All Questions
2,664
questions with no upvoted or accepted answers
5
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answers
122
views
Optimization for sampling multiple points of maximized minimum distance
I'm trying to find a way to sample new points that have maximum minimum-distance (maximin distance).
The current situation is where there are ns number of pre-existing sample points. I want N number ...
5
votes
0
answers
77
views
Why does the naive barycentric hodgestar fail?
The discrete exterior calculus is defined first using circumcentric dual cells, because the primal and dual edges are orthogonal and thus the dual cells are convex. This leads to a diagonal hodge star ...
5
votes
0
answers
107
views
Efficient way to find eigenvalues of complex symmetric matrix with real off-diagonal elements
My goal is to find all eigenvalues (and eigenvectors) in a given range of magnitudes of a complex symmetric matrix with real off-diagonal elements (only diagonal elements are complex). Currently I'm ...
5
votes
0
answers
377
views
Polynomial rooting - fast root finding
I need to solve a rooting problem of a polynomial (the order of which is 2(N-1), where N=48).
So far I'm using Python Numpy algorithm (that relies on computing the eigenvalues of the companion matrix)...
5
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0
answers
144
views
Explanation of subspace strategy regarding CG described in Golub's book
I was wondering about the last paragraph in Matrix Computations (4th edition) by Golub, Chapter 11 (11.3.3), specifically his explanation of subspace strategy for Conjugate Gradient.
Note that in ...
5
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answers
103
views
Why the two Gram-Schmidt algorithms produce different results for qr factorization?
For the qr factorization using classic Gram-Schmidt algorithm, I found the 2 different implementations below. The first one uses the for loop to compute the upper ...
5
votes
0
answers
97
views
Trying to understand splitting-based iterative method for 2D Laplace problem
I am trying to understand the theory behind a splitting based iterative method which uses the incomplete Cholesky factorization. Before giving the specific details, let me first give the problem ...
5
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answers
133
views
Levinson Recursion for Non Square Toeplitz Matrices
Given a rectangular Toeplitz Matrix $ H $, how could one solve:
$$ y = H x $$
For instance, $ H $ can be Linear Convolution Matrix of the filter $ h $:
$$ H = \begin{bmatrix}
{h}_{1} & 0 & ...
5
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answers
89
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Nonlinear least squares optimized Jacobian calculation
I have a nonlinear least squares problem, in which I am trying to minimize residuals which can be divided into four classes:
$$
\min_x ||\epsilon(x)||^2 + ||\xi(x)||^2 + ||\delta(x)||^2 + ||s(x)||^2
$$...
5
votes
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answers
101
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Evolutionary dynamics in vascularised tumors, PDE-ODE coupled system
I have to solve the following PDE-ODE system
$$ \displaystyle{\partial_{t} n = \bigl[a(s) - b(s)(y - h(s))^{2} - d\int_{\mathbb{R}} n \, dy \; + \; \beta \, \partial^2_{yy} n \;}
\\\\
\...
5
votes
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answers
1k
views
Symmetric sparse direct solvers in scipy
scipy.linalg.solve, in its newer versions, has a parameter assume_a that can be used to specify that the matrix $A$ is symmetric ...
5
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answers
116
views
Probability approximation: monte carlo VS sde
I have a probability measure $\mu$ (say, in $\mathbb{R}^{d}$, with density) and I want to approximate it numerically. Today I noticed that my measure is ergotic for a certain Stochastic Differential ...
5
votes
0
answers
80
views
Is there an open-source material database management GUI?
Does somebody know an open-source GUI for the management of a small material database?
I have a spreadsheet with some materials in it. Each materials has some temperature-dependent properties like ...
5
votes
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answers
155
views
Minimum of quadratic assignment (QAP) with convex objective
Suppose $A,B\succeq0$ and $C\in\mathbb R^{n\times n}$. I am hoping to solve an instance of the following optimization problem:
$$
\min_{\textrm{permutation matrices }P}
\mathrm{tr}(BP^\top AP+C^\top ...
5
votes
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answers
64
views
Evaluate Nth root of a rational to a correctly rounded float
Excuse my lack of vocabulary for I have no formal training in this field, which is also why I ask this question - it may be trivial or it may be impossible.
I want to evaluate an expression in the ...
5
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answers
92
views
Numerical methods for the continuity equation with Sobolev vector field
Consider the continuity equation
$$
\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,
$$
with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$.
...
5
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answers
178
views
Best way to numerically compute elliptic integrals of the third kind with complex arguments?
I need to compute elliptic integrals of the third kind with complex arguments, preferably in C++. Is there code out there to do this? I have discovered the Arb library, but that does much more than I ...
