# All Questions

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99 views

### How to compute over 1 billion particles?

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

### Motivation behind Collocation Method

In previous question "Motivation behind Galerkin method", Paul gives a good and easy-understanding explanation indicating that the Galerkin method is a kind of projection method. Can anyone explain ...
6 views

### How to implement the gmres method using Householder transformation instead of the Gram-Schmidt?

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

### Type of Rosenbrock method by its coefficients

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

### evaluating $\coth(x) - 1/x$ for real x, on 2 “pieces”

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 ...
54 views

### Inverting really big symmetric block matrix

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 ...
71 views

### Is the similar subdivision of a delaunay mesh still delaunay?

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 ...
41 views

### Why Householder transformation can not be chosen to be an identity matrix?

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 ...
88 views

### Givens rotation vs 2x2 Householder reflection

The usual story of Givens rotations vs Householder reflections is that Householder reflections are better if you want to map a long vector to $e_1$, while Givens is better if you want to map a 2-...
44 views

### Using adolc for the sign function in c++

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

### HSS preconditioner with gmres [closed]

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 ...
32 views

### Initial condition for Kuramoto-Sivashinsky

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 ...
90 views

### Efficient ways to numerically evaluate matrix exponentials

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 ...
34 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 ...
26 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, ...
106 views

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### Calculation of Mean Square Displacement for Brownian dynamics system with Lennard Jones interactions in python3

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 ...
35 views

### Avoid matrix multiplication in algebraic multigrid method

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 ...
62 views

### Solving a 1D diffusion equation with linear and nonlinear source terms

I would like to numerically solve the following equation: $$\frac{\partial \rho (z,t)}{\partial t} = B(N_D \rho (z,t) + \rho(z,t)^2) + D \frac{\partial^2 \rho (z,t)}{\partial z^2}$$ with the boundary ...
81 views

### Can one publish a new model and simulation without physical experiments?

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 ...
40 views

### How to improve the efficiency of periodicity detection for long time based lined and gapped datasets

Our data set has $10^4$ data points, but has a long baseline and many gaps. As the histogram shows, the horizontal-axis is time and most of the time, there are no data. The vertical-axis is data ...
29 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 ...
65 views

### Interpolation of Data Value using Optimized Weighting of Its Features

Assume I have a data set ${ \left\{ {x}_{i} \right\} }_{i = 1}^{N}$ which represents the value of each data point. For each data we have its features ${f}_{i} \in {\mathbb{R}}^{d}$. The model I ...