# Questions tagged [numerics]

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### Efficiency of developing PDE solvers using sparse matrices versus loops

I am new to solving PDEs, but have been looking at different implementations of finite difference and finite volume schemes. One thing I have noticed in different implementations is that some ...
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1 vote
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### A question on the Poisson equation with Neumann and periodic boundary conditions on a rectangular region

I am trying to solve the following PDE by using finite difference \begin{eqnarray*} \Delta u&=& f ~~on~~(0,1)\times(0,1)\\ \frac{\partial u}{\partial y}(x,0)&=&0=\frac{\partial u}{\...
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### Reference request for finite elements theory

Consider a domain $\Omega \subseteq \mathbb{R}^{2,3}$ which is non convex and with $C^2$ boundary. Could you recommend a good reference where it is explained how: without needing isoparametric ...
• 101
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### FEM applied to heat equation and incompatible conditions

Consider the problem $$u_t - \Delta u = f \text{ on } \Omega\times (0,T)\\u=0 \text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) \text{ on } \Omega$$ with $g$ NOT vanishing on the boundary. If I ...
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1 vote
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### Secant Method for finding $\sup f^{-1}(0)$

Let $f \in C^0[0, 1]$, and suppose $f \ge 0$. How can I compute $\sup f^{-1}(0)$ efficiently?
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### Numerical scheme for the heat equation on the icosahedral hexagonal grid

I have a predefined grid(like this) that is spawned from a regular icosahedradron. It consists of many hexagons and 12 pentagons (corresponding to icosahedradron vertexes). I can tweak the granularity ...
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### Are there radial basis functions $\phi_n(x)$ which can be factored as $f_n(x) x^{\alpha}$ for $0<\alpha<1$ and $n>0$?

I am looking for a set of radial basis functions $\phi_n(x)$ on $\mathbb{R}^+=[0;\infty[$ which satisfy some othogonality condition $$\int x \phi_n(x) \phi_{n'}^*(x) dx =\delta_{nn'}$$ and "...
126 views

### Numerically solving a 6th order non-linear differential equation in Matlab

I've posted yesterday a question about solving a non linear equation : it was not clear so I am reformulating my question. I am trying to solve a high-order non linear differential equation presented ...
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### Multigrid/Two-grid method restriction and prolongation of residual

Starting from the problem $Au=f$, I'm not sure that I understand why a coarse grid solution is implemented on the coarse grid residual $r_c^{(k)}=P^Tr^{(k)}$, with $r^{(k)}=f-Au^{(k)}$ and $P^T$ ...
43 views

### Interpolation and Restriction operators in Multigrid

I saw in several places that interpolation operator ($P$) and restriction operator ($P^T$) are usually transposes of each other (up to multiplication by a constant). As I understood it related to ...
70 views

### Numerical instability in the inverse Laplace transform

I have a problem with Laplace inversion and my function is not numerically stable for the Laplace inverse, but I do not understand the cause of this problem. Here is my code and graph of this problem. ...
65 views

### Largest singular value without using the adjoint

The square of the largest singular value of a linear map $A$ can be computed by using the power iteration for $A^TA$ and one advantage of this is that the iteration is matrix free, i.e. you only need ...
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### What does it mean for a finite difference method to be conditionally stable? Specifically when solving the diffusion equation

The diffusion equation is ∂u/∂t =∂^2u/∂x^2. Consider using an explicit finite difference method to solve it. What does it mean for that explicit finite difference method to be conditionally stable?
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### Are Python/MATLAB/Mathematica numerical eigenvectors affected by eigenvalue degeneracies outside region of calculation?

I have a discretized 2D mesh over which I calculate eigenvalues and eigenvectors of some Hermitian 2 x 2 matrix at each point along a closed loop parameterized by parameter t. The eigenvectors are ...
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### Solving Kepler's Equation with Newton-Raphson Method

Note (2022/03/07): This question is solved. Unfortunately, I'm not able to accept the correct answer by Lutz Lehmann, because I screwed up my registration and the account which posted this question is ...
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### Best method to solve this system of PDEs?

