A stabilized FEM with POD and autoencoder compression plus a learned solution operator reconstructs PDE solutions in unique continuation problems, with a first-order H1 error estimate in the linear case and numerical demonstrations for nonlinear cases.
Computational unique continuation with finite dimensional Neumann trace
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abstract
We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative error estimates we prove Lipschitz stability of the unique continuation problem in the global H1-norm. This stability is then leveraged to derive optimal a posteriori and a priori error estimates for a primal-dual stabilised finite method.
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Stabilizing and Solving Unique Continuation Problems by Parameterizing Data and Learning Finite Element Solution Operators
A stabilized FEM with POD and autoencoder compression plus a learned solution operator reconstructs PDE solutions in unique continuation problems, with a first-order H1 error estimate in the linear case and numerical demonstrations for nonlinear cases.