Problems in which an operator or function can be represented with asymptotically less data than the naive representation. Not limited to sparse matrices.

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55 views

Sparse linear system of certain type

Let $n_1,n_2 \in \mathbb{N}$ and $n=n_1n_2$ and $b\in \mathbb{R}^n$. I have a SPD-matrix $A=(a_{i,j})\in \mathbb{R}^{n \times n}$ with $a_{i,j}=0$ if $|i-j| \notin \{0,1,n_1\}$. Can we solve the ...
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1answer
61 views

Iterative algorithms for sparsity using a function for operator A in Ax=b

I am going to solve an linear iterative inverse problem. I have two functions in matlab which one of them play the forward and ...
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2answers
176 views

calculating eigenvector components of a given vector

I have some vector $V$ which can be decomposed into the eigenspace of the hermitian sparse operator $M$: $V = \sum_i v_i \hat{m}_i$ Is there a way to find the $\hat{m}_i$ (the eigenvector itself) ...
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1answer
66 views

What is the cost of factorization for one-dimensional sparse problems?

In Golub and Van Loan's book, Matrix Computations, page 606, it is stated that: With standard discretizations, 2-dimensional problems can be solved with $O(n^{3/2})$ work and $O(n \log{n})$ ...
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0answers
87 views

Phase retrieval via SDP (semidefinite program) of 2D test image (Matrix completion)

In SDP based phase retrieval we have intensity measurement (abs(FFT2(x))^2) of the form A(xo) = |ak,x|^2 =b^2, phase retrieval is then find x that obeys A(xo)=b The quadratic measurements can be ...
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1answer
46 views

Is there an upper bound for fill-ins for indefinite triangular factorization?

For $A=LU$, or $A=LDL^T$ factorization, bandwidth is preserved when there is no pivoting. This is true even for indefinite A, see question. However, when there is pivoting band structure is destroyed, ...
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2answers
58 views

Time complexity for sparse direct solver for SPD system with respect to number of equations, bandwidth, number of nonzeros?

I am looking for information on the time complexity for solving sparse system Ax=b with direct solver. This system results from a finite-element discretization of an elliptic problem. The matrix A ...
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2answers
66 views

Is the bandwidth of indefinite A equal to its factor L in LDL^T?

In George, Liu, and Ng's book Computer Solutions of Sparse Linear Systems, it has been shown that bandwidth of $A$ is equal to bandwidth of its factors in $LL^T$.(section 4.3) However, I guess this is ...
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0answers
72 views

how to pick a submatrix of a sparse matrix quickly(matlab)

I have a large sparse matrix A and i want to pick the submatrix of it to do block jacobi iterations. For the blocks, i have get a matrix Q that contains the index of nonzero entries in its Jth column ...
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0answers
62 views

Solvers for stiff initial value ODEs with sparse Jacobian

What ODE solvers are optimized for solving stiff systems with sparse Jacobian? Such systems appear, for instance, when a parabolic PDE is discretized in space using typical finite difference or finite ...
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2answers
132 views

Condition number from incomplete Cholesky factorization

I'm having difficulties patching together from what I read about obtaining the condition number of a real, symmetric, positive definite sparse matrix. In my code, I found that there is incomplete ...
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3answers
135 views

Parallel solver for sparse matrices on unstructured grids

I am trying to solve Euler equations on unstructured grids. Consequently, the problem reduces to solving Ax=b where A is a ...
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0answers
33 views

Does cyclic reduction require a square mesh for 2D FEM/FDM?

In a 2-dimensional finite difference method or finite element method (triangular elements, each node is connected to 6 neighbors), each node has a matrix entry to its horizontal and vertical ...
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1answer
108 views

Fastest linear solver for sparse positive semidefinite, striclty diagonally dominant matrix

What is the state of the art for fastest linear solver for sparse, positive semi definite and strictly diagonally dominant matrix with N varies from ~700 to ~3000, and about a 1/16 of the matrix is ...
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2answers
211 views

Solving Lx = b for big sparse Laplacian matrices

What algorithm is more practically suited in terms of performance for solving the $\mathbf{Lx=b}$ equation, where $\mathbf{L}$ is a generic Laplacian matrix (associated to a strongly connected graph, ...
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0answers
19 views

Finding correlations between sparse data with different temporal bases

The problem was one that came up during a standards committee meeting. If have one channel of data sampled every 100ms to record fault codes on a system (say an gasoline engine). It records value ...
5
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1answer
180 views

