Questions tagged [stochastic]

For questions regarding the numerical treatment of processes whose behaviors are determined by both deterministic (predictable) and non-deterministic (random) actions.

Filter by
Sorted by
Tagged with
11
votes
2answers
507 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 ...
8
votes
2answers
198 views

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-...
7
votes
4answers
5k views

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 $1/f$ ...
7
votes
1answer
341 views

Richardson extrapolation for strong rate of convergence of SDE

Is it possible to apply Richardson extrapolation with Euler-Maruyama scheme to improve strong rate of convergence of stochastic differential equations?
6
votes
2answers
1k views

What is the deterministic counterpart of Robbins-Monro algorithm?

From Wikipedia, assume that we have a function $M(x)$, and we want to solve the equation $M(x) = 0$. But we cannot directly observe the function $M(x)$, we can instead obtain measurements of the ...
6
votes
0answers
638 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, ...
5
votes
4answers
3k views

Simulated Annealing proof of convergence

I implemented downhill simplex simulated annealing algorithm. Algorithm is very hard to tune, w.r.t. parameters including cooling schedule, starting temperature... My first question is about ...
5
votes
2answers
1k views

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-...
5
votes
2answers
272 views

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 ...
5
votes
0answers
81 views

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 ...
5
votes
0answers
35 views

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, ...
4
votes
3answers
12k views

How to choose a good step size for stochastic gradient descent?

For the purpose of model fitting in a large time series dataset, I am using stochastic gradient descent of the negative log likelihood. The model is nonlinear and non-convex. Is there a thumb rule for ...
4
votes
1answer
204 views

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 ...
4
votes
1answer
430 views

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{ ...
4
votes
1answer
6k views

Differences between “least square”, “mean square” and “least mean square”?

I was wondering what differences are between the terminology: "least square (LS)" "mean square (MS)" and "least mean square (LMS)"? I get confused when reading in Spall's Introduction to Stochastic ...
4
votes
1answer
43 views

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 ...
4
votes
1answer
362 views

Convert ODE into discrete probabilistic model

how can I turn an ODE equation into a discrete probabilistic model? I take for example the Verhulst equation for the growth of a population. $$\frac{dP}{dt} = rP(1-P/K)$$ I was thinking to simulate ...
3
votes
2answers
317 views

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 ...
3
votes
1answer
99 views

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 ...
3
votes
1answer
60 views

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 ...
3
votes
1answer
278 views

Sampling strategies to solve a stochastic partial differential equation

Suppose I had a stochastic partial differential equation of the form: $\nabla^2U=F(x,D)$, where $x\in\Omega\equiv [0,1]$ and $F(x,D)$ is a function which depends on position $x$ and a uniform random ...
3
votes
1answer
967 views

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 ...
3
votes
1answer
152 views

Introduction for (numerical) linear algebra of random variables

I am in search of an introduction into numerical linear algebra - or, at least, pure linear algebra - that treats the case when the input data are random variables. A typical application would be to ...
3
votes
1answer
829 views

Where can I find coded examples of stochastic collocation applied to an elliptical PDE using smolyak sampling?

I'm having some troubles implementing a collocation method to solve a stochastic partial differential equation of the form: $\nabla (a(x,w)\nabla u(x,w))=f(x,w)$ in $D$, $u=g$ in $\partial D$ where $...
3
votes
2answers
198 views

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 ...
3
votes
0answers
121 views

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 ...
3
votes
0answers
81 views

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 ...
2
votes
2answers
140 views

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 $R$ ...
2
votes
2answers
2k views

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 ...
2
votes
2answers
306 views

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 ...
2
votes
1answer
73 views

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
1answer
124 views

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 $...
2
votes
1answer
71 views

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) ...
2
votes
1answer
158 views

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 ...
2
votes
1answer
348 views

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 ...
2
votes
1answer
409 views

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. ...
2
votes
1answer
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 $...
2
votes
1answer
99 views

Optimal sample size for Stochastic Steepest Descent

Suppose $g(x_{1:n})$ is the estimate of a gradient, which is calculated at each step of a Stochastic Steepest Descent algorithm. A dataset $x_{1:n}$ is simulated at each step, so if $n$ is small the ...
2
votes
1answer
47 views

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}_{\...
2
votes
0answers
70 views

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 ...
2
votes
0answers
47 views

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 ...
2
votes
0answers
40 views

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 ...
2
votes
0answers
66 views

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 ...
2
votes
0answers
94 views

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 ...
2
votes
0answers
82 views

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 ...
2
votes
0answers
65 views

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}^{-\...
1
vote
1answer
56 views

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
vote
1answer
1k views

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 ...
1
vote
3answers
286 views

Translating from SBML to Petri Net (matrix representation)

Is there a piece of software that can easily take a representation of a biochemical network from a SBML file and translate it into a Petri Net? Specifically, I'm looking for something that can give ...
1
vote
1answer
127 views

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 ...