Questions tagged [symmetry]
For questions about exploiting symmetries to solve computational problems. This could include seeking an algorithm for symmetric matrices or finding machine learning descriptors that are invariant under certain symmetry operations.
26 questions
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Axisymmetric jet boundary conditions
I am working on analyzing a flow downstream of an axisymmetric jet in a 2D slice i.e, in a rectangle (size L,R) of (z,r) coordinates so that velocity components are (w,v). I am confused about the ...
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Is there an efficient way to compute the inverse of several symmetric matrices that share the same structure?
I have many symmetric matrices to be inverted. They all share the form
$A^\top D_{x_1} A$, $A^\top D_{x_2} A$, ... with $A\in\mathbb{R}^{n\times p}$, $n>p$ and $D_x$ positive diagonal but with ...
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Symmetrization of Laplacian Matrix Operator (finite volumes)
The aim is to construct symmetric Laplacian matrix $L$ for 2D square domain.
I have the following discretization of second derivatives on non-uniform grid (will skip the steps of derivation, any ...
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Compatibility condition for Poisson equation in cylindrical symmetry
I'm trying to implement multigrid approach for a Poisson equation $\frac{1}{r}\frac{\partial}{\partial r}\left( r \frac{\partial H}{\partial r} \right) = f$ with all Neumann boundary conditions. ...
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Under what specific circumstances can a matrix lose symmetry?
This is a general question: In this blog post, the author points out that under a unitarily invariant norm, the closest symmetric matrix to the square matrix $M$ is $\frac 1 2 (M + M^T)$. The general ...
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Stable iterative solver for complex symmetric linear systems
I am interested in the iterative solution (preferably Krylov-type solvers) of a problem $\boldsymbol{A}x=b$, with $x,b\in\mathbb{C}^{n\times1}$ and $\boldsymbol{A}\in\mathbb{C}^{n\times n}$. $\...
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Choice of iterative solver for a sparse asymmetric matrix with symmetric structure
I have a sparse $n\times n$ matrix $A$ with a pretty interesting structure. It has a block structure with a symmetric structure but asymmetric blocks. Expressed mathematically the block $A_{jk} = A_{...
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Methods to improve the efficiency and the memory requirement of LU factorization for complex symmetric system matrix
I want to solve a linear set of equations (Ax=b) using LU decomposition. My "A" matrix is a complex matrix which is ...
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How property-invariance is imposed to neural nets?
I was wondering how specific symmetries or constraints such as property-invariance transformation are imposed on any (deep) neural net when they are trained.
I'll appreciate it if anyone can aware me ...
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Does a symmetric positive definite matrix also have $\mathbf{A} = \mathbf{L}^T\mathbf{L}$ (where $\mathbf{L}$ is a lower triangular matrix)?
As we know, for a symmetric positive definite (SPD) matrix $\mathbf{A}$, there is a theorem about the Cholesky factorization $\mathbf{A}= \mathbf{L}\mathbf{L}^T$, where $\mathbf{L}$ is a lower ...
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How can Navier--Stokes equations have asymmetric solutions such as Karman vortex streets
The Navier--Stokes equations are axially symmetric, so with symmetric boundary conditions, how can features such as Karman vortex streets develop?
I understand that in reality symmetry does never ...
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Conservation violation in axisymmetric Diffusion Equation
1d diffusion equation
Integrating the diffusion equation,
$$
\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2},
$$
with a constant diffusion coefficient D using forward Euler for ...
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CG question: is symmetry always necessary?
Consider the 1D Poisson equation
$$\nabla^2 u = f.$$
Using finite difference method on cell corner data and a uniform grid with ghost points, I think we can write the system of equations with Neumann ...
- 619
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Mapping from n x n complex symmetric tridiagonal to 2n x 2n real symmetric tridiagonal
In my program I have a complex symmetric tridiagonal matrix. In order to perform some algorithmic optimizations I am searching for a (ideally linear) mapping from $n\times n$ complex symmetric ...
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Why does conjugate gradient work with this nonsymmetric preconditioner?
In this previous thread the following multiplicative way to combine symmetric preconditioners $P_1$ and $P_2$ for the symmetric system $Ax=b$ was suggested:
\begin{align}
P_\text{combo}^{-1} :=& ...
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Which numerical methods preserve time reversal symmetry?
If I have a physical system which contains a time reversal symmetry (for example a Hamiltonian $H(x,p)=p^2/2m + V(x)$ with $V(x)$ real) and I want to solve the differential equations which describe ...
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Breaking symmetries in a (binary) integer program
I want to solve a integer programming problem with binary variables $x_1,\ldots,x_n.$
I have a permutation group $G \leq S_n$ such that for every $f \in G$ the vector $\overline{x}_1,\ldots,\...
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Are there any standardized file formats for point group character tables?
Character tables are an important tool for symmetry analysis in many computational chemistry software packages. Are there any standardized file formats for point group character tables?
This may seem ...
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Is my matrix symmetric?
I obtained a mass matrix through Finite Elements discretization. Now, I want to check if it is symmetric. To do that I subtract to my matrix $M$ its transposed $M^T$. The result is another matrix of ...
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Get symmetric Finite Difference matrix in non Laplacian settings
I would like to solve a system of differential equations $u+\nabla(\nabla\cdot u)=f$ or in more detail
$a+\partial_t^2a+\partial_t\partial_xb+\partial_t\partial_yc=f$
$b+\partial_x^2b+\partial_x\...
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(Fortran) Integrating/summing over complicated 3D domain
I have some function $F(k_x,k_y,k_z)$ that I wish to numerically integrate over a polygon domain - physically, I am integrating over the first Brillouin Zone (BZ) of the FCC lattice (a truncated ...
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Are the eigenvalues of the product matrix of two real symmetric square matrices also real values?
Suppose $A,B \in \mathbb{R}^{n\times n}; A=A^T, B=B^T$, let $C = AB, D =BA$,
If we have all the real eigenvalues of $A$ and $B$, e.g. the eigenvalue decomposition of them:
$A=P\Lambda_1 P^T$,
$B=Q\...
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What is the most efficient way to obtain the max eigenvalue of a specific symmetric matrix via Eigen C++
Suppose I have a symmetric matrix $A_{1000\times 1000}$, which can be represented by:
$A = J G J^T$
where $J$ in 1000x3 is full column rank dense matrix; $G$ in 3x3 is a nonsingular dense matrix.
...
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Is there congruent transform implementation for dense symmetric matrix in Eigen(C++)?
I need to determine whether a real dense symmetric matrix is positive definite or not.
One possible way is to obtain all the eigen values and check the sign of the minimum eigen value but requires ...
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Symmetric Matrix Computation (Eigen & C++)
I want to compute a symmetric matrix A from a vector b by: A = b * b'.
Does the Eigen library automatically take into account that it does not need to do all calculations to get A (because of the ...
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Are there any numerical advantages in solving symmetric matrix compared to matrices without symmetry?
I'm applying finite-difference method to a system of 3 coupled equations. Two of the equations are not coupled, however the third equation couples to both the other two. I noticed that by changing the ...
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