All Questions

Today, I have studyied Full Orthogonalization Method (FOM), one of the Krylov subspace method, using book Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, PA, second edition, ...

Recently, I have studied how to construct an orthonormal basis for Krylov subspace to solve $Ax=b$, where $A\in \mathbb{R}^{n\times n}$ is nonsingular. As we know, there are usually 4 ways to ...

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On a desktop scale computer, what are the most important bottlenecks (RAM vs. CPU, single vs. multithread) for numerical calculations? I'm specifically most interested in exact diagonalization and ...

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

The above graph represents reduced viscosity as a function of reduced temperature for several values of the ...

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_{...

What is a good way to sample parameters for performing global sensitivity analysis? Some methods are defined using integrals, some are use Monte Carlo. How do these compare?

A maximization problem reads as $$ J(y) = \sum_{k=1}^{K} \sigma_k(y)^q \mathop{\rightarrow}^{y} max$$ where $q \in [1,\infty]$ is a user-defined parameter and functions $\sigma_k, k=\{1,\dots,K\}$ ...

I'm reading in data from a system which has very high range of frequencies (MHz). I'm demodulating this signal and only interested in the 1/f noise behavior in the first few Hz of this system. Say ...

Well, actually my goal is solving ODEs with Explicit Adam-Bashforth-Moulton Seventh Order Steps with matlab code. I'm searching in the internet, and i can't find about the formula (Sadly). But if ...

Suppose I have $$f=\frac x 3+ \frac y 3 +\frac z 3$$ And I want to use collect(f, 1/3) And I wish it will displays $$f=\frac1 3(x+y+z)$$ But it doesn't work....

I'm trying to educate myself on modelling solute flows through pipe networks. This is a follow up of my previous post here $$\frac{\partial C}{\partial t} = - v\frac{\partial C}{\partial x}$$ While ...

I've been reading this paper recently, which talks about using Lattice Boltzmann methods and two way coupling. Specifically, it outlines fluid solid coupling, and solid fluid coupling, and how simply ...

I need an explicit Runge-Kutta 14th order formula. If you know about some reference that discusses at least 10th order (or higher, since I'm looking for the 14th) of Runge-Kutta and there is ...

This semester, I study the Krylov subspace iterative methods (about Ax=b) using the book H. A. Van der Vorst. Iterative Krylov Methods for Large Linear Systems, volume 13. Cambridge University Press, ...

So far, all the MG literature is regarding FDM.What changes if we use FEM techniques?How coarsening works(In FDM we reduce nodes in FEM do we reduce elements)?

I see images of steel connections, concrete dams, and other complicated 3D objects in papers which finite element analysis has been performed on them. My questions are: How these objects are created ...

Suppose I rasterize a rectangle of width 2.5 gridpoints and get the values as shown: =============== | 0 | 1 | 1 | 0.5 | 0 | Now I resample that ...

What is an efficient way of evaluating the column (col_ind) and the row (row_ptr) vectors for the CRS (Compressed Row Storage) storage format using the Connectivity Array? The Connectivity Array is a ...

Consider a square $N \times N$ block-matrix $\mathbf{A}$, where each $n \times n$ block $\mathbf{A}_{ii}$ is either a dense block or a zero-block. So, $N$ denotes the number of blocks, $n$ denotes the ...

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 $...

Every real matrix $A$ can be reduce to real Schur form $T = U^T A U$ using an orthogonal similiary transform $U$. Here the matrix $T$ is quasi-triangular form with 1 by 1 or 2 by 2 blocks on the main ...

I have to calculate in numpy the matrix-product of many matrices (~400). Are there common practices to increase numerical stability? If this is relevant, the matrices are $300\times 300$ orthogonal ...

Problem statement I implemented geometric multigrid for $-\nabla^{2}=f$ where $f=\frac{3\pi^{2}}{4}sin \frac{\pi x}{2} sin \frac{\pi y}{2} sin \frac{\pi z}{2}$ on $\Omega \in [0,1]$ on a unit cube. ...

I'm trying to evaluate the exact solution of heat diffusion in circular plate. I'm not able to use the natural logarithm inside expression. ...