Sparse matrix ordering in Python

I would like to implement custom, domain-specific algorithms for sparse matrix orderings. I am looking for Python packages for ordering sparse matrices. It would be nice to have: The underlying ...
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1answer
120 views

solving underdetermined system of equations with a sparse matrix as input

I am using Matlab to solve Ax=b and my A is very large, sparse, binary and also rectangular. I saw the Matlab backlash \ operator help and it states that if A is rectangular then it will use the QR ...
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5answers
542 views

20% performance penalty for a nice software design

I'm writing a small library for sparse matrix computations as a way to teach myself to make the best use of object-oriented programming. I've worked really hard on having a nice object model, where ...
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3answers
284 views

Eigenvalues of a sparse banded nonsymmetric matrix from an elliptic operator

I have a sparse matrix coming from the discretization of a 3D elliptic PDE. The matrix is banded with seven non-zeros diagonals. The sparsity pattern of the matrix looks like this (the actual matrix ...
4
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1answer
138 views

Supernodal vs. Multifrontal Matrix Decompositions

Reading through Tim Davis' book Direct Methods For Sparse Linear Systems, he says that matlab can use a supernodal Cholesky decomposition but never uses a multifrontal Cholesky decomposition. At least ...
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1answer
110 views

Solver for large non-linear system of equations

I am curently using R package nleqslv for solving a non-linear system of equations with 300 variables. I need to scale this to the system with ~50k variables and naturally this does not scale very ...
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1answer
70 views

GMRES: Making the matrix square without solving for boundaries

How do we define the matrix for GMRES, if we do not want to solve the boundary elements but only the interior ones. I am using pentagonal elements so in a row there are 6 elements (cell itself + 5 ...
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3answers
245 views

Efficient solver for a symmetric tridiagonal system where the upper/lower diagonals are offset

I'm looking for an efficient way to solve a symmetric tridiagonal system $Mx = d$, where the upper and lower diagonals of $M$ are offset from the main diagonal by $k$ rows/columns: $$ \begin{bmatrix} ...
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3answers
201 views

Dense distributed matrix

A dense matrix is distributed for parallel computation column-wise, then multiplied from left & right by sparse matrices. What would be appropriate c++ libraries for these tasks?
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0answers
254 views

How to solve singular non symmetric poisson equation with Neumann boundary condtions?

I am trying to solve 2D Poisson equations with Neumann boundary conditions. When the mesh is uniform, Poisson equation is singular and symmetric, so the method listed in Null Space Projection for ...
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4answers
448 views

Good Finite Element Library for a small project

I'm currently working on this project and I have a basic structural analyzer that uses the finite element method. Essentially, I turn each block into a set of trusses, construct a stiffness matrix ...
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1answer
270 views

computing the inverse of a large block diagonal sparse matrix in r

I would like to compute the inverse of some large block diagonal sparse matrix. The number of rows and columns is somewhat over 50,000. The blocks are 12 by 12 and are sparse (27 non zero elements). ...
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137 views

Assembling sparse matrix in PETSC for Poisson equation

I am a novice at PETSC, and I have been trying to write an FVM code for steady heat conduction in 2D using PETSC (square, regular grid, Dirichlet boundaries) Since the large matrix , say A, will be ...
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2answers
185 views

What sparse linear programming solver it is better to use?

I have the following LP problem: $$ \min \limits_{\varepsilon, x_{1}, \ldots, x_{n}}f(\varepsilon, x_{1}, \ldots, x_{n}) = \varepsilon \;\;\;\;\; s.t. \;\;C x \geq 0, \;\; x_{i}^{0} - \varepsilon ...
1
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1answer
228 views

Sparse matrix - matrix multiplication

How can a sparse matrix - matrix product be calculated? I know the 'classic' / mathematical way of doing it, but it seems pretty inefficient. I thought about storing the first matrix in CSR form and ...
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0answers
134 views

Givens method for sparse matrix

I have a (very) large sparse matrix in CSC form and I'm supposed to factorize it. I've read that between Givens and Householder transformations, Givens is better for a sparse matrix. The problem is ...
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44 views

Sparse matrix factorization of a rank deficient matrix by decomposition into linearly independent components

I've got a little conjecture I need to prove for a theoretical result related to causal Bayes net search with latent variables under sparsity constraints. If you're interested in the application ...
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1answer
40 views

an over view of sparsity promoting inversion techniques [closed]

I have a function called f(x) which is convex and I can have access to its first order derivative , my objective function is $$\ J(\bf{x}) = f(\bf{x}) + \lambda |\bf{x}|_0 $$ $$\ \bigtriangledown ...
7
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2answers
280 views

Which novel data structures are used in adaptive FEM?

