## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 3, Number 3 (1993), 897-913.

### Accelerating Gaussian Diffusions

Chii-Ruey Hwang, Shu-Yin Hwang-Ma, and Shuenn-Jyi Sheu

#### Abstract

Let $\pi(x)$ be a given probability density proportional to $\exp(-U(x))$ in a high-dimensional Euclidean space $\mathbb{R}^m$. The diffusion $dX(t) = -\nabla U(X(t))dt + \sqrt 2 dW(t)$ is often used to sample from $\pi$. Instead of $-\nabla U(x)$, we consider diffusions with smooth drift $b(x)$ and having equilibrium $\pi(x)$. First we study some general properties and then concentrate on the Gaussian case, namely, $-\nabla U(x) = Dx$ with a strictly negative-definite real matrix $D$ and $b(x) = Bx$ with a stability matrix $B$; that is, the real parts of the eigenvalues of $B$ are strictly negative. Using the rate of convergence of the covariance of $X(t)$ [or together with $EX(t)$] as the criterion, we prove that, among all such $b(x)$, the drift $Dx$ is the worst choice and that improvement can be made if and only if the eigenvalues of $D$ are not identical. In fact, the convergence rate of the covariance is $\exp(2\lambda_M(B)t)$, where $\lambda_M(B)$ is the maximum of the real parts of the eigenvalues of $B$ and the infimum of $\lambda_M(B)$ over all such $B$ is $1/m \operatorname{tr} D$. If, for example, a "circulant" drift $\bigg(\frac{\partial U}{\partial x_m} - \frac{\partial U}{\partial x_2},\frac{\partial U}{\partial x_1} - \frac{\partial U}{\partial x_3}, \cdots, \frac{\partial U}{\partial x_{m-1}} - \frac{\partial U}{\partial x_1}\bigg)$ is added to $Dx$, then for essentially all $D$, the diffusion with this modified drift has a better convergence rate.

#### Article information

**Source**

Ann. Appl. Probab., Volume 3, Number 3 (1993), 897-913.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005371

**Digital Object Identifier**

doi:10.1214/aoap/1177005371

**Mathematical Reviews number (MathSciNet)**

MR1233633

**Zentralblatt MATH identifier**

0780.60074

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J60: Diffusion processes [See also 58J65]

Secondary: 62E25 65C05: Monte Carlo methods 82B31: Stochastic methods 68U10: Image processing

**Keywords**

Diffusions convergence rate stochastic relaxation Monte Carlo method stability matrix Ornstein-Uhlenbeck process reversible process image analysis covariance matrix

#### Citation

Hwang, Chii-Ruey; Hwang-Ma, Shu-Yin; Sheu, Shuenn-Jyi. Accelerating Gaussian Diffusions. Ann. Appl. Probab. 3 (1993), no. 3, 897--913. doi:10.1214/aoap/1177005371. https://projecteuclid.org/euclid.aoap/1177005371