5
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answers
88
views
Solution of constrained system of ODEs
Can someone point me in a direction to solve this kind of integral constrained system of ODEs.
\begin{align}
&\int_0^{1/2}\dot{y}^2(t)=p\\
&2\lambda_1\ddot{y}(t)+\pi cos(\pi y(t))=0\\
&y(...
5
votes
0
answers
87
views
Is there a numerically stable way to take $\epsilon \rightarrow 0$ in integrals of the form $\int \frac{f(x)dx}{x+i\epsilon}$?
The Sokhotski-Plemelj theorem states,
$$\lim_{\epsilon\rightarrow 0^+}\int_a^b\frac{f(x)dx}{x+i\epsilon} = \mathcal P \int_a^b \frac{f(x)dx}{x} - i\pi f(0). $$
Is there a numerically stable way to ...
5
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0
answers
871
views
Any way to avoid catastrophic cancellation when computing the discriminant of a quadratic function?
Homework disclaimer...
The task:
We are using the following algorithm to solve the quadratic equation $x^2+2px+q=0$:
$x_1=|p|+\sqrt{p^2-q}\mathtt{;}$
$\mathtt{if}\,p>0\,\mathtt{then}\,x_1=-x_1\...
5
votes
0
answers
88
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Comparing sum of floating points
I am currently working on a numerical algorithm involving a lot of floating point arithmetic, involving some badly conditioned problem sets.
I am using the relation $|x - y| / (\max(|x|, |y|, 1)) \...
5
votes
0
answers
93
views
Improving convergence of Jacobi iteration to Schur form
I'm using SIMD processor arrays to compute the eigen-decomposition for large numbers of small (up to $32\times 32$) matrices. For assorted technical reasons, Jacobi iteration maps well to the SIMD ...
5
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answers
43
views
Stochastic conjugate directions to improve convergence in narrow valleys
My question concerns a specific statement in this paper:
N. N. Schraudolph and T. Graepel, "Conjugate Directions for Stochastic Gradient Descent," in Int. Conf. Artificial Neural Networks, Berlin, ...
5
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0
answers
164
views
How can Navier--Stokes equations have asymmetric solutions such as Karman vortex streets
The Navier--Stokes equations are axially symmetric, so with symmetric boundary conditions, how can features such as Karman vortex streets develop?
I understand that in reality symmetry does never ...
5
votes
0
answers
288
views
A good 2D finite difference for the continuity equation
How could I go about solving the continuity equation below in 2D?
$$\frac{\partial \rho}{\partial t} + \nabla \cdot \left(\rho u\right)=0$$
I saw that a similar question was posted here: A good ...
5
votes
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answers
562
views
Avoiding divergent solutions with `odeint`? shooting method
I am trying to solve an equation in Python. Basically what I want to do is to solve the equation:
$$
\frac{1}{x^2}\frac{d}{dx}\left(Gam \frac{dL}{dx}\right)+L\left(\frac{a^2x^2}{Gam}-m^2\right)=0
$$
...
5
votes
0
answers
153
views
Discrete sine and cosine transform for mixed derivatives
Using sine and cosine transforms to solve Poisson's equation with Dirichlet boundary conditions seem quite standard nowadays (see, e.g., here or Table 2 in this paper). In the case of Poisson's ...
5
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answers
80
views
Padua-type pointset for functions singular on line $x=y$
The Padua points $\mathrm{Pad}_{n} \subset [-1,1]^{2}$ are a unisolvent pointset with optimal growth of Lebesgue constant, described in detail here. With some work they can be used to generate a ...
5
votes
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answers
82
views
Complex Integral Equation Solution in MATLAB
I need to solve an integral equation in the form:
$$A(z)+\int\limits^{z_2}_{z_1}B(z') \frac{z^N}{z^N-z'^N} \frac{e^{i\beta}}{|z|}\mathrm{d}z'=0 $$
where $A(z)$ distribution is known and we are ...
5
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answers
116
views
Name for vectors in a Krylov space but not the preceding one
It seems to me that a useful concept to define when studying Krylov subspace methods is the idea of a vector $v$ that belongs to a Krylov subspace $\mathcal{K}_{n+1}(A,b)$ but not to the preceding one ...
5
votes
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answers
160
views
Multi-matrix orthogonal basis problem
Suppose we are given a set of symmetric, positive definite matrices $A_1,A_2,\ldots,A_k\in\mathbb{R}^{n\times n}$. Is there any numerical method or reduction to a known problem (e.g. eigenvalue ...
5
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answers
89
views
How to construct a eigensolver targeting a specific type of matrix
I need to diagonalize such kind of matrix during research: it's n-by-n, with it's upper-left (n-1)-by-(n-1) corner be diagonal while the nth row & column are dense. It's observed that the self ...
5
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answers
308
views
Roots of transcendental equation involving bessel functions
I need an efficient way to numerically find the first $n$ positive roots $\lambda_n$ of the transcendental equation
$$ \dfrac{J_0 (\lambda_n r) Y_1 (\lambda_n) - J_1 (\lambda_n) Y_0 (\lambda_n r)}{...
