Questions tagged [simulation]
The simulation tag has no usage guidance.
54
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From biased coins (and nothing else) to biased coins
Background
We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads ...
11
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3
answers
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On mathematical studies of the Mpemba effect
Since the days of Aristotle and Descartes, it has been known that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named ...
10
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1
answer
915
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exactly simulating a random walk from infinity
In diffusion-limited aggregation on the square lattice, one lets a particle do "random walk from infinity" until it hits the current aggregate, at which point the site occupied by the particle is ...
8
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1
answer
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What is the "Tangle" at the Heart of Quantum Simulation?
The following questions generalize and naturalize the question that was originally asked. Provisional answers largely due to Will Sawin are now included.
As was discussed in the question originally ...
8
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1
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Potts model simulation
I was wondering what were the state-of-the-art methods to simulate low temperature configurations of Potts-like models that exhibit a discontinuous phase transition. For models with a continuous phase ...
6
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1
answer
543
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Is this a Brownian motion?
I am building a 2D stochastic process as follows. I start with a point $P_0=(0,0)$. Then $P_k=(X_k,Y_k)$ is defined as follows, for $k>0$:
\begin{align}
X_k & =X_{k-1}+R_k \cos(2\pi\theta_k) \\
...
6
votes
1
answer
232
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Violating an order statistic inequality?
[Edit: for posterity, I'm adding two small comments to the code explaining how to fix it, in light of Iosef Pinelis' answer below. Look for "Should be:" to find the corrections.]
Suppose we ...
6
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2
answers
417
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how to sample a conditioned diffusion
there are several reasons why we could be interested in sampling conditioned diffusions:
if we observed a diffusion at discrete time and want to do some kind of inference on the parameters of the ...
5
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2
answers
2k
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Real world example of use of Monte Carlo method for high dimensional integrals
The Monte Carlo method for numerical integration is usually presented as a method invented to efficiently compute high dimensional integrals numerically. However, I haven't found any source which has ...
5
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1
answer
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How can I generate the simulated time series
I am curious how one can generate simulated time series data. I found a list of simulated series here and a similar tool for stock market. What is the best way to generate domain specific time series ...
4
votes
1
answer
153
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Intractability of an integral by deterministic numerical methods
Suppose $X_1,\ldots,X_n$ is an i.i.d. sample from a probability distribution with continuous c.d.f. $F.$ Let $F_n$ be the empirical c.d.f.
$$
F_n(x) = \frac 1 n \sum_{k=1}^n \mathbf 1_{X_n\le x} = \...
4
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1
answer
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How to simulate the fractional noncentral Wishart distribution?
I already asked this question on math.stackexchange but got no answer.
For a non-integer number of degrees of freedom $\nu > p-1$, one can simulate the central Wishart distribution $W_p(\nu, \...
4
votes
1
answer
659
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How to draw a random normal matrix?
I would like to pick a random real normal (i.e. commuting with its transpose) matrix and I wonder if it can be done easily. I thought whether it would be possible to use a similar trick to drawing a ...
4
votes
0
answers
225
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From biased coins to biased coins, as efficiently as possible
Background
We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads ...
4
votes
0
answers
106
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Designing Character Other Than Temperature for Simulated Annealing on Combinatorial Optimization
Many research on designing temperature for simulated annealing is carried out. We wonder if there is any research on designing general feature of the Hamiltonian used in Simulated Annealing.
For ...
4
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0
answers
728
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Monte Carlo sampling high dimensions with the halton sequence?
Referring to the Halton Sequence, Swiler et al 2006 state that
In cases where a large number of input variables are sampled,
Robinson and Atcitty recommend using a leaped sequence, where the
...
3
votes
2
answers
717
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Is it possible to reliably generate a particular approximation of an ideal knot via a simulated annealing approach?
Say I take a cord, tie a loose knot in three-dimensional space, and pull tightly on the ends to generate an approximation of an ideal knot. If the cord has a fixed knot topology and a random initial ...
3
votes
1
answer
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What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?
This question was posted in MSE but is still open hence posting in MO.
The area of the largest triangle that can be inscribed in a circle of raidus $1$ is $\displaystyle \frac{3 \sqrt{3}}{4}$ for a ...
3
votes
2
answers
164
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Drawing random variates from a partially described probability distribution
I have a probability distribution over $\{0,1\}^n$ but instead of knowing the full joint distribution $p(x_1,\dots,x_n)$, I only know $p(x_i=x_j)$ for each $i,j$. How could I draw a random binary ...
