All Questions

I am trying to simulate the propagation of a gaussian beam through a lens using an FFT approach. I tried to implement the approach described by Couairon in this paper at page 43: https://link....

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For a project in my advanced numerical method class I have to solve the 1D Kuramoto-Sivashinsky equation. $$ u_t + u u_x + \lambda u_{xx} + \eta u_{xxxx} = 0. $$ As explained here I will solve it ...

For Generalized Minimal Residual method GMRES, we usually use the Modified Gram-Schmidt MGS to generate an orthonormal basis of ...

A Fortran code that solves stiff PDE systems contains the following arrays of Rosenbrock-Wanner method coefficients: ...

The function $\coth(x) - 1/x$ has a removable singularity at 0. Its Taylor series is: $$ \coth(x) - 1/x = \frac{x}{3} - \frac{x^3}{45} + \frac{2x^5}{945} + \ldots $$ I would like to evaluate the ...

For Householder transformation, we know that $H = I-uu^T$, where $\|u\|_2=\sqrt{2}$. When it acts on any vector $x$, $Hx$ and $x$ is symmetric with respect to $span(u)^T$. But I have read a ...

For a project in my advanced numerical method class I have to solve the 1D Kuramoto-Sivashinsky equation of which I know little. I just know that it was derived the equation to model the diffusive ...

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 ...

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, ...

What are some computationally efficient ways to solve matrix exponentials, i.e. functions of the form : f(X)=$e^{X}$, where X is a square matrix ? So far I have been able to diagonalise some ...

I have a question about HSS preconditioner with GMRES method. For implementing the HSS preconditioner with GMRES, we need to solve the linear system of the form (I + H)(I + S)z =r, for a given r at ...

Here is an implementation of the sign function in C++ using Adolc librairy for automatic differentiation. ...

In quantum mechanics, the wavefunction of N electrons is given by a determinant. I am working on a Monte Carlo algorithm. At each Monte Carlo step, I need to add or remove an electron, which means ...

I have a really big symmetric 7.000.000 X 7.000.000 matrix that i would like to invert. The matrix is extremely sparse and it can be rearranged as to become a block matrix. The biggest blocks are ...

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 ...

I am beginner in MATLAB and similar. I sow and discussed with my professors doing simulations some times: they wrote down a lot of calculus, most of them using Crank-Nicolson Method and so implement ...

I want to simulate human erythrocytes in capillaries. I calculated, that for a 1 meter long and 1 mm in diameter capillary there are about 3 billion blood cells. Erythrocytes are actually discs, but ...

I write a simple MATLAB code for solving solid FEM problem. The problem looks like that (1) (2) x-------x | / | | / | | / | x-------x (3) (4) ...

$A$ and $B$ are $n \times n$ matrices and $v$ is a vector with $n$ elements. $Av$ has $\approx 2n^2$ flops and $A+B$ has $n^2$ flops. Following this logic, $(A+B)v$ should be faster than $Av+Bv$. Yet,...

I have a delaunay triangulation for a 2d box with say an airfoil inside. If I uniformly refine this mesh by subdividing each triangle in the mesh into 4 triangles by halving each edge, is the ...

I am using the Armadillo library to solve a 3d heat conduction problem on 3d unstructured grid system,the gradient of the T field is determined by the least square method. I have created a matrix ...

Sorry it's bit abrupt, but recently I am caught up in some symbolic calcualtion which is tedious and almost impossible with mere human hands, so just wondering is it possible to solve the double ...

I am using Comsol to model a really hard problem. I am using the sweep for one variable (width), and as the problem state (Length=2*width). When I use the sweep the value of Length does not update. ...

According to the survey paper on autodiff (linked) Autodiff works on inputs that cannot be specified in closed form but can be described by a sequence of code, each component of which is ...

