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4
votes
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75 views

Order of convergence of Scrodinger eq. with CN scheme

I'm trying to solve numerically the 1-dim time dependent Schrodinger equation using the Crank Nicolson scheme and the Thomas algorithm to solve the tridiagonal matrix. The physical system consists of ...
4
votes
0answers
678 views

Generating pseudo-random orthonormal bases for random projection

I am performing series of random projections i.e. projecting the input matrix onto randomly generated orthonormal bases (of much lower dimensionality). The projection is just a matrix multiplication ...
4
votes
0answers
83 views

Non-convergance when calculating temperature/heat flows through a section of rock

I am attempting to calculate temperature of section of rock in the earth as a function of vertical position in the rock and time. Along with it I am calculating the heat flow through the rock as a ...
4
votes
0answers
500 views

Fenics: Result of Steady state dynamic linear elastic doesn't match with actual values

I solved the steady state dynamic linear elastic model in a solid. My equation is a function of frequency and the strong form is: $$\operatorname{div}(\operatorname{stress}(\vec x, w)) + w^2 \rho u( \...
4
votes
0answers
219 views

Adaptive mesh data structure for Fast Marching Method to overcome RAM limit

On an uniform mesh of positions in space $\ (x_i,y_j,z_k)$: $$\ x_i = x_0 + i\Delta x,\quad i=0,\ldots,n_x$$ $$\ y_j = y_0 + j\Delta y,\quad j=0,\ldots,n_y$$ $$\ z_k = z_0 + k\Delta z,\quad k=0,\...
4
votes
0answers
169 views

Solving diffusion PDE using finite differences

I need some hints on how to solve this diffusion equation ($\alpha, k_1,k_2$ and $k_3$ are constants): $$ {\partial P \over \partial y} + k_1 {\partial P \over \partial t} + \alpha P = {1 \over k_2} ...
4
votes
0answers
174 views

Find maximum distance between elements given constraints on some

I have a list of numbered elements 1 to N that fit into positions on a number line starting with 1. I also have constraints for these elements: The element 1 is in position 1, and element N must be ...
4
votes
0answers
105 views

Existing software/scripts for spiral graphs?

I am looking for existing software or scripts to generate spiral graphs from cyclical (time) data, as presented by Webber and Muller. The graphs shown in the paper look like a great means of ...
4
votes
0answers
622 views

Perron-Frobenius theorem on general real symmetric matrices

From the Perron-Frobenius theorem, it might be concluded that the spectral radius is the largest eigenvalue for positive matrices, ie, matrices with strictly positive entries. In other words, the ...
4
votes
0answers
99 views

Time-stable spectral decomposition algorithm

Consider an $n \times n$ real, time-dependent matrix $A(t)$ such that $A(t) = A(t)^{T} > 0$ and $A(t)$ is continuous on $[a,b]$. Then it is posible to specify a matrix $S(t) \in SO(n)$ such that $S(...
4
votes
0answers
394 views

Why is my lower convex hull extraction algorithm not working?

Recently, I wrote an algorithm to obtain a delaunay triangulation of a random point set in $I=[-10,10]$x$[-10,10] \subset R^2$ by projecting these points onto the 3 dimensional paraboloid $z=x^2+y^2$, ...
4
votes
0answers
134 views

Probabilistic algorithms for matrix approximation

Considering regular matrix approximation inequality || $A - QQ^TA $|| < e where we try to approximate matrix $A$ by a lower rank orthonormal matrix $Q$. I've read an article on probabilistic ...
4
votes
0answers
252 views

Convergence rate of Monte-Carlo variance estimate

What is the convergence rate for Monte-Carlo variance estimate for a random variable $X \in {L^q}(\Omega ,R),2 < q < 4$?
3
votes
0answers
48 views

Comparing Two Matrix Notation

I have two matrix A and B, I want to find pattern B in matrix A. So I get 2 pattern similar like pattern B. What the name of this operation? and How I write this in mathematics notation? Thank you in ...
3
votes
0answers
45 views

Solving saddle point problem having non-invertible top-left block with a PETSc nested matrix

My system is a symmetric FE problem with lagrange multipliers: $Z=\begin{pmatrix}A & C^T \\ C & 0\end{pmatrix}$ The matrix $A$ is positive semi-definite, non-invertible. The whole matrix is ...
3
votes
0answers
32 views

Hit-n-Run Monte Carlo on convex polytope

So, I'm currently trying to implement a MCMC to uniformly sampling hyper-points from the polytope defined as $\mathbb{K}=\{x\in\mathbb{R}^{n}\;\;\text{s.t.}\;\; A\,x=b \}$ in the specific case where, ...
3
votes
0answers
33 views

