Questions tagged [stochastic]
For questions regarding the numerical treatment of processes whose behaviors are determined by both deterministic (predictable) and non-deterministic (random) actions.
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Simulate Jump-Diffusion $dX_t = \mu(X_t)dt + \sigma(X_t)dW(t) + \int_{\{|c| <1 \}}g(X_t,c)\tilde{N}(dt,dc) + \int_{\{|c| \ge 1 \}}h(X_t,c)N(dt,dc)$
I would like to be able to model an SDE having the form
$$dX_t = \mu(X_t)dt + \sigma(X_t)dW(t) + \int_{\{|c| <1 \}}g(X_t,c)\tilde{N}(dt,dc) + \int_{\{|c| \ge 1 \}}h(X_t,c)N(dt,dc).$$
where $W$ is a ...
- 155
4
votes
1
answer
76
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Stochastic cellular automata - algorithm limited by 1 cell per timestep
Context
Let's say I am trying to model the spread of mold in a petri dish, using a stochastic cellular automata approach. The petri dish can be thought of as a grid of 1mm x 1mm squares, each called ...
- 197
1
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0
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102
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Stochastic differential equation system (SDE) : overflow encountered in double-scalars
I'm trying to integrate the following SDE system from Dekker et al. [1]
$$\begin{cases}
\frac{dx}{dt}=a_1x^3+a_2x+\phi+\zeta_x\\
\frac{dy}{dt}=b_1z+b_2(\kappa(x)-(y^2+z^2))y+\zeta_y\\
\frac{dz}{dt}=...
- 153
1
vote
1
answer
73
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Rate of convergence - Stochastic Euler Method
The absolute error criterion of the pathwise approximation of an Ito process $X$ by an Euler approximation $Y$ is:
$$
\epsilon=E\left(\left|X_{T}-Y(T)\right|\right)
$$
We shall say that a time-...
- 111
1
vote
1
answer
53
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Numerical stability of taking the `mean` of outputs from the simulation of a discrete stochastic dynamical system
I am writing a simulation for a discrete stochastic dynamical system. Since the simulation is stochastic, I need to run the simulation multiple times and then average the values of each timestep. I ...
- 245
4
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1
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48
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Guidelines for publishing data from a stochastic simulation
So, my question is if one should ideally keep a record of all seeds that are used when publishing numerical work that involves one or more random number generators (e.g. a stochastic simulation), and ...
- 161
2
votes
1
answer
78
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Adaptive Runge-Kutta for Stochastic (Projected) Gross-Pitaevskii Equation
I am using the XMDS library for solving the stochastic (projected) Gross-Pitaevskii equation
$$i \hbar \partial \Phi\left(\mathbf{r},t\right)_t=\hat{\mathcal{P}}\left\{(1-i \gamma)\left(\hat{H}_{\...
- 21
0
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1
answer
136
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Numerically solving nonlinear parabolic stochastic PDEs
For a project I'm doing, I have to numerically solve a nonlinear parabolic stochastic partial differential equation, of the form
$$
u_t = u_{xx} + f(u)(u_x)^2 + a(u) + b(u)W(t, x),
$$
where primes ...
1
vote
0
answers
25
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Effective fitness formula in the Moran process on a game
I was recently reading some literature about evolutionary game theory and I got particularly interested in Moran process linked to prisoner's dilemma as a model of evolution of two sub-populations.
...
- 111
5
votes
0
answers
88
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Probability approximation: monte carlo VS sde
I have a probability measure $\mu$ (say, in $\mathbb{R}^{d}$, with density) and I want to approximate it numerically. Today I noticed that my measure is ergotic for a certain Stochastic Differential ...
- 121
2
votes
0
answers
104
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Numerical integration of SDE: choice of $dt$ and algorithm
I am working on the following Stochastic Differential Equation (SDE) in the Quantum Mechanics context:
$$dX_{t} = a X_{t} dt + b X_{t} dW$$
where $X_{t}$ is my stochastic varible, $dt$ is my ...
- 21
1
vote
1
answer
60
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What is the right way to set up two random tensor fields which have an identical average diffusivity
I want to compare some properties of traveling waves through two randomly diffusive media. The traveling waves follow the fisher equation:
$$\frac{du}{dt} = \nabla(\mathbf{D}_{\gamma} \nabla u) + u(1-...
- 1,936
3
votes
2
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207
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How to optimize sampling for parameter estimation
I have a computer model with a number of parameters that need to be calibrated based on experimental results. It's also important to understand the sensitivity of the results to each parameter ...
- 41
2
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1
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97
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How to get started with numerically solving a Stochastic Navier Stokes equation
I originally posted the question on the math stackexchange, and was told I should try here.
I’m researching Stochastic PDE, in particular the Navier Stokes Equation, and would like to estimate the ...