5
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0
answers
95
views
Minimize interesting objective function with knowledge of gradient nonlinearity?
I plan on using a Quasi-Newton method (L-BFGS) to minimize a non-linear objective function.
$$ f: \mathbb{R}^n \rightarrow \mathbb{R}$$
The gradient is kind of interesting: as the values of the ...
5
votes
0
answers
118
views
Inverse problems with a discrete set of known parameters
What are the techniques on inverse problems to discover the distribution of parameters from a discrete set of values? For instance, I know that my domain where the PDE is defined is made up of ...
5
votes
0
answers
725
views
Order of accuracy of FVM discretization
I've recently got interested in CFD and started a small project by solving the radial Reynoldsequation. Why the Reynoldsequation? I recently encountered it through my studies and somehow got stuck :)
...
5
votes
0
answers
530
views
Optimisation of matrix exponential
I have a 7000x7000 sparse matrix (scipy), which I want to exponentiate. I've tried using scipy.sparse.linalg.expm, which works quite well for smaller matrices (takes a few seconds for a 1000x1000 ...
5
votes
0
answers
196
views
How to optimally choose points for multivariable Hermite interpolation?
I have a multi-variate, continuous function $f$ from $R^n$ to $R$, which I can query for its output for any input.
I would like to create interpolation polynomial for it.
In one-dimensional case ...
5
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0
answers
246
views
Preconditioning technique for large sparse non-hermitian matrix
I am attempting to solve a computational acoustics problem that involves solving an underlying sparse matrix. The size of the problem varies with grid size (3D) and fill-in's obviously make direct ...
5
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180
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Geometric Multigrid for Conform and Non–″ Elements: Restriction Operators
First of all, let me set up some notations.
Suppose we have a hierarchy of meshes $\mathcal{T}_0 \subset \mathcal{T}_1 \subset \dots \subset \mathcal{T}_n$. For simplicity I restrict myself here to $...
5
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answers
271
views
Stability analysis for a hyperbolic PDE on staggered grid
I am trying to understand the stability of a finite difference equation on the staggered grid.
I could understand the Von Neumann stability analysis for the collocated grid for a simple acoustic ...
5
votes
0
answers
506
views
Optimization on the manifold of stochastic matrices
So I have an optimization problem of the form
$$\text{maximize}\hspace{3mm}f(A):{\bf R}^{K\times K}\rightarrow{\bf R}$$
$$\text{subject to}\hspace{19mm}A^T{\bf 1}=\bf{1}$$
$$\hspace{33mm}A\geq 0$$
...
5
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0
answers
121
views
Using entropy functions for increasing numerical stability
Regarding the numerical stabilization of two-dimensional advection equation,
\begin{equation}
\dfrac{\partial f}{\partial t} + \Big(\dfrac{d\varepsilon_1(k)}{dk}\Big)\dfrac{\partial f}{\partial z}
- \...
5
votes
0
answers
139
views
Are there any benefits of computable analysis to numerical algorithms
Computers can work only with computable numbers, while most of the algorithms are based on analysis of real numbers (real analysis).
When I heard of the existence of computable analysis I ...
5
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answers
134
views
ADMM for Linear Program over graph
I want to use ADMM to solve a LP defined over a graph.
According to
Distributed optimization and statistical learning via the alternating direction method of multipliers
S. Boyd, N. Parikh, E. ...
5
votes
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179
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Difference of convex functions optimization problem in R
I am seeking of any already written R package which could help in an optimization technique which is called Difference of convex functions. This technique is sketched here and could be very useful ...
5
votes
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answers
2k
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CFL Condition and Convection Diffusion Equation in 2D
I am solving the convection-diffusion equation in 2D using Finite Differences with the $\theta$ scheme. The velocity of the fluid and the diffusion coefficient is low in my case (in the range of $10^{-...
5
votes
0
answers
625
views
Numerically calculating the divergence of a set of oriented points
Say I have a set of oriented points at locations $\vec{v_i}$ with each some direction $\vec{n_i}$, in practice they represent the normals of some surface that has been non uniformly sampled. How would ...
5
votes
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answers
110
views
FFT-based Image Rotation Algorithms More Accurate Than Chirp-Z?
We're currently using a Chirp-Z based implementation:
R. W. Cox and R. Tong, "Two- and three-dimensional image rotation using the FFT," IEEE Trans. Image Processing, vol. 8, no. 9, pp. 1297–1299, Sep....
5
votes
0
answers
127
views
Load balancing/partitioning with unknown weights
For a grid-based numerical simulation, I am looking for a load balancing/partitioning algorithm that not only distributes my grid elements, but also determines (approximates) their respective weights. ...