Quasi-Monte Carlo for unbounded integrands with importance sampling
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929581099843584 |
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| author | Ouyang, Du Wang, Xiaoqun He, Zhijian |
| author_facet | Ouyang, Du Wang, Xiaoqun He, Zhijian |
| contents | We consider the problem of estimating an expectation $ \mathbb{E}\left[ h(W)\right]$ by quasi-Monte Carlo (QMC) methods, where $ h $ is an unbounded smooth function on $ \mathbb{R}^d $ and $ W$ is a standard normal distributed random variable. To study rates of convergence for QMC on unbounded integrands, we use a smoothed projection operator to project the output of $W$ to a bounded region, which differs from the strategy of avoiding the singularities along the boundary of the unit cube $ [0,1]^d $ in 10.1137/S0036144504441573. The error is then bounded by the quadrature error of the transformed integrand and the projection error. If the function $h(\boldsymbol{x})$ and its mixed partial derivatives do not grow too fast as the Euclidean norm $|\boldsymbol{x}|$ goes to infinity, we obtain an error rate of $O(n^{-1+ε})$ for QMC and randomized QMC (RQMC) with a sample size $n$ and an arbitrarily small $ε>0$. However, the rate turns out to be $O(n^{-1+2M+ε})$ if the functions grow exponentially with a rate of $O(\exp\{M|\boldsymbol{x}|^2\})$ for a constant $M\in(0,1/2)$. Superisingly, we find that using importance sampling with t distribution as the proposal can improve the root mean squared error of RQMC from $O(n^{-1+2M+ε})$ to $O( n^{-3/2+ε})$ for any $M\in(0,1/2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_00650 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quasi-Monte Carlo for unbounded integrands with importance sampling Ouyang, Du Wang, Xiaoqun He, Zhijian Numerical Analysis We consider the problem of estimating an expectation $ \mathbb{E}\left[ h(W)\right]$ by quasi-Monte Carlo (QMC) methods, where $ h $ is an unbounded smooth function on $ \mathbb{R}^d $ and $ W$ is a standard normal distributed random variable. To study rates of convergence for QMC on unbounded integrands, we use a smoothed projection operator to project the output of $W$ to a bounded region, which differs from the strategy of avoiding the singularities along the boundary of the unit cube $ [0,1]^d $ in 10.1137/S0036144504441573. The error is then bounded by the quadrature error of the transformed integrand and the projection error. If the function $h(\boldsymbol{x})$ and its mixed partial derivatives do not grow too fast as the Euclidean norm $|\boldsymbol{x}|$ goes to infinity, we obtain an error rate of $O(n^{-1+ε})$ for QMC and randomized QMC (RQMC) with a sample size $n$ and an arbitrarily small $ε>0$. However, the rate turns out to be $O(n^{-1+2M+ε})$ if the functions grow exponentially with a rate of $O(\exp\{M|\boldsymbol{x}|^2\})$ for a constant $M\in(0,1/2)$. Superisingly, we find that using importance sampling with t distribution as the proposal can improve the root mean squared error of RQMC from $O(n^{-1+2M+ε})$ to $O( n^{-3/2+ε})$ for any $M\in(0,1/2)$. |
| title | Quasi-Monte Carlo for unbounded integrands with importance sampling |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2310.00650 |