The quadratic knapsack problem (QKP) $$\max_x x^TPx$$ $$\mathrm{s.t.}\;\;w^Tx\leq c,\; x\in\{0,1\}$$ where $P\geq0, w\geq0$ elementwise, is well studied and has existing solvers. My problem below ...

I use software for pollutant propagation on rivers that takes as input a set of parameters ($p_1,p_2,\ldots,p_n$) and creates an output file which is basically a matrix where on each row the ...

Last week, I have learned the details of the robust iterative methods of PCG, MINRES, GMRES, which will converges to the exact solution $x^*$ of nonsingular system within $N$ steps for $A\in \mathbb{R}...

The strong form of a PDE requires that the unknown solution belongs in $H^2$. But the weak form requires only that the unknown solution belongs in $H^1$. How do you reconcile this?

I would like to plot the points contained in this file with Paraview, but can't seem to figure out how to do so. Each column in this file corresponds to a set of 2048 points on a 64x32 grid. Each ...

I am developing a solver that gets a CAD model as entry and does the topological optimisation calculation on it. My solver is inspired by the open source codes presented in literature. Since it is ...

Given an N-dimensional matrix A, I want to find an M<N dimensional index array I such ...

I am trying to run an MD simulation using this generalized version of Lennard Jones. $$ U(r) = \left(\frac{r_0}{r}\right)^A -\frac{A}{B}\left( \frac{r_0}{r} \right)^B $$ However, I do not know how ...

Since many computers today do not come with a CD Drive, am wondering if there is a way to rework the recovery software, which allowed for a system image backup using a recovery USB but it only allows ...

I have a computer model with a number of parameters that need to be calibrated based on experimental results. It's also important to understand the sensitivity of the results to each parameter ...

Could anyone suggest the most computationally efficient method for finding amplitude at a given frequency having a noisy wide band signal. To be more specific about a task. I have some physical ...

My question is regarding solving the conservative form and the non-conservative form of the governing-equations (GE), like continuity or the navier stokes equation, using finite difference method (FDM)...

As the title says, I am trying to code in FEM a plate structure that either has a hole in one of the layers or one of the layers is made of patches of plates, rather than one whole plate. However, ...

I have a linear system where I am given 2 matrices, $A$ and $B$, and 2 vectors, $v$ and $c$, and I need to solve for the vector $x$. $A$ is $n\times n$, $B$ is $n \times n \times n$, and the vectors $...

For example, I have a function to optimize: $$f_1(x,y) = x^2+y^2, \quad x_{lb}\le x\le x_{ub},\quad y_{lb}\le y\le y_{ub}$$ Then I apply rotation by $\theta$ plus translation by $x_0$ and $y_0$: $$f_2(...

I've been reading Albright's Orbital Interactions in Chemistry. In the chapter on solids, he provided a general approach to find the band structure of a solid state system Now if we are to model a ...

This semster, I have been studying the iterative methods for large sparse matrix system. But I have some questions. For large sparse matrix, we must use an economic storage to store them. The most ...

In principle the PDE Toolbox in MATLAB can handle multi-domain 3D geometries as noted here. This feature and the associated function geometryfromMesh were introduced in MATLAB R2018a. The associated ...

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 ...

I was wondering what would be an efficient way to produce compatible displacements for mesh nodes/vertices if the computed data is volume shrinkage of each element/cell in the unstructured mesh? ...

I am looking for a fast algorithm for computing eigenvalues and eigenvectors from sparse stiffness and mass matrices in MATLAB. The eig(K, M) doesn't work with ...

I am trying to find out if the known QR algorithm to find the eigenvalues of a real matrix, which can be found in the book Fundamentals of Matrix Computations, can also be used for complex matrices ...

I'm trying to understand how convection-diffusion equations are solved in pipe flow modules available in CFD solvers. $$ \frac{\partial C}{\partial t} + \nabla \cdot (\mathbf{v} C) = \nabla \cdot (D \...

I have a linear system $$Ax=b$$ which I'm solving approximately, and I need to take the frechet derivative of x with respect to z. Were I solving the problem exactly (either analytically or to machine ...