I'm trying to compute efficiently the following \begin{equation} A_j = \sum_{l'=1}^{\infty}\sum_{k= 0}^{K-1} L_{l'}T_ke^{2\pi i \frac{k}{K}j}\epsilon_{l',k} \end{equation} for $j = 0,1, \ldots, K-2,K-...

Can a subfield such as the representation theory of Lie algebras be studied computationally / numerically -- is there an interplay between the abstract and the concrete? I would be grateful for an ...

The governing equation of transient heat transfer analysis is described as follows: $$C \frac{dT}{dt}+K T = Q$$ When using backward difference scheme for the discretization of the time we get the ...

Octave calculations is too slow specially when you deal with scientific calculations that can span a very very large matrix, even just iterating. I have to vectorize it to speed up the calculation. ...

I'm trying to solve the 1D heat equation with Neumann boundry conditions numerically using finite differences: $$u_t = \alpha u_{xx}$$ $$u_{x}(0, t) = u_{x}(L, t) = k$$ $$u(x, 0) = u_0(x)$$ The main ...

I have a cloud of points scattered in a rectangle and some data in this points, a bit like this $$ x(:)=[x(1), x(2), ..., x(N)] \\ y(:)=[y(1), y(2), ..., y(N)] \\ u(:)=[u(1), u(2), ..., u(N)] $$ ...

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 ...

This question is more oriented around suggestions for simulation tools and how to approach simulating an object in orbit. So high level I am trying to simulate the concept of a Sky Hook. What are the ...

Currently when I try to solve a linear algebra system of the form of $A x =b$ I use the algebraic multigrid method. The algebraic multigrid method uses a Galerkin product to form the coarse grid ...

When I read strong papers from, say, the Journal of Fluid Mechanics, a simple model, simulations and physical experimental results are given, showing good agreement. Can one publish a new model and ...

I have a problem getting a sensible result for the Mean Square Displacement (MSD) for a simulation of $N$ particles under Brownian dynamics with Lennard-Jones interaction between them with or without ...

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 ...

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 & ...

I am trying to implement Minimum Discarded Fill (MDF) Ordering algorithm for incomplete matrix factorization. The algorithm description is here on page 60 Preconditioning Techniques for a Newton–...

More specifically, when training a neural network, what reasons are there for choosing an optimizer from the family consisting of stochastic gradient descent (SGD) and its extensions (RMSProp, Adam, ...

I'm trying to simulate the heat diffusion in a 3D piston. I marked the boundaries on GMSH. I have used a Dirichlet BC of 300 on the top face of piston. But the results look abnormal. There is a ...

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 ...

This is a follow up to my previous post I've been trying to compare the diffusion time obtained from theoretical derivation(answered in my previous post) and what is obtained computationally, for a ...

I am solving the steady, two-dimensional adjoint Euler equations, $$A_x^T \partial_x \Psi + A_y^T \partial_y \Psi = 0$$, where $A_x = \partial F_x/\partial U$ and $A_y= \partial F_y/\partial U$ are ...

I was reading the book: The Finite Element Method: Theory, Implementation, and Applications by Mats Larson and Fredrik Bengzon, in page 140 of this book they say this: "The Lax-Milgram Lemma is one ...

I was recently reading some literature about evolutionary game theory and I got particularly interested in Moran process linked to prisoner's dilemma as a model of evolution of two sub-populations. ...

I want to learn a very simple PDE to gain physical and mathematical intuition—a PDE with just one spatial dimension $x$ and a time dimension $t$. And, I want to write code for it in Matlab and use a ...

Some days ago I began to study the local discontinuous galerkin (LDG) method, this is my first time working with a discontinuous method, so I decided to solve the poisson equation in 1D to learn the ...

I am trying to simulate a case in ANSYS FLUENT for combustion of methane and oxygen where oxygen enters the domain in the form of droplets. There is a change in temperature but there is no formation ...

I encountered some odd results when using the function fix() in Octave and Scilab. The following is input and output of the Scilab console but the exact same thing happens in Octave. I start with the ...