3
votes
1
answer
85
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Finite differencing scheme for Hamilton's equation with planar linkages
I am trying to simulate the movement of a planar linkage in the plane whose position and momentum obey Hamilton's equations, which is to say that $${{dq}\over{dt}} = {{dH}\over{dp}}$$ and $${{dp}\...
3
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1
answer
820
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Sampling from a particular multivariate probability distribution
Given $3$ real variables $x_1, x_2, x_3 \equiv \bf{x}$, consider their probability density function (PDF)
\begin{equation}
P({\bf x}) = C \, p(x_1) \cdots p(x_3) \exp[f({\bf x})],
\end{equation}
where ...
3
votes
0
answers
355
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A conjecture on consistent monotone sequences of polynomials in Bernstein form
A Conjecture
In the following, a polynomial $P(x)$ is written in Bernstein form of degree $n$ if it is written as— $$P(x)=\sum_{k=0}^n a_k {n \choose k} x^k (1-x)^{n-k},$$ where $a_0, ..., a_n$ are ...
2
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1
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How far away is $\max_{x: x \in \{0, \ldots, N\}} |W(x/N)|$ from $\max_{0 \leq t \leq 1} |W(t)|$ ($W(t)$ a Wiener process)?
How far away is
$$\max_{x: x \in \{0, \ldots, N\}} \left|W\left(\frac{x}{N}\right)\right|$$
from
$$\max_{0 \leq t \leq 1} |W(t)|$$
In other words, if you simulate a Wiener process over a finite ...
2
votes
1
answer
463
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Multiple Wiener-Ito integral distribution
Distribution of standard Ito integral is well known: $$I_1(f) = \int_0^T f(t)dB(t) \sim \mathcal{N}\bigg(0, \int_0^T f^2(t)dt\bigg).$$
Is it possible to find the distribution of multiple Wiener-Ito ...
2
votes
1
answer
2k
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Design a Galton Board to simulate a uniform distribution
This link http://mathworld.wolfram.com/GaltonBoard.html suggests that a certain specific placement of pegs can be used to simulate a binomial or a normal distribution. Is there a specific peg ...
2
votes
0
answers
37
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Discrete approximation of continuous determinantal point processes
(throughout, "DPP" denotes "Determinantal Point Process")
TL;DR: Discrete DPPs are straightforward to compute with, continuous DPPs less so. Can we approximate continuous DPPs well ...
2
votes
0
answers
126
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Multiple Wiener integral as Witt polynomial of Brownian motion
I know that if i have a Brownian motion $W_t$ the multiple Wiener integral
$\int_0^t \int_0^{\xi_1}...\int_0^{\xi_n} dW_{\xi_1}...dW_{\xi_n}$
can be expressed as $H_n(\int_0^t dW_s)$ where $H_n$ is ...
2
votes
0
answers
514
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Convergence Based on Recurrence Relation
I am studying a sequence based on the following recurrence:
$$X[t] = \sqrt{\alpha X[t-1]^2+(X[t-1]^2-\alpha X[t-2]^2)\frac{(2-X[t-1])^2}{X[t-1]^2}}$$
$$X[0]=0$$
$$X[1]>0$$
$$\alpha \in (0,1)$$
I ...
2
votes
0
answers
53
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brownian motion of 100 nm spherical particles in evenly spaced arrays [closed]
Generally looking for perspective from the computational experts. Question comes down to how tractable is the following problem. Let's say at time 0, I have a 2-D array of $N = 10$ spherical particles ...
2
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0
answers
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Customers and Anti-Customer Queueing Problem: What is the Customer delete probability
Hello may I ask for your help?
First the setting:
I have got a problem with some queueing theory. The whole problem would be a grid of nodes, all nodes have an operation intensity $\mu_{i,j}$. ...
2
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0
answers
153
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A question on discrete numerical simulation on fluids mechanics
I read the paper "Stable, circulation-preseving simplicial fuids" by Elcott, et al: http://www.cs.jhu.edu/~misha/Fall09/Elcott07.pdf. It gives a structure preseving discretization of fluids. I have ...
1
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4
answers
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Will a random walk on [0, inf) tend to infinity? [closed]
Consider a random walk on [0, inf) where you start at 0. With probability p = 0.5, you increase by 1. With probability (1-p) = 0.5, you decrease by 1, but not below 0.
As time goes to infinity, will ...
1
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2
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Another question on provable non-existence of an efficient deterministic numerical method
Herewith I submit what may or may not be considered a simpler version of this question.
The question is whether it is provable that there is no efficient deterministic numerical method for a ...