2
votes
1
answer
1k
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Simultaneously maximize and minimize
I am virtually new to optimization (saw it years ago in a very shallow course) and now I came across a problem that I believe would require from it. The problem is I don't know exactly how to proceed.
...
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2
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0
answers
74
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How to determine the order of convergence of the Euler-Maruyama method?
This question is originally posted in Quant.StackExchange but has been unanswered for some time so I ask in here.
To make this simple let us consider the Geometric Brownian Motions (GBM).
My ...
- 255
2
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0
answers
46
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Is this a form of stochastic gradient descent?
I want to minimize the following with respect to parameters $B$.
$$\sum_{k = 1}^{K} f(A_{k}, B)$$
where $A_k$ are $K$ different data-sets and $B$ is a matrix of parameters.
Can I do this by a ...
- 163
5
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0
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42
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Stochastic conjugate directions to improve convergence in narrow valleys
My question concerns a specific statement in this paper:
N. N. Schraudolph and T. Graepel, "Conjugate Directions for Stochastic Gradient Descent," in Int. Conf. Artificial Neural Networks, Berlin, ...
- 51
2
votes
1
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171
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Jump-Diffusion process: practical solver beyond Euler method?
A jump-diffusion process is a stochastic process where both continuous noise (in my case complex Wiener noise $dZ,dZ^*$ such that $dZ^2=dZ^{*2}=0,|dZ|^2=dt$) and discrete Jumps (in my case Poissonian $...
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3
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1
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Computation of the heat kernel from Brownian motion
This question is rather simple but I have some difficulties to find code. Let us suppose that I wrote a routine, in a given language, that computes the evolution of a particle doing Brownian motion in ...
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0
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1
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51
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Implementation of stochastic cellular automata
In my problem, I have a lattice with a stochastic cellular automaton. In order to simplify a bit, let's say it is 1D. In my system, each node can be type A, B or C. A way to represent the system and ...
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1
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1
answer
54
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Open source solver for continuous-time stochastic non-linear DAEs (SDAEs)
I am trying to solve a system of non-linear index-1 DAEs in which the derivatives of the state variables, $x(t)$ are corrupted by additive noise, $w(t)$ (whose covariance matrix is known).
$\dot x(t) =...
3
votes
2
answers
410
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Runge Kutta and Milstein – system of second-order coupled differential equations with noise
I would like to solve a system of second-order differential equations to describe the dynamics of a system of particles.
Two Newton-like forces are responsible for the motion of each particle $i$: A ...
- 31
3
votes
1
answer
62
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Linear programming with stochasticity?
Suppose I have implemented an LP, where some constraint coefficients are implemented as the mean of some probability distribution.
Now, I would like to solve the same problem but with stochasticity ...
- 133
1
vote
1
answer
139
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How is KDE used in stochastic tomography
I am currently writing my masters thesis and my topic also touches on Stochastic Tomography for volume reconstruction presented in this
paper. Now i understand most of the process described, but i ...
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8
votes
2
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208
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What’s so great about derivative-free solvers for SDEs?
I am trying to familiarise myself with SDEs and have been reading a few review papers on the topic. They leave the impression that a great deal of work has been put into solvers that are derivative-...
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Stochastic gradient descent for large deterministic optimization problems
The Wikipedia page for SGD describes optimizing a function $f = \sum f_i(\theta;x_i)$ by successively approximating gradients from random subsets of the data, while most literature poses the problem ...
- 233
2
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0
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79
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Global optimization with known distributions of some variables
I'm solving simple single-objective multidimensional global optimization problem using various stochastic algorithms like Monte-Carlo, GA and other evolutionary approaches. The task is formulated as ...
- 21
2
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1
answer
83
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From deterministic to stochastic LP formulations
I am having a hard time understanding the very first example in "A Tutorial on Stochastic Programming".
More specifically the authors show that one can formulate the stochastic variant of (1.2) ...
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2
votes
2
answers
170
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What kind of optimisation algorithm is suitable for a computationally expensive function?
I have a reference value $R$ and a modelled value $M$. $M$ is generated using a stochastic algorithm with parameters $a$ and $b$.
The objective is to tune $a$ and $b$ so that $M$ is as close as $R$ ...
- 23
2
votes
1
answer
220
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Problem with Richardson extrapolation method for weak convergence in SDE
I have implemented the Richardson extrapolation of the Euler-Maruyama method to 4th order, to estimate the moments of SDE.
The Euler-Maruyama works, and I would expect the Richardson extrapolation to ...
- 123
1
vote
1
answer
2k
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Solving second order SDE with Gaussian white noise for first time derivative in Matlab
I'm having trouble solving a second order differential equation with Gaussian white noise. The equation I'm solving follows the form:
$$Ax'' + Bx' + \sin(x) = i + i_{n}$$
where $i_{n}$ is the ...
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2
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0
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non convex, non linear optimization involving matrix differential equation solution
I'm trying to develop an inferential procedure for a multivariate dependent Markov process. Basically, the procedure could be considered as a non linear regression, with a known dependence structure ...
