All Questions
2,750
questions with no upvoted or accepted answers
14
votes
0
answers
535
views
Sequential approach to solving coupled PDEs
I'm dealing with a coupled system of three transient, non-linear convection-diffusion equations. Let's just say to simplify the problem that they take the following form:
$$
-\nabla\cdot(D_{1}(u_{2},...
- 947
14
votes
0
answers
455
views
Operator Splitting methods for DAEs
After doing some research, I've found that most of the literature on operator splitting methods (e.g. Strang Splitting, Fractional Step, etc.) are specifically designed for a standard problem type of ...
- 12.1k
13
votes
0
answers
734
views
Fast Eigenvalue and SVD Solver for Structured Matrices
I am looking for a fast Eigenvalue and SVD solver for small dense structured matrices (Hankel and Toeplitz). I have searched for efficient implementations in libraries like MKL but I am not able to ...
- 241
12
votes
0
answers
4k
views
Optimized open source BLAS / LAPACK package
I was wondering what is a more optimized open source BLAS/LAPACK package with respect to modern multi-core processors (Haswell and beyond). Is there any distribution that can attain performance close ...
- 121
12
votes
0
answers
139
views
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 ...
- 221
11
votes
0
answers
350
views
Numerical integration using interval arithmetic, nowadays
Is there now a package for rigorous numerical integration that uses interval arithmetic and has access to a well-developed library of special functions?
By "well-developed", I mean something that, at ...
- 279
10
votes
0
answers
180
views
What's the most computationally efficient implementation of Kalman Filter
I know there are many formulations of the Kalman Filter. A few I can name are:
Classical Covariance Form
Informational Filter Form
Square-Root Form or Factor Form
But somehow it's hard for me to ...
- 467
10
votes
0
answers
882
views
Implementing std::nextafter: Should denormals-are-zero mode affect it? If so, how?
This might be the wrong stackexchange site for this question. math.SE, cs.SE, programmers.SE, and of course stackoverflow are all possibilities. I'm hoping to reach an audience that might actually ...
- 310
10
votes
0
answers
240
views
Time advance in Adaptive Mesh Refinement method
I am working on solving complex system of 2D PDEs governing the behaviour of plasma in a gas lamp during discharge. Recent tests have shown that because of steep gradients in temperature field and ...
- 131
9
votes
0
answers
235
views
Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system
Problem
Solving a non-linear system of equations.
The number of variables is the same as the number of equations.
When I fix a set of variables (say $\vec{y}$) and keep another set free (say $\vec{...
- 163
9
votes
0
answers
144
views
Review of modern homotopy methods and practical techniques
I'm hoping someone can recommend recent literature concerning homotopy methods for solving systems of nonlinear equations. Already by the time of Layne Watson's 1986 paper there were a lot of methods,...
- 297
9
votes
0
answers
539
views
Updating matrix diagonal with Woodbury matrix identity and maintaining numerical accuracy
I have a dense matrix A and its corresponding inverse $A^{-1}$. The Woodbury matrix identity states:
$$ (A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1} $$
I wish to perform small ...
- 1,320
9
votes
0
answers
439
views
Simple turbulence model appropriate for buoyancy-driven cavity like problem
Which turbulence model is suitable for resolving incompressible buoyancy-driven flow of a fluid within an cylindrical ampoule?
I prefer turbulence model which is sufficiently simple so that fully ...
- 927
9
votes
0
answers
451
views
What's a good numerical/optimization software package for solving the 2-D optimal stopping problem?
I am looking for a numerical software package to help me solve the 2-dimensional "free boundary" PDEs that arise in optimal stopping problems. In one dimension a standard optimal stopping problem in ...
9
votes
0
answers
167
views
Fast algorithms to solve Markov Decision Processes
In my master thesis I used an Algorithm called Approximative Dynamic Programming [1] to solve equations of the form
$$
\max_{\pi}\mathbb{E}^{\pi}\left\{\sum_{t=0}^{T}\gamma^tC_t^{\pi}(S_t,A_t^{\pi}(...
- 191
8
votes
0
answers
303
views
Stable alternatives to "condition number"?
A number of numerical problems are easy to solve when condition number $\kappa$ of the problem is low. For instance, conjugate gradient descent complexity scales as $O(\sqrt{\kappa})$.
However
"...
- 2,735
8
votes
0
answers
106
views
How do we approximate the numerical error a numerical scheme (e.g Runge Kutta, Euler etc) makes without having access to an analytical solution?
So I recently encountered this question in my head while taking my Scientific Computing class, where the lecturer talked about computing numerical error of a scheme.
My guess would be that we take a ...
