**Numerical modeling of PDEs for multi-physics, multi-scale and hybrid-dimensional problems **

- Conforming, Non-conforming and Discontinuous Finite Element methods
- Finite Volume methods
- Isogeometric analysis
- Non-standard Finite Element methods: Virtual Elements, Mimetic Finite Differences, XFEM, CutFEM
- Spectral methods and Spectral Element methods
- Time integration methods: exponential and multirate methods

An ever-increasing need for new modern technologies is leading modern science to tremendously boost research into numerical modeling of physical processes and the corresponding numerical treatment. The MOX lab has a long-standing history of developing and analysing innovative methods to extend the frontiers of scientific computing, thereby permitting to tackle increasingly complex problems.

**Physics-based grey-box models bridging ****Machine Learning and Scientific Computing**

- Data-driven model order reduction
- Data-driven discovery of differential equations
- Neural Networks-based acceleration of numerical methods
- Physics-Informed Neural Networks
- Surrogate models and emulators

The research carried out at MOX creates a concrete bridge and synergies between Machine Learning and Scientific Computing: we complement physics-based methods and data-driven models, to improve knowledge of real phenomena. For example, models based on PDEs can serve as regularizers for Machine Learning algorithms, and, conversely, data-driven methods can complement traditional models where knowledge of physics is lacking or not fully understood.

**Reduction of complexity and uncertainty**

- Mesh adaptation
- Uncertainty quantification
- Projection-based model order reduction

Real-life problems are often characterized by high (geometric) complexity and uncertainty, both in the knowledge of the models and the data of the problem itself. At MOX, we develop methods aimed at reducing the computational cost associated with the numerical approximation of differential models, thereby mitigating their complexity. This allows not only for the real-time solution of physically-sound problems, but also for the control of the accuracy of the reduced models and for the study of how data uncertainty propagates into model outputs.

**Fast solution techniques for large-scale problems**

- Algebraic methods
- PDE-based methods

Real-life applications generally require the solution of large-scale algebraic systems arising, for example, from the discretization of partial differential equations. In this context, the fast and accurate solution of large systems of (linear or nonlinear) equations is crucial for enabling new discretization methods. The MOX lab has a long tradition in the development of fast solution techniques either based on parallelizable divide-and-conquer strategies, such as domain decomposition methods, or highly scalable multilevel techniques, such as geometric and algebraic multigrid methods.

**Numerical optimization**

- Inverse problems and parameter estimation
- Optimal control
- Shape optimization
- Topology optimization

Many problems in Engineering, Life sciences, Natural Sciences, and Finance are modeled as optimization problems, often governed by differential equations. The solution of these problems requires efficient numerical optimization strategies, usually achieved by coupling optimization algorithms with, e.g., appropriate numerical modeling of the governing differential equations and fast solution techniques. This research line addresses shape optimization problems, optimal control problems, inverse problems, and parameter estimation problems and takes full advantage of the synergy between the competencies of the other MOX pillars and actively contributes to their development.