I have a system of PDEs constituting an initial value problem (IVP) consisting of three coupled PDEs: \begin{align} \partial_t \rho + \partial_x(\rho v) &= \left(k_A (1-\phi) + k_B \phi \right)...
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### How to troubleshoot numerical instability using finite difference for steady-state non-linear heat conduction equation

I have a problem which I believe is numerical instability when trying to solve a heat conduction equation using finite difference. The short version is that when the parameter $I=80.3$ I get the blue ...
• 510
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### How do we approximate the numerical error a numerical scheme (e.g Runge Kutta, Euler etc) makes without having access to an analytical solution?

So I recently encountered this question in my head while taking my Scientific Computing class, where the lecturer talked about computing numerical error of a scheme. My guess would be that we take a ...
59 views

### Contact analysis does not converge due to the projection falls outside valid domain

I implemented Node-To-Surface contact algorithm (Wriggers, Peter, Computational contact mechanics., Berlin: Springer (ISBN 3-540-32608-1/hbk). xii, 518 p. (2006). ZBL1104.74002.). The code is done by ...
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### Gridline cutouts after FFT and iFFT on Python

EDIT: I think I messed up on the coordinates of $(p,q)$. Num was missing a multiple of $2\pi/N$. Assuming my interpretation of DFT isn't wrong. I am currently using FFT to run Fresnel Diffraction as ...
1 vote
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### Anyway to escape ODEintWarning (scipy)?

I am trying to fit a differential equation to some data and obtain the parameters of the underlying model. This requires me to try out various parameter values, but this often gives me an ...
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1 vote
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### How do people deal with resized grid steps while numerically integrating using discrete Fourier Transform?

I am trying to simulate light propagation on python using FFT following the Fresnel diffraction equation given on Wikipedia: The problem with this is that the output matrix from the DFT would be ...
46 views

### State change of input-output system

Edited Given a computer model $F:\mathbb{R}^3 \to \mathbb{R}$, with inputs $x, w$ and $z$, and output $y=F(x,w,z)$, where for any input, we are able to evaluate the output, my goal is to tune the ...
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### Importance Sampling for multidimensional integrals and random numbers from multivariable pdf's

I am aiming to get a numerical value for a five-dimensional integral using Monte Carlo Integration. I am getting good results using the Mean Value Method, but I would like to try to use Importance ...
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### Numerical Partial Differentiation Check

In my computer vision course, we are working on extracting a 3D surface from a chain of 2D images taken under several conditions. This procedure is known as Photometric stereo. Prior to extracting the ...
• 155
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### What is the best method to do a MC Integration of a multidimensional integral where the integration limits depend upon other variables?

What is the best method to do a Monte Carlo Integration of a multidimensional integral where the integration limits depend upon other variables? I am interested in getting a numerical value of a 5 ...
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### Difference between asymptotic and non-asymptotic convergence in optimization?

I am reading some optimization methods and I am facing some issues with two terms "asymptotic and non-asymptotic convergence". What is the difference between them?
85 views

### Implicit integrator for ODE with quadratic right-hand side

I have an ODE for an unknown $x(t):[0,\infty)\to\mathbb R^n$ of the following form: $$x_i'(t)=a_i^\top x(t) + x(t)^\top Q_i x(t),$$ for $i\in\{1,\ldots,n\}$. Here, the vectors $a_i\in\mathbb R^n$ ...
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### Is this a legit way to sample a random matrix spectrum?

In order to undergird a theoretical model concerning many body physics, I want to have exponentially large eigenvalue spectra from the random matrix GOE ensemble. its properties are mainly (i) a ...
1 vote
116 views

### Fast evaluation of trigonometric polynomials

Suppose you have a trigonometric polynomial of the form \begin{equation*} x(t) = \sum_{k = 0}^N a_k \cos(2 \pi k f_0 t). \end{equation*} Using Clenshaw algorithm, one can evaluate this polynomial in \$...
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