Y$
Suppose I have $k$ pairs of $(a,b)$ where $a$ and $b$ are vectors in $\mathbb{R}^d$, $Y$ is $d\times d$ and I need least squares solution for $X$ in the following
$$\sum_{(a,b)}^k b a^T (b^T X a) = Y$...
- 2,799
2
votes
0
answers
47
views
How to use a preconditioner estimated from a subset of data?
Suppose I'm solving $Ax=b$ using row-action method like Kaczmarz for $m\times n$ matrix A with $m\approx \infty$ and have $H_k=\frac{1}{k}A_k^T A_k$ which is an estimate of the Hessian obtained from ...
- 2,799
0
votes
0
answers
238
views
Right blocked linear equation solver on Dense Algebra and Sparse Algebra
I have implemented 1D mesh parallel QR decomposition and LU decomposition,I would like to ask if a linear equation Ax=b,b is a large matrix and I need to shard b or Shard A,b at the same time. Is ...
5
votes
1
answer
107
views
Updating QR decomposition for geometrically similar least squares problem
Let's say we have a weighted least squares problem under the same matrix $A$ such that,
$$\hat{x} := \arg \min_x ||A x - b||_{W_1}$$ where $|| \cdot ||_{W_1}$ is the Euclidean norm weighted by ...
- 151
1
vote
0
answers
81
views
preconditioning least square in python?
For a nonsymmetric matrix, we can solve { A^T @ A x = A^T b } by lsqr or cgls or something else. Usually it will be slow, so we need a preconditioner either ilu, multigrid or something else. Is there ...
- 11
2
votes
2
answers
491
views
Iteration counts of AMG solver changes in parallel
I am solving the linear elasticity equation within a FEM library with a complex 3D geometry. The resulting linear system is solved with CG, preconditioned by AMG (Algebraic Multigrid). The computed ...
- 435
1
vote
2
answers
255
views
Are there good block sparse matrix solver libraries?
There are some great libraries with linear solvers for sparse matrices - SuiteSparse is the obvious one. The methods work on sparse matrices with scalar entries.
However, often in optimization ...
- 111
1
vote
1
answer
338
views
Why does scipy Conjugate Gradient solver fail to converge for non-steady heat equation using Crank-Nicolson method
Could someone please explain why my implementation of the Crank-Nicolson method applied to the non-steady heat equation won't converge? There shouldn't be any nonlinear aspects to my implementation ...
- 13
2
votes
0
answers
114
views
Efficient heat diffusion implementation with varying coefficients
I have the following heat diffusion equation:
\begin{alignat}{3}
\partial_t u(t, \vec{x}) &= g(\vec{x})\Delta u(t,\vec{x}), &\quad& \vec{x} \in\Omega, \, t\in(0,\infty],\\
\partial_n u(t,\...
- 2,892
11
votes
1
answer
1k
views
Is using iterative methods to solve a linear system always superior to inversing the matrix?
I have a silly question. Is it always more computationally efficient to use iterative methods to solve for some matrix $A$, $Ax=b$, where $x$ and $b$ change but $A$ stays constant, compared to ...
- 131
5
votes
1
answer
107
views
Prediction of sphere (i.e. roast) core temperature heated in an oven
The real-life problem
Assume I put a spherical roast with initially constant temperature of start_temp=25 (°C) into an oven with ...
- 161
1
vote
1
answer
308
views
2D Heat equation solved with finite element method converges in skewed way
I tried to solve the 2D heat equation with the finite element method, using triangles as elements. Currently generated by a Delaunay triangulation. The base function I'm currently using is basically ...
- 63
2
votes
1
answer
186
views
Numerically stable way to implement Cramer's rule analog
Problem statement
Let $A$ be an $n\times n$ matrix and $b$ an $n$-dimensional vector. For $j\in \{1, \dots, n \}$, let $A_j$ be the matrix where we take $A$ and replace the $j^{\rm th}$ column with $b$...