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votes
4
answers
7k
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Algorithm for high quality 1/f noise?
How can I generate arbitrarily high quality $1/f$ noise, for use in a model?
My model involves a lot of feedback, over a large number of iterations, with a very high bandwidth, so I'd like the $1/f$ ...
- 173
2
votes
0
answers
86
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How to fix time intervals to store data in a stochastic simulation (continous time markov chain)
I am using FORTRAN to implement Gillespie's stochastic simulation algorithm. I would be running many simulations in parallel (both parallel instances with different seed and parallel functions); if I ...
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1
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How to integrate numerically a function with error bars?
Typically, the function that one wants to integrate numerically, $f$, is given, i.e. its values for various points $\{x_i\}$ are known precisely. The resulting error is due to the fact that we chose a ...
- 151
1
vote
0
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2k
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How to solve an ode with stochastic time-dependent input
I am trying to repeat an example I found in a paper.
I have to solve this ODE:
$25 \ddot{x} + 15 \dot{x} + 330000 x = p(t)$
where $p(t)$ is a white noise sequence band-limited into the 10-25 Hz ...
- 382
2
votes
1
answer
369
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Is Langevin thermostat/equation correct when trying to model time-dependent behaviour of a molecule?
I've been taught that when simulating a biomolecule in thermal equilibrium, it's best to use the Langevin thermostat - an algorithm which produces a trajectory, which is a realization of a stochastic ...
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0
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Solving a nonlinear equation with a Markov process and RVs
Assume that we have the following equation and the following assumption. The scope is to solve for some particular variables expressed later.
Update
$$E_{t}\left[ b(A_{t+1})^{1-\gamma} *R_{t+1}^{-\...
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4
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1
answer
240
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Solving a nonlinear equation with random variable
I would like to solve an equation that looks like this
UPDATE
$E[(R^{1-\gamma})(r_k+\theta-r_z)]=0$ , where $R=\phi r_z+(1-\phi)(r_k+\theta)$ and $\phi\in[0,1]$,
$\theta$, is a random variable ...
- 235
2
votes
2
answers
359
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Extracting time scales information from empirical cumulative distribution function
I have a stochastic process (a Markov chain actually) that has two absorbing states. I am using a difference equation to calculate the first passage time to either of the absorbing states. There are ...
- 269
2
votes
2
answers
2k
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visualization of 3D probability flow
I have a master equation for $P(N_A^+,N_B^+,N_C^+,t)$, with $N_A^+,N_B^+,N_C^+$ all discrete. The numerical integration is done by this Matlab program using Euler's method. Despite the crude Euler's ...
- 269
4
votes
1
answer
438
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Slight mistake in Stochastic Galerkin code
I'm following Paul Constantine's Primer on Stochastic Galerkin Method, Section 3.1 (2D Poisson Example). In this matlab code, the example attempts to solve the PDE
$$\alpha(w)(u_{xx}+u_{yy})=1 \text{ ...
Paul♦
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3
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0
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87
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Stochastic Collocation for time evolving ODE
For an Stochastic Differential Equation, e.g.,
$$
\frac{du}{dt} = \alpha*\sin(u*t)
$$
where $\alpha$ is normally distributed with nonzero mean, I am trying to use a stochastic collocation approach ...
- 31
7
votes
2
answers
1k
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Examples of numerical solution of stochastic differential equation(SDE)?
I want to simulate a nonlinear stochastic differential equation
$$
{\rm d}X_t = f(X_t) {\rm d}t + g(X_t){\rm d}B_t
$$
where $f,g \in C^{\infty}({\mathbb R}^n ,{\mathbb R})$ and $B_t$ is one-...
11
votes
2
answers
528
views
Numerical method for equation solving that works on stochastically computed functions
There are many well known numerical methods for solving equations of the type
$$ f(x) = 0, \quad x \in \mathbb{R}^n,$$
e.g. bisection method, Newton's method, etc.
In my application $f(x)$ is ...
- 2,550
5
votes
2
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294
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Convergence tests in Markov Chain Monte Carlo
For a relatively simple Markov chain Monte Carlo process, such as using Metropolis to find calculate thermal averages for an Ising model, how is it possible to determine whether quantities have ...
Matthew Matic
2
votes
1
answer
172
views
How to solve this numerical technique problem?
Well, in a numerical technique test we were given the following problem:
A physical phenomenon is modeled such that,
$F(f,d) = A(f)/d^2 + L$;
Where, $F$ is a function of frequency $f$ and distance $...
- 121
6
votes
0
answers
678
views
Stochastic Galerkin projection approach for using generalized polynomial chaos expansion (GPCE) in solving PDE
I want to know if there is any way to define the test and trial function in the way that I want instead of using the default functions. So if I want define the polynomial and basis and coefficient, ...
- 111