Research

My research focuses on high-order numerical methods for high-dimensional and multiscale systems, data-driven and machine learning acceleration of principle-based numerical approaches, and applications such as electromagnetic waves, radiative transfer, and inverse problems. Below are selected research highlights.

Full publication list: see publications

Element learning: a systematic approach of accelerating finite element-type methods via machine learning

My research in this area centers on two closely related questions:

  1. What are the advantages and limitations of machine learning based methods compared to classical numerical methods (e.g. finite element)?
  2. How to combine their advantages?

Thinking about the above two questions has led to a recent discovery of a new computational framework which we call element learning. This new approach can be used to systematically accelerate finite element-type methods through machine learning while still retaining the many advantages of classical finite element-type methods, such as proficient handling of complex geometry and boundary conditions, reliability, etc.

See the following figures for one of our numerical results and a visual explanation of the core idea of element learning.

Element learning - results
Numerical results comparing element learning (EL) with a standard discontinuous Galerkin method, showing speed-up while maintaining accuracy.
Element learning - core ideas
Core idea of element learning: local element computations in a finite element-type method are replaced by a trained neural network.

Adaptive-mesh inversion: a goal-oriented hp-adaptive mesh refinement method for solving inverse radiative transfer problem

Inverse radiative transfer serves as the mathematical foundation for a wide spectrum of applications such as optical tomography and remote sensing. Despite its broad applications, devising numerical methods for inverse radiative transfer is notoriously challenging due to the high-dimensional nature of the forward problem. Standard discretizations that meet the accuracy standards often demand significant memory resources, making the solver inefficient and sometimes impractical. This challenge amplifies in the inverse problem scenario, where numerous iterations of the forward problems are necessary. As such, there's a pressing need to design numerical methods that are memory-efficient and have fewer degrees of freedom (DOFs) without compromising on accuracy.

To address the challenges previously mentioned, we introduced the so called adaptive-mesh inversion method. This is a goal-oriented hp-adaptive mesh refinement (AMR) technique tailored for solving inverse problems. A distinctive feature of the approach is the integration of two optimization processes: one for inversion and the other for mesh adaptivity. By leveraging the connections between duality-based mesh adaptivity and adjoint-based inversion techniques, we are able to devise a goal-oriented error estimator. This estimator is cost effective to compute. The numerical tests suggest it can efficiently steer mesh-refinement to effectively address the inverse radiative transfer problem. The main idea of the methodology can be visually represented as in the following figure.

Adaptive mesh inversion - core ideas
The adaptive-mesh inversion framework: two nested optimization loops for inversion and goal-oriented mesh adaptivity.
Goal-oriented adaptive mesh Standard uniform mesh
Left: goal-oriented adaptive mesh refined near the sensor region. Right: standard adaptive mesh refinement for comparison.

Symplectic Hamiltonian finite element methods for electromagnetics

Symplectic Hamiltonian finite element methods (SHFEM) are a special type of numerical methods for numerically approximating the solution of Hamiltonian partial differential equations. These methods use Galerkin space-discretizations which result in a system of ODEs displaying a discrete version of the Hamiltonian structure of the original system. The resulting system of ODEs is then discretized by a symplectic time-marching method. This combination results in high-order accurate, fully discrete methods which can preserve the invariants of the Hamiltonian defining the ODE system. The following figure shows an example with electromagnetic waves propagating in a non-homogeneous medium. It shows that our SHFEM method conserves the energy of the semidiscretization in space up to 10 digits of precision, and does not allow the exact energy to drift in time.

SHFEM energy conservation
Energy conservation in SHFEM for electromagnetic waves in a non-homogeneous medium: the method preserves the discrete Hamiltonian to 10 digits of precision with no drift over time.

Generalized projection-based analysis of hybridizable discontinuous Galerkin methods

Hybridizable discontinuous Galerkin methods (HDG) lie in between the two most important groups of finite element-type methods: (1) conforming/mixed methods and (2) discontinuous Galerkin (DG) methods. This enables HDG to combine the many advantages from these two groups.

Despite the many advantages of HDG methods, their analysis was also separated into two groups: for those more alike mixed methods, they were analyzed similar to the classical analysis of mixed methods, by the so-called projection-based analysis; on the other hand, for those more alike nonconforming/DG methods, they were analyzed by ad hoc approaches. This separation of analysis has several undesired consequences: (1) Those ad hoc analyses are not as concise and simple as the projection-based analysis, (2) Ad hoc analysis becomes difficult for complicated problems (e.g., time-dependent problems, coupling problems), (3) Analysis is hard to reuse because of the separation.

To address this issue, we developed the so-called generalized projection-based analysis, extending the classical projection-based analysis to those HDG methods that do not admit M-decomposition (e.g. Lehrenfeld-Schoberl HDG), thus enabling a unified analysis. With these new analysis tools, we are able to devise/analyze new HDG methods or improve the analysis of existing ones. The applications include time-harmonic and transient elastic waves, static and time-dependent Maxwell's equations and Stokes equations.

HDG projection-based analysis
The generalized projection-based analysis framework unifies the error analysis of HDG methods across both mixed and nonconforming/DG families.