Hendrik Kleikamp

Hendrik Kleikamp, Tizian Wenzel

In this contribution, kernel approximations are applied as ansatz functions within the Deep Ritz method. This allows to approximate weak solutions of elliptic partial differential equations with weak enforcement of boundary conditions using Nitsche's method. A priori error estimates are proven in different norms leveraging both standard results for weak solutions of elliptic equations and well-established convergence results for kernel methods. This availability of a priori error estimates renders the method useful for practical purposes. The procedure is described in detail, meanwhile providing practical hints and implementation details. By means of numerical examples, the performance of the proposed approach is evaluated numerically and the results agree with the theoretical findings.

Björn Schembera, Frank Wübbeling, Hendrik Kleikamp et al.

In applied mathematics and related disciplines, the modeling-simulation-optimization workflow is a prominent scheme, with mathematical models and numerical algorithms playing a crucial role. For these types of mathematical research data, the Mathematical Research Data Initiative has developed, merged and implemented ontologies and knowledge graphs. This contributes to making mathematical research data FAIR by introducing semantic technology and documenting the mathematical foundations accordingly. Using the concrete example of microfracture analysis of porous media, it is shown how the knowledge of the underlying mathematical model and the corresponding numerical algorithms for its solution can be represented by the ontologies.

Björn Schembera, Frank Wübbeling, Hendrik Kleikamp et al.

Mathematical models and algorithms are an essential part of mathematical research data, as they are epistemically grounding numerical data. In order to represent models and algorithms as well as their relationship semantically to make this research data FAIR, two previously distinct ontologies were merged and extended, becoming a living knowledge graph. The link between the two ontologies is established by introducing computational tasks, as they occur in modeling, corresponding to algorithmic tasks. Moreover, controlled vocabularies are incorporated and a new class, distinguishing base quantities from specific use case quantities, was introduced. Also, both models and algorithms can now be enriched with metadata. Subject-specific metadata is particularly relevant here, such as the symmetry of a matrix or the linearity of a mathematical model. This is the only way to express specific workflows with concrete models and algorithms, as the feasible solution algorithm can only be determined if the mathematical properties of a model are known. We demonstrate this using two examples from different application areas of applied mathematics. In addition, we have already integrated over 250 research assets from applied mathematics into our knowledge graph.