Questions tagged [numerics]

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Is it ok to get the parameter of a random variable, from a random variable itself?

I want to generate the parameter, for a Poisson distribution, from a lognormal distribution. Has there been literature on if this is proper in probability & statistics, or are there nuances to ...
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1answer
86 views

Numerical solution of ill-conditioned differential equation

I want to solve the following Cauchy problem \begin{equation} y' = y^2 + \frac{t^4 - 6t^3 + 12t^2 - 14t + 9}{(1+t)^2} \end{equation} with initial condition: $y(0) = 2$ for $t \in [0,1.6]$ using a 3 ...
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1answer
55 views

How does the diffusion of a finite volume method with a WENO scheme compare with that of spectral methods?

I know that, in general, finite volume (FV) methods are more (numerically) diffusive than spectral methods. However, I can't find any information on how the advection scheme changes that. For example, ...
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1answer
56 views

Numerical solution of the advection equation with Crank–Nicolson finite difference method

I need to implement a numerical scheme for the solution of the one-dimensional advection equation $$\\\frac{\partial u}{\partial t} + C(x, t) \frac{\partial u}{\partial x} = 0 \\\\$$ $$ \\ C(x,t) = \...
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2answers
129 views

Using the BDF and RK4 methods to solve this coupled system of ODEs in C++

I'm trying to solve a system of ODEs using the BDF order 4 method. I find the first 3 points using RK4, then for the implicit part of the BDF, I use Newton-Raphson iteration. Unfortunately my solution ...
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65 views

understanding the form of Steklov-Poincare operators

I am reading this paper https://paola-gervasio.unibs.it/Publications/pg_cmame.pdf and I fail to understand the local Steklov-Poincare operators defined in $3.2$. Please help me on that. In my ...
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49 views

Book recommendation on numerical methods for solving Integro-Differential equations

I was wondering if anyone could recommend a good book or resource on numerical methods for solving integro-differential equations? Of course I am familiar with the methods for solving ODEs and PDEs ...
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1answer
58 views

Trouble with backwards time integration in Python

I am struggling with a rather basic numerical integration task: Using Python's scipy.integrate.solve_ivp module to integrate an ODE sytem backwards in time. As a test, I am using the following ODE ...
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97 views

A Question About a Claim from 1991 Computational EM paper about the Cancellation of certain Boundary Terms

Please let me know if this is not the appropriate site for this question. I found questions regarding EFIE/MFIE/CFIE on this site, so I thought my question might fit. I am studying the paper by Putnam ...
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2answers
96 views

Software to build a mesh of a surface from points on the surface

I have a set of points $(x_i,y_i,u(x_i,y_i))\in\mathbb{R}^3$, $i=1,\dots N$, over a surface $S$ (from experimental data). I need to calculate the integral of a function $F$ over that surface. If the ...
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1answer
23 views

Numerical stability of taking the `mean` of outputs from the simulation of a discrete stochastic dynamical system

I am writing a simulation for a discrete stochastic dynamical system. Since the simulation is stochastic, I need to run the simulation multiple times and then average the values of each timestep. I ...
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1answer
62 views

How to include negative number in the log-sum-exp?

I want to know summation of some small numbers, such as {e^-1000, -e^1001, e^1002...} If all numbers are positive, I can use log-sum-exp algorithm. But unfortunately, negative numbers are also ...
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1answer
83 views

preconditioner for $u''(x)=\sin(x)$

I am interested in finding preconditioner to solve the problem for one dimensional problem $u''(x)=\sin(x), u(0)=u(1)=0$ using Dirichlet-Neumann method. The preconditioner $M$ coming from Dirichlet-...
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54 views

Upper bound on condition number in linear preconditioning

I'm studying iterative methods for solving linear system, and I find the following setting in Wikipedia: Consider a matrix splitting $A = M-N$, where $A,M,N$ are all symmetric and positive definite ...
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1answer
92 views

Numerical Linear Algebra: When to use Direct methods versus iterative methods to solve a linear system - for PDEs in particular

I am reading the Chapra and Canale book on numerical methods, and was working through the chapters on solving linear systems. Now the book goes through direct methods including Gaussian Elimination, ...
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17 views

Problem with recursive implementation of Subspace Iteration method in Numpy

I am having trouble with implementing the method of subspace iteration to find the eigenvalues and vectors of a random, symmetric matrix, A that is mxm with m = 10. ...
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0answers
265 views

Minimax optimization with an oracle

I have an optimization problem of the following form: $$\min_y\left[\max_x f(x,y)\right].$$ It is fairly straightforward to minimize $f(x,y)$ over $y$ with $x$ fixed, and similarly to maximize $f(x,...
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1answer
132 views

Conserve energy by message passing?

