Abstract
We study a scalar elliptic problem in the data driven context. Our interest is to study the relaxation of a data set that consists of the union of a linear relation and single outlier. The data driven relaxation is given by the union of the linear relation and a truncated cone that connects the outlier with the linear subspace.
Highlights
The data driven perspective is new in the field of material science and partial differential equations, we mention [18] and [6] as the two fundamental contributions of this young field
We investigate a special data set: Dloc is the union of DlAoc and DlBoc, where DlAoc is as in (1.3) and DlBoc is a one-point set of a single outlier
We ask what effective laws can be obtained in the limit. The warning about this description is that DB is not a linear relation and does not describe a material law in the classical setting of homogenization
Summary
The data driven perspective is new in the field of material science and partial differential equations, we mention [18] and [6] as the two fundamental contributions of this young field. In the data driven perspective certain laws of physics are accepted as invariable, e.g. balance of forces or compatibility. On a more formal level, one introduces a set E of functions that satisfy the invariable physical laws. A second set D denotes those functions that are consistent with the data. In this setting, the aim is to find functions in E that minimize the distance to the data set D. In the data driven perspective, the material law is replaced by a data set D. In the data driven perspective, the task is: Find a pair (G, J ) ∈ E f that minimizes the distance to the set D. and the corresponding set of functions D A as in (1.2). Three different types of questions can be asked: 1. Minimality conditions: When E f ∩ D is empty, what are conditions for minimizers of the distance?
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