In this paper, we propose a variant of the Empirical Mode Decomposition method to decompose multiscale data into their intrinsic mode functions. Under the assumption that the multiscale data satisfy certain scale separation property, we show that the proposed method can extract the intrinsic mode functions accurately and uniquely.

We propose a data-driven stochastic method (DSM) to study stochastic partial differential equations (SPDEs) in the multiquery setting. An essential ingredient of the proposed method is to construct a data-driven stochastic basis under which the stochastic solutions to the SPDEs enjoy a compact representation for a broad range of forcing functions and/or boundary conditions. Our method consists of offline and online stages. A data-driven stochastic basis is computed in the offline stage using the Karhunen-Loève (KL) expansion. A two-level preconditioning optimization approach and a randomized SVD algorithm are used to reduce the offline computational cost. In the online stage, we solve a relatively small number of coupled deterministic PDEs by projecting the stochastic solution into the data-driven stochastic basis constructed offline. Compared with a generalized polynomial chaos method (gPC), the ratio of the computational complexities between DSM (online stage) and gPC is of order O((m/Np) 2 ). Here m and Np are the numbers of elements in the basis used in DSM and gPC, respectively. Typically we expect m Np when the effective dimension of the stochastic solution is small. A timing model, which takes into account the offline computational cost of DSM, is constructed to demonstrate the efficiency of DSM. Applications of DSM to stochastic elliptic problems show considerable computational savings over traditional methods even with a small number of queries. We also provide a method for an a posteriori error estimate and error correction.

In this paper, we present an adaptive, analysis of variance (ANOVA)-based data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique. To improve the slow convergence of the generalized polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the data-driven stochastic method (DSM) for speed up. An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions.Our ANOVA-DSM consists of offline and online stages. In the offline stage, the original high-dimensional stochastic problem is decomposed into a series of low-dimensional stochastic subproblems, according to the ANOVA decomposition technique. Then, for each subproblem, a data-driven stochastic basis is computed using the Karhunen-Loève expansion (KLE) and a two-level preconditioning optimization approach. Multiple trial functions are used to enrich the stochastic basis and improve the accuracy. In the online stage, we solve each stochastic subproblem for any given forcing function by projecting the stochastic solution into the data-driven stochastic basis constructed offline. In our ANOVA-DSM framework, solving the original highdimensional stochastic problem is reduced to solving a series of ANOVA-decomposed stochastic subproblems using the DSM. An adaptive ANOVA strategy is also provided to further reduce the number of the stochastic subproblems and speed up our method. To demonstrate the accuracy and efficiency of our method, numerical examples are presented for one- and two-dimensional elliptic PDEs with random coefficients.

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