A lot of adaptive FEM libraries use more advanced mesh data structures to handle adding/removing nodes, edges, triangles, tetrahedra, etc. For example, the p4est library uses octree data structures ...
2
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2answers
194 views

Poisson-Nernst-Planck equations with ill-conditioned sparse matrix

I am trying to solve Poisson-Nernst-Planck system of equations for ions diffusion problem using finite volume method. Nernst-Planck equation for mass transport and Poisson equation for electrostatic ...
3
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2answers
230 views

Solving Newton-Raphson step with ill-conditioned sparse matrix

I am trying to build a complex simulator for the transport of mass & heat in porous media. I am currently following coarsely the algorithm laid out in an older software and have got the simulator ...
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0answers
41 views

Markowitz Pivoting to reduce size of a dense integer system

I am dealing with a large sparse integer matrix that I need to find the nullspace of. I've seen Markowitz Pivoting come up in several places discussing similar problems such as here: ...
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0answers
40 views

MLLL algorithm for sparse, integer bases to find a nullspace

I am trying to find a suitable algorithm that can find a basis for the nullspace of a sparse, integer matrix. Reading A Course in Computational Algebraic Number Theory by Cohen, an algorithm based ...
3
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2answers
188 views

Consumer GPUs for sparse matrices (e.g. FEA)

Are consumer grade GPUs (like the NVidia Geforce series) used for solving sparse linear equations systems in a professional setting? For example, by engineers performing finite element analysis? ...
4
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1answer
288 views

Nullspace algorithm for a sparse matrix

I am dealing with large, sparse, rational matrices that I need to determine the nullspace of. Currently, I have one that is about 12000x12000 (but not square), where one in every 2000ish elements is ...
5
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2answers
147 views

Inverting a pressure matrix for fluid simulation

I am implementing a fluid simulator as my numerical methods course project and I have to compute pressure at each simulation step. Basically, that means solving an equation $Ap = b$, where $A$ is a ...
2
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3answers
175 views

Sparse, underdetermined system of linear equations

I'm looking for an algorithm to solve the underdetermined system of linear equations $$\mathbf{A}\,\mathbf{x} = \mathbf{b}$$ with $\mathbf{A} \in \mathbb{R}^{n\times n}$, $\mathbf{b} \in ...
2
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2answers
214 views

Cropping in Sparse Matrix

Let $A$ be an $M \times N$ sparse matrix stored in compressed column format, in a C-like programming environment. I am interested in the best solution to get a sub-matrix of $A$. In MATLAB notation, ...
2
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1answer
88 views

Sparse matrices origins

I am using the sparse matrices provided by the University of Florida Sparse Matrix Collection and most matrices are accompanied with little description of the problem or discipline from which the ...
3
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1answer
171 views

indirect method for least-squares with inequality constraints

I aim to find $x \in \mathbb{R}^n$ that $\min_x |D \cdot F \cdot x|^2$ subject to $x_i = X_i$ and $x_j \geq X_j$ , $i \in I, j \in J$ and I and J partition ${1\cdots N}$ into two sets. it is ...
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1answer
208 views

Generalized eigenvalue problem using ARPACK

Is it possible to solve the eigenvalue problem: $$Ax = \lambda Mx$$ using ARPACK when $A$ and $M$ are both non-symmetric complex matrices? According to this documentation, the function ...
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1answer
289 views

Sparse matrices that represent common stencil operations

I am not sure if this is the correct place to ask this question! Is there a data set such as the University of Florida Sparse Matrix Collection which is produced from stencil operations? Or is ...
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1answer
779 views

How to use Lanczos method to compute eigenvalues and eigenvectors

I have a sparse and symmetric matrix A(n x n). The method Lanczos tranforms matrix A into tridiagonal and symmetric matrix T and the Lanczos vectors in matrix V. From there how do I compute k ...
2
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1answer
97 views

Numerically efficient way to compute sparse-matrix arithmetic on GPU?

Can anyone tell me some very good/efficient numerial algorthims for GPU/CUDA to compute multiplication/ between sparse matrices (its good if you can recommend me some research papers)? I googled ...