Summary
We combine the idea of sampling inequalities and Galerkin approximations of weak formulations of partial differential equations. The latter is a wellestablished tool for finite element analysis. We show that sampling inequalities can be interpreted as Pythagoras law in the energy norm of the weak form. This opens the way to consider regularization techniques known from machine learning in the context of finite elements. We show how sampling inequalities can be used to provide a deterministic worst case error estimate for reconstruction problems based on Galerkin type data. Such estimates suggest an a priori choice for regularization parameter(s).
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Acknowledgement
The author would like to thank Barbara Zwicknagl and Robert Schaback many helpful and stimulating discussions.
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Rieger, C. (2011). Sampling Inequalities and Support Vector Machines for Galerkin Type Data. In: Griebel, M., Schweitzer, M. (eds) Meshfree Methods for Partial Differential Equations V. Lecture Notes in Computational Science and Engineering, vol 79. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16229-9_3
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DOI: https://doi.org/10.1007/978-3-642-16229-9_3
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