There are $N$ particles with positions $x_i(t)$ and velocities $v_i(t)$ and mass 1. There is a potential function $U_{i,j}(x_i, x_j)$ between each pair of particles, which is $0$ unless the particles ...
4
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1answer
148 views

Time Reversibility of Velocity Verlet Algorithm

I'm very new to computational Physics and am finding conflicting statements on whether the velocity Verlet algorithm, defined as: $\begin{align} x_{n+1} &= x_n + v_n \Delta t + \frac{1}{2} a_n \...
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0answers
32 views

How can I improve the accuracy of the calculation of the magnetic field in Gmsh/GetDP?

I need to calculate the magnetic field along a straight line in proximity of an array of 6 magnets. I used the tutorial files "magnets" included in Gmsh and I slightly modified the file in ...
4
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2answers
82 views

Dividing functions over a wide range-

I try to solve a system of coupled equations, where a very nasty division operation occurs. In fact, I need to compute a derivative of two exponential decaying functions. Let's illustrate this with ...
1
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1answer
52 views

2-DOF Robotic Manipulator Trajectory Tracking Simulation

I am trying to simulate a 2-DOF planar robotic manipulator (have its joints follow a predefined trajectory) that's described by its dynamic model: $$ M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q) = \tau $$ ...
2
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1answer
81 views

Best practice for ADTs in computational science with Fortran

I have been writing a software package in Fortran for solution of the Vlasov-Poisson system in 2D2V. I want this software to be useful beyond its current application (e.g. systems with different ...
3
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1answer
68 views

Cauchy Lorentzian simulation on FFT with oscillation

Recently I do simulation on Lorentzian Function with FFT Lorentzian Function is 2a/(x**2+a**2) ...
2
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2answers
136 views

Numerical Methods of solving a non-linear ODE?

I want to solve the nonlinear equation $\frac{d^2x}{dt^2} + k\sin x = 0$, numerically. I found that solving this elliptic integral would be cumbersome, so is there a numerical method i could use to ...
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0answers
81 views

Algorithm for computing inner products multiple times

I am taking a computational linear algebra course and i got stuck during a homework problem concerning the computation of inner products. I am supposed to compute the inner product:$$\mathrm{a}_{\...
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35 views

discretizing advection equation with variable wave speed + stability

I currently have a code that solves $u_t+ cu_x=0$ with periodic boundary conditions, and constant $c$ (using an upwind method). I'm wondering how I would alter this code to solve something of the form ...
2
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1answer
79 views

Rank of a double-precision augmented matrix

Let $A$ be a matrix with real entries, and let $A_+$ be $A$ augmented by a single column. From linear algebra we know \begin{equation} \operatorname{rank}(A_+) = \operatorname{rank}(A) \hspace{10pt} ...
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1answer
122 views

Numerical integration problem: IntegrationWarning The integral is probably divergent, or slowly convergent

I am trying to get the numerical integration of a function using scipy's integrate.quad as follows. $$ \begin{equation} G (\alpha) = \frac{4\alpha}{\pi}\int_0^{\...
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0answers
65 views

Ill-conditioned stiffness matrix

I am writting a Fem code in c++ for a 2d plane stress model. My question is regarding the assembly stiffness matrix.I noticed that some elements of the matrix are not exactly zero but insted a number ...
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0answers
65 views

Error curve does not oscillate in between reference points using Remez

Using the Remez algorithm, implemented using multi-precision library, in certain functions that I want to approximate, the error curve does not oscillate in between reference points, and so no roots ...
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1answer
46 views

Linearization of Remez algorithm rational case

In the rational case, we are interested to find polynomials $P(x)$ and $Q(x)$ s.t. $f(x_k)-P(x_k)/Q(x_k)=(-1)^kE$ for $k=1,2,\ldots, N$ where $N=deg(P)+deg(Q)+2$ This can be rewritten as $$ (1)~~~~~~(...
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1answer
75 views

Least Square fit of a system of equations

I've a matrix equation $\bar z^{(r)}(k + 1) = \bar{DF} ~ \bar z^{(r)}(k)$. Now the $\bar z$'s and $\bar{DF}$ are $d \times d$ matrices. Now $\bar{DF} = \begin{pmatrix} 0 & 1 & 0 & 0 & \...
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22 views

offline Bin Packing problem with multiple size bins

As per my research on stack overflow communities, This is probably known as cutting stock problem / multiple Knapsack problem (a variant of the bin packing problem) which is NP hard. here are the ...
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0answers
79 views

