We study quasi-Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of depends on and the dimension . Strong tractability means that it does not depend on and is bounded by a polynomial in . The least possible value of the power of is called the -exponent of strong tractability. Sloan and Wozniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the -exponent of strong tractability is between 1 and 2. However, their proof is not constructive.
In this paper we prove in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with -exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Wozniakowski's assumption. We show that quasi-Monte Carlo algorithms using Niederreiter's -sequences and Sobol sequences achieve the optimal convergence order for any 0$"> independent of the dimension with a worst case deterministic guarantee (where is the number of function evaluations). This implies that strong tractability with the best -exponent can be achieved in appropriate weighted Sobolev spaces by using Niederreiter's -sequences and Sobol sequences.
Quasi-Monte Carlo (QMC) methods have been playing an important role for high-dimensional problems in computational finance. Several techniques, such as the Brownian bridge (BB) and the principal component analysis, are often used in QMC as possible ways to improve the performance of QMC. This paper proposes a new BB construction, which enjoys some interesting properties that appear useful in QMC methods. The basic idea is to choose the new step of a Brownian path in a certain criterion such that it maximizes the variance explained by the new variable while holding all previously chosen steps fixed. It turns out that using this new construction, the first few variables are more “important” (in the sense of explained variance) than those in the ordinary BB construction, while the cost of the generation is still linear in dimension. We present empirical studies of the proposed algorithm for pricing high-dimensional Asian options and American options, and demonstrate the usefulness of the new BB. 相似文献
Lateral-flow assays (LFAs), with its convenience and low cost, promise to become the in-home test format for early diagnosis and monitoring of tumor marker. However, the insufficient signal intensity was generated by signal reporters reducing the sensitivity of this format. In this study, a novel nanoscale signal reporter capable of amplifying the fluorescence signal is fabricated by encapsulating quantum dots (QDs) into modified tri-copolymer (poly(tert-butyl acrylate-co-ethyl acrylate-co-methacrylic acid)) (ODA-g-tri-copolymer). The amplified signal varied by simply adjusting the ratio of QDs to the ODA-g-tri-copolymer for obtaining QD nanospheres with high QD loading. They exhibits outstanding stability compared to the individual QDs both in the biological buffer and strong acid solutions. Here, human chorionic gonadotrophin (HCG) is employed as the model protein of LFAs. The results show that the detection limit of the QD nanospheres is pushed down to 0.016 IU/L, which is about ~?38.5 times enhanced compared to the individual QD-based LFAs without any signal amplifying. The ultrasensitive LFAs were attributed to the signal amplification strategy, and their efficiency and robustness demonstrated the great potential in clinical applications.
