2015/11/17 by Abdul-Lateef Haji-Ali, Haji-Ali, Abdul-Lateef, Fabio Nobile +5 · 1 citation
Computer Science · Decision Sciences · Economics, Econometrics and Finance · Mathematics · #41A10 #65C20 #65N05 #65N30 #Advanced Multi-Objective Optimization Algorithms #FOS: Mathematics #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Spatial and Panel Data Analysis #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.1511.05393
openalex publication_date 2015/11/17 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We analyze the recent Multi-index Stochastic Collocation (MISC) method for\ncomputing statistics of the solution of a partial differential equation (PDEs)\nwith random data, where the random coefficient is parametrized by means of a\ncountable sequence of terms in a suitable expansion. MISC is a combination\ntechnique based on mixed differences of spatial approximations and quadratures\nover the space of random data and, naturally, the error analysis uses the joint\nregularity of the solution with respect to both the variables in the physical\ndomain and parametric variables. In MISC, the number of problem solutions\nperformed at each discretization level is not determined by balancing the\nspatial and stochastic components of the error, but rather by suitably\nextending the knapsack-problem approach employed in the construction of the\nquasi-optimal sparse-grids and Multi-index Monte Carlo methods. We use a greedy\noptimization procedure to select the most effective mixed differences to\ninclude in the MISC estimator. We apply our theoretical estimates to a linear\nelliptic PDEs in which the log-diffusion coefficient is modeled as a random\nfield, with a covariance similar to a Mat 'ern model, whose realizations have\nspatial regularity determined by a scalar parameter. We conduct a complexity\nanalysis based on a summability argument showing algebraic rates of convergence\nwith respect to the overall computational work. The rate of convergence depends\non the smoothness parameter, the physical dimensionality and the efficiency of\nthe linear solver. Numerical experiments show the effectiveness of MISC in this\ninfinite-dimensional setting compared with the Multi-index Monte Carlo method\nand compare the convergence rate against the rates predicted in our theoretical\nanalysis.\n