Integer partition algorithms

I am familiar with and have written MathCad algorithms for the partition functions 𝑝(𝑛,𝑘),which gives the number of ways of partitioning 𝑛 into 𝑘 parts, 𝑞(𝑛,𝑘), which gives the number of ways ...
3
votes
0answers
50 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 & ...
3
votes
0answers
52 views

What is the reason for this finite-difference high errors on non-uniform grid?

tl;dr Using a Taylor-matched method to find coefficients for the discretized equation $ \mathbf{A} \vec{f}'' = \mathbf{B} \vec{f} $, a Fortran code has been implemented to find the second derivative ...
3
votes
0answers
60 views

Computation of Troullier-Martins pseudowavefunctions

The computation of Troullier-Martins pseudowavefunctions has been described in [1]. The pseudowavefunction $R^{\textrm{PP}}_l$ is defined by $$ R^{\textrm{PP}}_l(r) = \left\{ \begin{array}{ll} R^{\...
3
votes
0answers
67 views

Large-scale optimization of nonlinear equations

I'm looking to find a computationally efficient solution to a large system of nonlinear equations. I'm trying to maximize the following function: $$ f(\vec{x}) = \sum_i^N C_i (x_i-A_i)x_i^{\epsilon_{...
3
votes
0answers
50 views

Methods to approximate obective function gradients from point cloud

Problem statement: Assume that I have an objective function $f(x)$ which takes as input a $D$-dimensional vector $x\in\mathbb{R}^D$, and that $f(x)$ is sufficiently smooth. Assume further that I ...
3
votes
0answers
96 views

What is the fastest algorithm for computing log determinant of a PSD matrix? (All possible PSD matrices)

I am diving into some literature to understand which is the best algorithm for computing the log-determinant of a PSD matrix. More generally, I am interested in a list of resources to read, which ...
3
votes
0answers
65 views

Numerical integration with singularity term

In https://www.johndcook.com/blog/2012/02/21/care-and-treatment-of-singularities, the author explains the subtraction method to get rid of singularities when performing numerical integration. The ...
3
votes
0answers
42 views

Subspaces for Iterative methods

In the original paper of Conjugate Gradients, the authors mention that if we pick the canonical basis $\{e_1,e_2,\ldots,e_n\}$, to obtain A-orthonormal vectors, we end up with the Gaussian elimination ...
3
votes
0answers
50 views

Why GA convergence curves continue as two parallel lines?

I'm working on a optimization problem and using GA algorithm (in MATLAB, ga function). As you know MATLAB plots GA result with two curves, one for the best values and other to show the mean values ...
3
votes
0answers
47 views

Singular vectors of s1 for tiny dense matrices

I have a function whose main bottleneck is finding a(ny) singular vector pair in the space of the largest singular value, along with the singular value itself. This is done a huge number of times. ...
3
votes
0answers
44 views

Efficient evaluation of spherical harmonic expansions

Assume I know that I can express an approximation of a function by $$ \sum_{l=0}^{k}\left( \sqrt{A_l} z_{l,0}^1 L_{l,0}(\theta)+\sqrt{2A_l}\sum_{m=1}^l L_{l,m}(\theta)(z_{l,m}^1 \cos(m\phi)+z_{l,m}^2\...
3
votes
0answers
45 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 ...
3
votes
1answer
62 views

Differences between Discrete Fourier Transform and Continuous Fourier Transform?

I am trying to visualize the time dependence of a free particle given an initial wave-function using Python and I just wanted to know if I could use the in built FFT implementation from NumPy to find ...
3
votes
1answer
68 views

Scipy Spline Interpolation Parameter

Documentation in scipy.interpolate (found at https://docs.scipy.org/doc/scipy/reference/tutorial/interpolate.html) states: "The parameter variable is given with the keyword argument, u, which ...
3
votes
0answers
56 views

Is it possible to retain sets (physical tags) after remeshing the model in Gmsh?