- 101
8
votes
0
answers
176
views
Is there a graphical interpretation or explanation of automatic differentiation compared to numerical differentiation
I have been looking at automatic differentiation for solving differential equations lately. I understand the basic ideas of using Dual numbers and such for finding derivatives, etc. However, I feel ...
- 297
8
votes
0
answers
253
views
Why not use the preconditioned residual as termination criterion for preconditioned CG?
I have a Poisson equation with wildly varying material parameters (1 .. 1000), wildly varying element sizes (5 nm .. 100 um) and some quite anisotropic (tetrahedral) elements (100 nm x 100 um). I use (...
- 2,141
8
votes
0
answers
114
views
How to construct an effective preconditioner for this particular problem
A quick introduction to my problem
I am currently developing a method for simulation of water waves in three dimensions based on potential flow theory. The computational bottleneck of the method is ...
- 115
8
votes
0
answers
5k
views
Good C++ optimization library for BFGS
To implement maximum likelihood estimators, I am looking for a good C++ optimization library that plays nicely with Eigen's matrix objects. Eigen has some capability of interfacing of its own but if ...
- 183
8
votes
0
answers
138
views
Finding the smallest root of a function on $[0, \infty)$
I would like to find the smallest real root of a 1-D real-valued function $f(x)$ on the domain $x\in [0,\infty)$. In this problem, I can make the following guarantees on $f$:
$f$ does have a root at ...
- 735
8
votes
0
answers
653
views
Speed and accuracy of Strassen vs Winograd matrix multiplication algorithms
I am doing work which requires as fast matrix multiplication as possible and just want to double-check with this community that the Winograd variant of Strassen's MM algorithm is the fastest practical ...
- 91
8
votes
0
answers
478
views
Eigenvalue with largest imaginary part
Iterative eigensolvers such as ARPACK, give the option to find a subset of the eigenvalues which have the largest imaginary part. My question is how do these algorithms work.
As I understand it, ...
- 243
8
votes
0
answers
293
views
What can be done with Finite Element Method and not with the Finite Volume Method, and vice versa?
What are some applications where you would absolutely go for either FEM, but not FVM, or vice versa? What are some applications where both methods are equally suited?
I worked with the FEM so far and ...
- 1,463
8
votes
0
answers
907
views
Numerical implementation of the Dirichlet-to-Neumann map
I am solving the Dirichlet problem
$$
\begin{cases}
\Delta u = 0, \\
u|_{\partial D} = f,
\end{cases}
$$
in a $2d$ domain $D$ using the finite element method. What I want to get is the ...
- 445
8
votes
0
answers
180
views
Tucker factorisation to compare multiple PCA decompositions?
This is an entry-level question for multiway matrix decompositions. I have a set/population $k$ of entities (here biological cells) for each of which I also have a number ($l$) of flavours of length $...
- 203
8
votes
0
answers
591
views
DIIS method to accelerate SCF convergence for stretched geometries
I am implementing from scratch an Hartree-Fock calculation in the STO-3G basis set to perform Born-Oppenheimer molecular dynamics. I have a Restricted Hartree-Fock procedure that can reproduce very ...
user18279
8
votes
0
answers
174
views
Accelerated convergence for Sparse NMF
In the Non-Negative Matrix factorization (NMF), you basically compute an approximation of a given matrix $V \in \mathbb{R}_{+}^{n \times m}$ into matrices $W$ and $H$ such that $W \in \mathbb{R}_{+}^{...
- 253
8
votes
0
answers
822
views
What is the source of the error in the Sherman-Morrison formula application?
The Sherman-Morrison formula
$$ (A+uv^T)^{-1} = A^{-1} - \frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u} $$
results in small errors in relation to the standard matrix inverse operation after each application, ...
- 180
8
votes
0
answers
150
views
Potential Reduction and Primal Path following methods
In both the potential reduction and primal path following interior point methods for linear programming, a barrier function is constructed which contains the terms $-\sum \log x_j$ where $x_j$ are the ...
- 341
8
votes
1
answer
143
views
Choosing how many iterations to use in VEGAS
I'm using VEGAS integration, specifically the GSL implementation, for some QCD calculations, and I've been investigating the behavior of the algorithm for various numbers of iterations in an attempt ...
- 3,403
7
votes
0
answers
304
views
Matrix-free FEM references
I've seen that many people are using matrix-free fem codes in my community (mechanical engineering). I have to admit that I googled a bit and I didn't manage to find a good reference for the subject. ...
- 405
7
votes
0
answers
89
views
Choice between using Moore-Penrose inverse and G2 inverse
Moore-Penrose inverse for an arbitrary matrix $X\in \mathbb{R}^{n \times p}$ is defined by a matrix $X^\dagger$ satisfying all of the Moore-Penrose conditions, namely
\begin{align}
(1) \;\;\;& XX^\...