- 31
0
votes
1
answer
65
views
Reverse engineering phase shift and numerical damping
I've been trying to validate the physics behind a particle system framework, but I'm having some difficulties.
A particle system is a set of lumped masses connected by spring-damper elements. Linear ...
2
votes
1
answer
181
views
Solution of linear system doesn't work, in parallel
I'm solving $Ax = b$ with PETSc, $A$ sparse and asymmetric.
I'm using BCGS or FGMRES or TFQMR as a solver, and ILU as a preconditioner.
When I use 1 core, everything works as expected. But with 8 ...
- 259
0
votes
1
answer
143
views
How to get a normalized gradient with FreeFem++?
I am trying to use FreeFem++ to solve the heat geodesics algorithm.
The algorithm is:
solve $\dot u = \Delta u$ at a specific time $t$.
compute $X = \frac{\nabla u_t}{|\nabla u_t|}$
solve $\Delta\phi ...
- 379
1
vote
0
answers
124
views
Schur complement formulation of linear system
Consider a system of the following form:
$$(A+K)x=b$$
where $A$ is symmetric, positive definite and block diagonal (in fact, a block diagonal matrix made of stiffness matrices arising from FEM ...
- 259
1
vote
1
answer
84
views
Powers of convergent DPR1 matrices in $O(d)$ time?
Suppose $u$,$v$ are vectors and $A$ is a convergent $d\times d$ diagonal + rank-1 matrix.
How do I estimate $u^T A^k v$ in $O(d)$ time?
Powers of convergent diagonal $D$ can be computed in $O(d)$ time ...
- 2,799
1
vote
0
answers
124
views
Which dense matrices are hard to invert?
Suppose I'm solving $Ax=b$ for dense $m\times d$ matrix $A$. For which $A$ is this hard to do?
More concretely, is there any work on estimating the error after $k$ steps of iterative solver, $k\le d$, ...
- 2,799
1
vote
0
answers
115
views
Accuracy of the Crank-Nicolson method for non-linear, inhomogeneous heat equation
I am currently coding a solution to the following PDE:
$\frac{\partial T }{\partial t} =\frac{\partial}{\partial \theta}(A(\theta ,\phi )\frac{\partial T }{\partial \theta}) +\frac{\partial }{\partial ...
- 11
2
votes
0
answers
113
views
Conceptual doubt regarding 2D conjugate heat transfer modelling (COMSOL and Mathemtica)
I have been dealing with some conceptual flaws in my understanding of modelling, which I will elaborate herein. I am modelling conjugate heat transfer of a reciprocating fluid, which flows with ...
- 41
1
vote
0
answers
52
views
FEM Basis for functions of two variables in $\mathbb{R}^2$ (Applied to Linear Full Stokes Equations)
I'm trying to do FEM for a very basic version of the linear full Stokes equations in two dimensions. Say we are working in the grid $[0,1]\times[0,1]$ in the $xy-$plane.
To solve an FEM problem for a ...
- 111
5
votes
1
answer
444
views
Block-Tridiagonal Matrices with tridiagonal blocks
The Setup
Using finite differences to discretize the 2d diffusion equation $$\partial_tu=\partial_x\left(A\partial_xu+B\partial_yu\right)+\partial_y\left(B\partial_xu+C\partial_yu\right)$$ we get a ...
- 153
0
votes
1
answer
101
views
How to combine multigrid preconditioner with jacobi preconditioner?
I have not found any relevant information in the literature on the following rather simple problem:
How to combine (geometric) multigrid preconditioned conjugate gradient (MGPCG) with an additional ...
- 101
2
votes
0
answers
36
views
How to exploit QR factorization implicitly
I meet a problem when I try to develop an iterative method for discrete inverse problem
$$Ax+e=b$$
where $A\in\mathbb{R}^{m\times n}$ and $e$ is a noise. I want to approximate the true solution $x_{...
- 21