Báo cáo viên: Đinh Dũng
Thời gian: 9h30, Thứ 3 ngày 6/3/2018 Địa điểm: Phòng 201, Nhà A5, Viện Toán học, 18 Hoàng Quốc Việt, Cầu Giấy, Hà Nội Tóm tắt: We analyze the complexity of the sparsegrid interpolation and sparsegrid quadrature of countablyparametric functions which take values in separable Banach spaces with unconditional bases. Under the provision of a suitably quantified holomorphic dependence on the parameters, we establish dimensionindependent convergence rate bounds for sparsegrid approximation schemes. Analogous results are shown in the case that the parametric solutions are obtained as solutions of corresponding parametricholomorphic, nonlinear operator equations as considered in [A. Cohen and A. Chkifa and Ch. Schwab: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, Journ. Math. Pures et Appliquees 103(2) 400428 (2015)] by means of stable, finite dimensional approximations, for example nonlinear PetrovGalerkin projections. Error and convergence rate bounds for constructive and explicit multilevel, sparse tensor approximation schemes combining sparsegrid interpolation in the parameter space and general, multilevel discretization schemes in the physical domain are proved. The results considerably generalize several earlier works in terms of the admissible multilevel approximations in the physical domain (comprising general stable PetrovGalerkin and discrete PetrovGalerkin schemes, collocation and stable domain approximations) and in terms of the admissible operator equations (comprising smooth, nonlinear locally wellposed operator equations). Additionally, a novel, general computational strategy to localize sequences of nested index sets is given for the anisotropic Smolyak scheme realizing best nterm benchmark convergence rates. We also consider Smolyaktype quadratures in this general setting, for which we establish improved convergence rates based on cancellations in gpc expansions due to symmetries of the probability measure [J. Zech and Ch. Schwab: Convergence rates of high dimensional Smolyak quadrature, Report 201727, SAM ETH Z¨urich].
Several examples illustrating the abstract theory include domain uncertainty quantification (“UQ” for short) for general, linear, second order, elliptic advectionreactiondiffusion equations on polygonal domains, where optimal convergence rates of FEM are known to require local mesh refinement near corners. For these equations, we also consider a combined sparsegrid scheme in physical and parameter space, affording complexity similar to the recent multiindex stochastic collocation approach. Further applications of the presently developed theory comprise evaluations of posterior expectations in Bayesian inverse problems.
