| dc.description.abstract | In many applications arising from geosciences, one needs to solve problems with multiple scales. For examples, simulating the complex flow, porous media applications and so on. To capture the multiscale features of a problem with heterogeneous property by the traditional finite element method, one need to use the fine mesh. Consequently the computation is accurate but the efficiency is compromised because of the large number of the degree of freedoms used. One idea is to improve the computation efficiency while preserve the accuracy is to design basis which can resolve multiscale features of the problem on coarse mesh. A class of methods have been proposed based on this methodology and in this work, I am going to develop the multiscale methods on solving some important problems. The first two applications of the multiscale modeling in this work are based on the Quasi-gas-dynamic (QGD) model. In particular, we study QGD model in a multiscale environment. This is not only because of the challenges in solving and analyzing the equation; but also because of its wide applications in solving other types of equations, for example, paraxial wave equations. It should be noted that the key step in the multiscale methods is to find the multiscale basis defined on the coarse mesh. This is the most time consuming step which usually involves constructing snapshot or auxiliary space, and solving spectral problems. In particular, there are heavy computations when one is dealing with nonlinear or time dependent problems. Besides basis are problems dependent, i.e., given a new heterogeneous input, one need to evaluate a new set of basis; hence it is not flexible to apply the methods on solving stochastic problems. Deep learning is a branch in the machine learning and started to show its power in computer science since 2010's. It is accurate meanwhile very efficient in many computer vision and language processing applications.
Deep learning usually consists of two steps: training and testing. Given a model which is well trained by a data set, one can evaluate the new example which has common features as the training data set very efficiently and accurately. This motivates us solving the multiscale problems with the deep learning approach. In the second part of this work, we will show some problems which are solved by combining the multiscale methods and deep learning. The deep learning approach indeed improves the efficiency of the traditional method. | en |
