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How to understand the choice of Krylov subspace orthonormal basis?

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, ...
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Block-matrix: optimal fill-in reduction for LU factorization

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 ...
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Numerical stability in the product of many matrices

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 ...
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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 ...
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Prove that the set of maximizers are independent of parameter in the objective function

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 $\sigma_k, k=\{1,\dots,K\}$ ...
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Necessary information that a toplogical optimisation solver needs to collecte from a pre-processed CAD model

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 ...
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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 ...
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Fast algorithm for computing lower mode shapes and natural frequencies in MATLAB using sparse stiffness and mass matrices

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 ...
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Produce vertex displacements from volumetric shrinkage data on unstructured meshes

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? ...
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When should I write a matrix-vector function to handle the sparse matrix vector multiplication?

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 ...
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Plot sinewave on ZX axis [duplicate]

I am trying to plot a sinewave with a bit of 3d perspective along the ZX axis instead of the XY axis. I have so far been unable to get anything that works, and have been unable to locate any examples....
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A lot of identical staff in Comsol material database?

I got a lot of elements in Material Browser of Comsol Multiphysics of Optics section. ...
51 views

Solving differential equation in Python with discretized variable coefficients

I am trying to solve a differential equation with discretized variable coefficients which are calculated from a time serie. In this case the Runge-Kutta step size is fixed by the frequency in the time ...
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Question regarding 1D implementation of the DG method

I'm pretty new to the DG method and have been writing a 1D code to help me understand the coding aspect. With respect a reference, I've been following these notes https://www3.nd.edu/~zxu2/...
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Why do not we choose the error solution norm as an iterative method's criterion?

For solving linear system $$Ax=b,$$ using iterative mehods, we often use the terminate criterion as follows: $$\frac{\|r_k\|}{\|r_0\|}=\frac{\|b-Ax_k\|}{\|b-Ax_0\|}<eps.$$where $x_0$ is the ...
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Using MATLAB to simulate the Ising Model

I am using MATLAB to simulate a 1D Ising Chain. I am running into an issue where when trying to find heat capacity, my system has a tremendous amount of noise. I'll post my code and an image of the ...
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Derive the Finite Difference equation corresponding to the steady state PDE. Explain how to implement Dirichlet and Neumann boundary conditions

A rod of length L = 1m, is made of copper, with thermal conductivity κ = 380W/m/K. A uniform heat supply h = 10kW/m3 is applied to the rod. The left boundary of the rod is kept at constant temperature ...
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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 ...
131 views

Consumer hardware for scientific computing?

I'm interested in problems around probability, statistics, and statistical mechanics, and often I find it useful to perform simulations to get some sense of the underlying phenomena. Example ...
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Why does the initial guess for linear system usually choose by zero vector?

For solving linear system $$Ax=b,$$ using iterative mehods, we often use the terminate criterion as follows: $$\frac{\|r_k\|}{\|r_0\|}=\frac{\|b-Ax_k\|}{\|b-Ax_0\|}<eps.$$where $x_0$ is the ...