I am currently working on a remeshing algorithm for Abaqus using Gmsh as part of my Bachelors thesis and I seem to be stuck at some point. I am able to remesh .geo and .msh files using the script ...
3
votes
1answer
78 views

Compute the function between two images

Take an image $f$ with some characters on it (below, hjFu3). Let's apply a filter $h$ on it to obtain a second image $g$ where the text is not visible. Is there a way to compute what kind of filter $...
3
votes
0answers
84 views

numerical instabilities in Fluid Dynamics, Finite Element Method

I'm looking for references to understand where the numerical instabilities come from in hydrodynamics in general, and notably when the Péclet number: $Pe>1$. I'm using the finite element method. ...
3
votes
0answers
35 views

Sparsity-Promoting Convex Optimization Over Simplex

Say we want to find a sparse approximate minimizer to the function $f(x) : \mathbb{R}^d \to \mathbb{R}$. Then in line with the work in the field of compressed sensing, we can instead minimize $$f(x) + ...
3
votes
0answers
106 views

A maximization problem, with motivation in machine learning

Consider the minimization problem described this paper. Let $f_{\lambda}$ be the minimizer. As a part of extending my work, I am able to show the following facts $$\lim_\limits{\lambda \to 0}\|f_{\...
3
votes
0answers
27 views

How to set up and solve acceleration-limited trajectory optimization problems?

I've been trying to learn how to solve simple acceleration-limited trajectory planning problems. I'm working in C++ and I've been using the Eigen library to do linear systems solving. I'm doing the ...
3
votes
0answers
55 views

Strain propagation from surface to bulk in COMSOL

I am trying to simulate strain propagation from the surface into the bulk. I have a rectangular semiconductor block (~2 μm thick) on top of which metal gates (~25 nm thick) are deposited as seen in ...
3
votes
0answers
20 views

Sequence planning with 3 machines

together! First of all, I have to mention that because of my background as an Industrial Engineer, I have limited abilities in mathematics, but am disciplined enough to expand myself from ...
3
votes
0answers
43 views

Is it Grid/Cluster/Cloud Computing? How are those terms defined?

There are three very connected and widely used terms: Grid and grid computing Cluster and cluster computing Cloud and cloud computing In many situations, it is not obvious which term to use, as I ...
3
votes
0answers
39 views

Derivative of Whittaker-Shannon interpolant

Last time we looked at how to improve the accuracy of Whittaker-Shannon interpolation, where user njuffa demonstrated that judicious use of sin_pi could greatly ...
3
votes
0answers
48 views

Constraint solver vs Bayesian optimizer for fast discontinuous processes

I have a complex domain-specific process that accepts inputs: 10-500 inputs, where each input is of type: enum: choice between multiple string or numeric values int: integers float: floating point ...
3
votes
0answers
50 views

Calculating depth mask from different lighting

I have a object which is static, the camera is static and light source is moving. How can the depth mask be calculated ? Concept is to use - calculate height from shadow length Lets imagine a have ...
3
votes
1answer
93 views

Poisson image blending artifacts

I am trying to implement Poisson image blending as in the paper Poisson Image Editing. This is the task of filling in a masked region of an image by minimizing $$\min_f\int_\Omega \left | \nabla f - \...
3
votes
0answers
62 views

Calculate the Bloch wave

The eigenvalue problem $$\frac{d^2u}{dx^2}+2i k\frac{du}{dx}-[k^2-6\sin(x)^2]u(x)=-\mu u(x)$$ gives the first five eigenvalues with $k=0$ or $k=1$ which are $2.06$, $2.26$, $5.16$, $6.81$, and $7....
3
votes
0answers
71 views

Solve ODE with non-negative and maximization constraints

My task is to solve $$\eta_k\frac{d^2C_k}{dz}(z)=-e_k, k = 1,2,3$$ $$C_k\ge0$$ $$C_1(0)=0, C_2(0)=A, C_3(0)=0$$ $$C_1(L)=B, \frac{dC_2}{dz}(L)=0, \frac{dC_3}{dz}(L)=0$$ with $$e_1 = -\beta_1-\beta_3$...
3
votes
0answers
49 views

Use of non-typical values of $\theta$ in theta-methods

The theta-method is a popular solution for solving time-transient PDEs (or ODEs), which consists of solving the general equation for each time step: $$ \frac{u^{n+1} - u^{n}}{\Delta t} + (\theta f(u^{...
3
votes
0answers
31 views

Detecting blocks in non-linear system of equations

When solving systems of non-linear equations using Newton's method, it is often observed that the system has an independent sub-system, e.g. : $$ f(x,y) = 0 $$ $$ g(x,y) = 0 $$ $$ h(x,y,z) = 0 $$ If ...
3
votes
0answers
63 views

Structural Analysis Library

Can anyone recommend a structural analysis library that satisfies the following requirements: C++ API Simulate both beam elements and shell (slab) elements Both static and dynamic analysis Free and/...
3
votes
0answers
65 views

Numerical analysis, pivoting and incomplete LU decomposition

When doing LU decomposition, the algorithm will break down if any of the diagonal element $x_{ii}$ is zero. Therefore, we can use pivoting on the matrix such that $x_{ii}$ is no longer zero. That is ...

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