Project description
Optimal computational methods for dynamic systems
In disciplines like computational physics, engineering, finance, or machine learning, accuracy without excessive computing is key. However, when it comes to time-dependent problems, existing methods often falter, unable to balance precision and efficiency. With this in mind, the ERC-funded OPTIMAL project aims to develop precise, yet computationally efficient, adaptive algorithms for these dynamic scenarios. Combining new mathematical insights with cutting edge computational tools, OPTIMAL seeks to revolutionise how we design and analyse adaptive algorithms. The goal is to develop provably optimal algorithms, ensuring unparalleled accuracy and efficiency in computational simulations. More than just pushing boundaries, OPTIMAL aims to redraw them, promising a transformative leap forward in the field.
Objective
The ultimate goal of any numerical method is to achieve maximal accuracy with minimal computational cost. This is also the driving motivation behind adaptive mesh refinement algorithms to approximate partial differential equations (PDEs).
PDEs are the foundation of almost every simulation in computational physics (from classical mechanics to geophysics, astrophysics, hydrodynamics, and micromagnetism) and even in computational finance and machine learning.
Without adaptive mesh refinement such simulations fail to reach significant accuracy even on the strongest computers before running out of memory or time.
The goal of adaptivity is to achieve a mathematically guaranteed optimal accuracy vs. work ratio for such problems.
However, adaptive mesh refinement for time-dependent PDEs is mathematically not understood and no optimal adaptive algorithms for such problems are known. The reason is that several key ideas from elliptic PDEs do not work in the non-stationary setting and the established theory breaks down.
This ERC project aims to overcome these longstanding open problems by developing and analyzing provably optimal adaptive mesh refinement algorithms for time-dependent problems with relevant applications in computational physics.
This will be achieved by exploiting a new mathematical insight that, for the first time in the history of mesh refinement, opens a viable path to understand adaptive algorithms for time-dependent problems. The approaches bridge several mathematical disciplines such as finite element analysis, matrix theory, non-linear PDEs, and error estimation, thus breaking new ground in the mathematics and application of computational PDEs.
Fields of science
- natural sciencesphysical sciencesastronomyastrophysics
- natural sciencesphysical sciencesclassical mechanics
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencescomputer and information sciencesartificial intelligencemachine learning
- natural sciencesmathematicsapplied mathematicsnumerical analysis
Programme(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Funding Scheme
HORIZON-ERC - HORIZON ERC GrantsHost institution
1040 Wien
Austria