1
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1
answer
181
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Sampling uniformly in a ball of radius $\epsilon$ in the space of dicrete r.v. of m modalities for the total variation metric
I am looking for some reference or an algorithm that allows to sample uniformly in the ball centered at a discrete random variable of n modalities in the TV distance.
For the record for 2 discrete ...
1
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1
answer
353
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[Numerical Mathemtics] How to solve hexagonal central differences
I want to simulate a 2d linear wave equation on a circle ($\displaystyle\frac{\partial^2 z(x,y,t)}{\partial t^2}=v^2\cdot\left(\displaystyle\frac{\partial^2 z(x,y,t)}{\partial x^2}+\displaystyle\frac{\...
1
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1
answer
2k
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Monte Carlo sampling from correlated empirical distributions
I have a dataset that contains six correlated variables, and I want to sample new data so that each variable has the same marginal distribution as the original data, and the correlations are also the ...
1
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1
answer
92
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Choose uniformly from fixed-length paths in $[0,n]\cap\mathbb{Z}$ with fixed start and end
Let $X_k$ be a symmetric (discrete time) random walk on $\mathbb{Z}$ and let $m,n\in\mathbb{N}$. I want to chose uniformly from the paths of $X_k$, which
start at $0$
stay in $[0,n]\cap\mathbb{Z}$ ...
1
vote
1
answer
295
views
Condition Number and CFL Condition in Finite difference Methods [closed]
when applying a Finite Difference scheme for an IVP, two factors come to mind when considering stability:
One factor would be the condition number of the approximation operator. The other factor ...
1
vote
1
answer
147
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Staggered timing on 2-D random walks by multiple agents
In 2-D lattice random walks by multiple drunks who can't step onto each other, mathematically I would just say the whole cellular automaton updates "at once".
But to simulate this on a computer, I ...
1
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0
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77
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Conditioned random walk over a graph
I want to solve for a conditioned random walk over a graph. I have a directed graph $G$. The random walkers start at a fixed node, Source. They all need to end up at fixed node, Sink. So the random ...
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0
answers
40
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Langevin dynamics or stochastic gradient flow for grand canonical ensemble
We know that for a measure exp(-U(X)) (canonical ensemble), we can use the dynamic dX=-DU(X)+ noise to sample the measure as t goes to infinity.
Is there any dynamic corresponding to the grand ...
1
vote
1
answer
71
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What new fractional brownian motion (fBm) simulation methods have emerged since 2010? [closed]
I want to describe new methods for simulating fBm, as in the work of Coerjolly and Dieker, but new methods are not very easy to find.
1
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0
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77
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Real life applications of distributions through models or simulations [closed]
What are the areas we can apply distributions in classical harmonic analysis? I don't mean probability distributions but distributions that are continuous linear functionals on the space of test ...
1
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0
answers
165
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A mathematical biology reference request
Is there any mathematical articles that describe the differential equation modelling of locomotion of amoeba using pseduopodia? I am looking for physics based models of pressure difference modeling of ...
1
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0
answers
115
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Monte Carlo Simulation - efficient simulation of tail outcomes [closed]
When running Monte Carlo type simulations in situations where you're only interested in tail outcomes, do you know of a way to only simulate those outcomes, so that you can come up with more reliable ...
1
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0
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88
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Importance sampling for bernoulli-sequence, favouring long sequences of ones
Assume we have a sequence of i.i.d. bernoulli-distributed random variables of length $n$.
I'm interested in doing rare event simulation and my event depends, among other random factors, on the ...
1
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0
answers
505
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9-point stencil "equivalent" for advection equation [closed]
So I inherited from some people a code that solves the advection-diffusion-reaction equation for a particular system. The original code was first implemented in 1D which worked fine in cartesian ...
0
votes
1
answer
695
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Generating random variables from the Cantor Distribution [closed]
I am looking for a method (exact, if possible, but at least asymptotically correct) for generating random variates from a Cantor Distribution? It seems like its abstract definition prevents this. In ...
0
votes
1
answer
200
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How are epidemic models simulated in case of mobility?
I am not a mathematician but out of curiosity I am trying to implement the SIS epidemic model when the nodes have mobility to understand how it will change the results. I understand how to perform ...
0
votes
1
answer
68
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Simulation of multivariate logistic distribution conditional to a plane
For an algorithm, I have to simulate $X_1, \ldots, X_n \sim_{\text{iid}} \text{Logistic}(0,1)$ conditionally to the event $(X_1, \ldots, X_n) \in P$, where $P$ is an affine plane in $\mathbb{R}^n$.
I ...