12 research outputs found
Sparse Generalized Multiscale Finite Element Methods and their applications
In a number of previous papers, local (coarse grid) multiscale model
reduction techniques are developed using a Generalized Multiscale Finite
Element Method. In these approaches, multiscale basis functions are constructed
using local snapshot spaces, where a snapshot space is a large space that
represents the solution behavior in a coarse block. In a number of applications
(e.g., those discussed in the paper), one may have a sparsity in the snapshot
space for an appropriate choice of a snapshot space. More precisely, the
solution may only involve a portion of the snapshot space. In this case, one
can use sparsity techniques to identify multiscale basis functions. In this
paper, we consider two such sparse local multiscale model reduction approaches.
In the first approach (which is used for parameter-dependent multiscale
PDEs), we use local minimization techniques, such as sparse POD, to identify
multiscale basis functions, which are sparse in the snapshot space. These
minimization techniques use minimization to find local multiscale basis
functions, which are further used for finding the solution. In the second
approach (which is used for the Helmholtz equation), we directly apply
minimization techniques to solve the underlying PDEs. This approach is more
expensive as it involves a large snapshot space; however, in this example, we
can not identify a local minimization principle, such as local generalized SVD
Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
In this paper, we discuss a general multiscale model reduction framework
based on multiscale finite element methods. We give a brief overview of related
multiscale methods. Due to page limitations, the overview focuses on a few
related methods and is not intended to be comprehensive. We present a general
adaptive multiscale model reduction framework, the Generalized Multiscale
Finite Element Method. Besides the method's basic outline, we discuss some
important ingredients needed for the method's success. We also discuss several
applications. The proposed method allows performing local model reduction in
the presence of high contrast and no scale separation
Non-stationary demand forecasting by cross-sectional aggregation
In this paper the relative effectiveness of top-down (TD) versus bottom-up (BU) approaches is compared for cross-sectionally forecasting aggregate and sub-aggregate demand. We assume that the sub-aggregate demand follows a non-stationary Integrated Moving Average (IMA) process of order one and a Single Exponential Smoothing (SES) procedure is used to extrapolate future requirements. Such demand processes are often encountered in practice and SES is one of the standard estimators used in industry (in addition to being the optimal estimator for an IMA process). Theoretical variances of forecast error are derived for the BU and TD approach in order to contrast the relevant forecasting performances. The theoretical analysis is supported by an extensive numerical investigation at both the aggregate and sub-aggregate level, in addition to empirically validating our findings on a real dataset from a European superstore. The results demonstrate the increased benefit resulting from cross-sectional forecasting in a non-stationary environment than in a stationary one. Valuable insights are offered to demand planners and the paper closes with an agenda for further research in this area. © 2015 Elsevier B.V. All rights reserved