Remez algorithm convergence

I have implemented the Remez algorithm in Python where all calculations were done with the Python mpmath library. I have noticed that sometimes the $|E_{max}|$ and $|E_{min}|$ do not monotonically ...
2
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1answer
151 views

Finite element (1D) for steady state non-linear problem

I need to solve with linear finite elements the equation $$\frac{\partial }{\partial x}\Bigl(\text{sgn}(x) u \Big) +\frac{\partial}{\partial x} \Bigl[ \sqrt{u} \frac{\partial u}{\partial x} \Bigr] =0$...
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1answer
115 views

Non-Linear advection diffusion with nondifferetiable advection term

I'm looking at Murray's book: Mathematical biology: an introduction , first volume, pag. 404 In particular, I'm interested to solve the following PDE: $$\partial_t u = \partial_x (\text{sign}(x) u) + \...
2
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1answer
115 views

Ill-condioned Linear System and Gaussian Elimination

Suppose that I have a linear system $Ax=b$ such that $A$ is ill-conditioned. Can I say that it is dangerous to find a solution with Gaussian Elimination for this system, or does there exist some class ...
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0answers
65 views

How can I practice multivariable root-finding?

Recently, I've been reading up on various root-finding / optimization algorithms such as the Levenberg-Marquardt method, Gauss-Newton, Conjugate Gradient, trust-region and trust-region-dogleg. I've ...
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1answer
61 views

Imposing pressure variation instead of Dirichlet boundary conditions on Finite Element Method

I always see Finite Element codes solving PDE with Dirichlet or Neumann boundary conditions. But, I have a problem now consisting of a straight cylinder with a circular base (a simple 3D tube), with ...
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0answers
103 views

Solution of Cahn-Hilliard equation

I need to solve the Cahn-Hilliard equation $$\frac{\partial u}{\partial t} = \Delta(f(u) - \epsilon^2\Delta u), \hspace{.5cm}(x, t)\in \Omega\times(0, T],$$ using mixed formulation \begin{equation}\...
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0answers
45 views

Norm estimates if adjoints can't be computed

Assume that you have two linear maps $A$ and $V$. For a given $x$ (of appropriate dimension) you can compute $Ax$ numerically, and for any $y$ (of appropriate dimension) you can calculate $V^Ty$ ...
5
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1answer
114 views

Accurate and efficient computation of the logarithm of the ratio of two sines

For exploratory work related to special function implementations, I need to compute $\log \frac{\sin y}{\sin x} $, where $0 \le x \le y \le 2x < \frac{\pi}{2}$. Cases with $x \approx y$ in ...
2
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1answer
63 views

Parity for artificial dissipation term in a finite-difference solution

I have a doubt regarding the signal of the dissipative term in a finite difference solution for an equation of the form $$ \frac{\partial u}{\partial x}+f(x)=0, u(0)=0 $$ In which $f$ is an odd ...
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0answers
48 views

numerical solution to pde on an ellipse

Looking for advice on discretization (preferably finite difference) schemes for pdes on curves in general, but in this case it is an ellipse (so given by $(a\cos(r), b\sin(r)$). The problem is the ...
3
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1answer
73 views

Find quadrature points and weights

I'm struggling with the following problem: What is the maximum degree of exactness that we can obtain with the following quadrature >formula $$\int_0^1 f(x)\frac{1}{\sqrt{x}}dx \approx w_0 f(x_0) +...
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1answer
71 views

Interpreting multivariable root-finding results from Matlab's fsolve algorithm

Edit: So I was able to get the same value of r that's given, when coding up the sum of squares of function values directly in the script file, rather than on the Command Window. So, maybe there's a ...
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2answers
132 views

No flux Neumann boundary condition for non-stationary PDE equivalent to Dirichlet boundary?

When using no flux Neumann boundary conditions (i.e. zero derivative to the normal on the boundary) in a non-stationary PDE, I don't seem to recognize the difference to using Dirichlet boundary ...
3
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1answer
162 views

How to use numerical integration to calculate the surface area of a superellipsoid?

I am working in an application in which I need to calculate the surface area of a superellipsoid. I have read that there is no closed form solution (see here), so I am trying to compute it using ...
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2answers
58 views

Grid Independence Study

Is the change in time step necessary for the grid independent study? As the CFL is based on the relation between dt and dx. In mesh independent study, only change should be mesh i.e, dx isn't it so?

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