MATHICSE Technical Report : Analysis of discrete least squares on multivariate polynomial spaces with evaluations at low-discrepancy point sets

We analyze the stability and accuracy of discrete least squares on multivariate poly- nomial spaces to approximate a given function depending on a multivariate random variable uniformly distributed on a hypercube. The polynomial approximation is calculated starting from pointwise noise-free evaluations of the target function at low- discrepancy point sets. We prove that the discrete least-squares approximation, in a multivariate anisotropic tensor product polynomial space and with evaluations at low-discrepancy point sets, is stable and accurate under the condition that the number of evaluations is proportional to the square of the dimension of the polynomial space, up to logarithmic factors. This result is analogous to those obtained in [7, 22, 19, 6] for discrete least squares with random point sets, however it holds with certainty instead of just with high probability. The result is further generalized to arbitrary polynomial spaces associated with downward closed multi-index sets, but with a more demanding (and probably nonoptimal) proportionality between the number of evaluation points and the dimension of the polynomial space

Duality theory and propagation rules for higher order nets

AbstractHigher order nets and sequences are used in quasi-Monte Carlo rules for the approximation of high dimensional integrals over the unit cube. Hence one wants to have higher order nets and sequences of high quality.In this paper we introduce a duality theory for higher order nets whose construction is not necessarily based on linear algebra over finite fields. We use this duality theory to prove propagation rules for such nets. This way we can obtain new higher order nets (sometimes with improved quality) from existing ones. We also extend our approach to the construction of higher order sequences

被積分関数の高層偏微分のL1ノルムの増大度が高々指数的である場合の多次元数値積分の加速的な収束と計算容易性

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 坪井 俊, 東京大学教授 山本 昌宏, 東京大学教授 吉田 朋広, 東京大学准教授 長谷川 立, 東京大学准教授 齊藤 宣一, 東京大学名誉教授 楠岡 成雄, 広島大学教授 松本 眞University of Tokyo(東京大学