Smart Mathematical Toolkit: New Adaptive Techniques Focus Computational Effort for Precision and Resource Savings

• Dr Judit Muñoz Matute has focused on the design of stabilised space-time adaptive techniques based on the Discontinuous Petrov-Galerkin (DPG) methodology.
• This sophisticated methodology, developed for the efficient handling of transient Partial Differential Equations (PDEs)—particularly those involving advection-dominated-diffusion and wave propagation problems—is crucial.
• The final application of these resultant methods is to improve seismic imaging, a critical step that makes deep underground carbon capture and storage (CO₂), a vital process against climate change, safer and more reliable.
• The results have been published in Results in Brief published article on CORDIS

Underground storing of carbon dioxide (CO₂) is considered an indispensable tool in the global fight against climate change. However, the viability of these projects is dependent upon engineers having detailed, accurate knowledge of the geological formations beneath the surface.

This need was addressed by the EU-funded GEODPG project, coordinated by the Basque Center for Applied Mathematics (BCAM). Led by researcher Judit Muñoz Matute (ESM at BCAM and Ikerbasque Research Fellow at EHU), GEODPG set out to design smarter ways of solving the complex equations that describe physical processes underground. Supported by the Marie Skłodowska-Curie Actions programme, the project ran from 2022 to 2025.

The core of the project lies in the development of advanced methods for solving partial differential equations (PDEs), particularly those describing wave propagation. A central outcome has been the design of highly efficient and precise numerical techniques, including a novel ‘space-time adaptive’ Discontinuous Petrov–Galerkin (DPG) method.

As Muñoz Matute explains, “Some regions or moments are more complicated and require finer resolution than others. Space-time adaptive techniques automatically concentrate computational effort where it is most needed, saving time and resources elsewhere. The DPG method ensures that the results remain accurate, even for very challenging problems.” This approach enables a highly efficient use of computational resources while preserving robustness and accuracy.

While these techniques can be applied to several fields —geophysics among them— such applications represent potential use cases rather than the main focus of the research. In this broader context, the mathematical and computational innovations are expected to support diverse scientific and engineering applications, such as producing sharper reconstructions of subsurface structures or identifying subtle anomalies, without being limited to any single domain.

For Muñoz Matute, the project reflects her passion for using mathematics to address urgent global issues. “The same ideas that explain how waves move can also help us fight climate change. I find it rewarding to contribute mathematical tools that protect our planet.”

Her next steps involve extending the methods to more complex problems and exploring synergies with artificial intelligence. Over the next decade, she hopes this work will deliver powerful new tools not only for geophysics, but also for areas such as computational biology.

The success of GEODPG highlights BCAM's strength in advanced applied modelling. Judit Muñoz Matute was a postdoctoral researcher in the centre's Mathematical Design, Modelling and Simulations line during the project. The work aligns with the research of other prominent BCAM mathematicians, such as Professor David Pardo, who leads groups focusing on numerical methods for geophysical flow.

The findings have been published in a CORDIS "Results in Brief" article (Nov. 2025), illustrating a successful translation of abstract mathematics into tangible tools for practical climate action.