Paper Deep Dive: Can We Actually Compute the Solution to a PDE?

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Partial differential equations (PDEs) are fundamental models of physical phenomena, yet most PDEs of practical interest require numerical approximations. This raises a natural question: given a PDE, can its solution be computed efficiently, or does the equation exhibit an inherent complexity blow-up?

In our recent work, accepted to Numerische Mathematik, we develop a novel variational framework that links the structural properties of PDEs, discrete gradient flows, and convergence rates to the computational complexity of their solutions.

Our analysis identifies:

• Classes of PDEs that admit polynomial-time computability — where solutions can be efficiently approximated.
• Regimes exhibiting complexity blow-up — where solutions become substantially harder to compute than the input data suggests.

The framework provides a new perspective on how regularity, variational structure, and optimization dynamics shape the computational complexity of PDE solutions, bridging ideas from variational calculus, approximation theory, and computability theory.

👉 Read the full paper

Authors: Juan Esteban Suarez, Holger Boche, and Gitta Kutyniok