- 173
7
votes
0
answers
144
views
Can we sparse solve a few eigenvalues specified by index range?
I need to solve a few eigenvalues of a large sparse matrix specified by their index range. These indices are according to the whole eigenspectrum sorted in algebraic (not absolute value) ascending ...
- 303
7
votes
0
answers
301
views
Finding points inside cells of power (generalized Voronoi) diagram
Suppose we have a set of points $p_1,\ldots,p_n\in\mathbb R^d$ as well as a set of weights $w_1,\ldots,w_n\in\mathbb R$. Recall that the power cell associated to the pair $(p_k,w_k)$ is given by:
$$\...
- 1,423
7
votes
0
answers
205
views
Is a complete bacteria simulation with an exascale supercomputer possible?
Will it be possible to simulate a complete (at least simple) bacteria atom by atom on an exascale supercomputer? or is it possible already today with the largest systems?
Here, I've read that ...
- 193
7
votes
0
answers
489
views
fastest way to compute many small dot products
I have two n-by-3 blocks contiguous in memory ("n vectors of length 3") and I'd like to compute the dot product between each of the rows as fast as possible. In numpy, using ...
- 3,136
7
votes
0
answers
612
views
How to check if my stiffness matrix is correct
I built the stiffness matrix for the Poisson equation on a 2-dimensional domain with the shape of "almost" an octagon, using pyramid basis functions. I used almost to intend the fact that I have an "...
- 71
7
votes
0
answers
184
views
"Geometry of ill-conditioning" for least-squares problems
It is an idea that dates back to Demmel, 1987 that the condition number of a problem is often related to the distance to the closest ill-posed problems. In Section 3 of the above paper, the author ...
- 12k
7
votes
0
answers
358
views
Implementation of Lanczos method that returns tridiagonal matrix
The Lanczos method can be used to obtain extremal eigenpairs of sparse symmetric or hermitian matrices. I know there are several implementations of the Lanczos method (as well as Arnoldi, Davidson, ...
- 171
7
votes
0
answers
632
views
Is there any catch on using `zgemm3m` vs regular `zgemm`?
I've just (to my embarrassment) encountered a BLAS-like extension of a matrix-matrix product subroutine gemm in Intel MKL: gemm3m...
- 8,712
7
votes
0
answers
98
views
Quadrature methods for peaky integrands
Consider
$$
I = \int_{-L}^L f(x)dx,
$$
where $f(x)$ is real-valued and analytic on $[-L,L]$, but it has a pole in the complex plane whose real part lies in $[-L,L]$. Call it $z_0$, and assume it is a ...
- 735
7
votes
0
answers
228
views
Solving a coupled eigen value problem
I have the following problem:
$$\begin{bmatrix}A &B \\C& D\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}\lambda I_m & 0 \\ 0& \mu I_n\end{bmatrix}\begin{bmatrix}x \\y\...
- 171
7
votes
0
answers
165
views
Solve ill-posed linear system without transposing matrices?
I am attempting to use an iterative solver to solve $p$ in
$$
Jp = -r
$$
where $J$ is an $m\times m$ matrix ($m$ is in the order of $10^5$ and never explicitly stored). $J$ is a dense matrix ...
- 173
7
votes
0
answers
932
views
Sparse matrix format and sparse-matrix sparse-matrix multiplication
I'm having some performance problems with my code dealing with the multiplication of big sparse matrices (stiffness and aerodynamic influence coefficient matrices).
Mainly I have to multiply such ...
- 71
7
votes
0
answers
344
views
Compute sparsity pattern of $A^2$
Suppose we have a sparse matrix $A$. Is there any way to compute just the sparsity pattern of $A^2 = A \cdot A$ (I do not actually need to know what exactly the nonzero value are) faster than to ...
- 131
6
votes
1
answer
152
views
Slope limiting with implicit time integration
I am solving the advection problem with high order numerical methods, using the method of lines. The boundary conditions and initial condition are selected in a way where I know that the exact ...
- 131
6
votes
0
answers
147
views
Runge-Kutta methods, higher derivative methods, and collocation methods
Consider an ODE system
$$\dot x = f(t, x), \quad x(0) = \xi.$$
A collocation method to solve this ODE (1) assumes that $x$ can be approximated as a polynomial
$x(t) \approx \sum_kx_kp_k(t)$ and (2) ...
- 10.4k
6
votes
0
answers
122
views
References on the theory of Petrov-Galerkin methods for more "basic" problems
In my reading on various aspects of FEM, Petrov-Galerkin methods often arise in the study of solutions of convection-dominated systems, such as Hughes' work on Navier-Stokes, or systems where optimal ...
- 2,936