Research
Research directions
A structured overview of the themes that organize the group's work in nonlinear PDE analysis, uncertainty-aware modeling, and mathematical imaging.
Anisotropic reaction-diffusion models
PDE models with structured diffusion, uncertainty-aware state variables, and biomedical interpretation.
This direction studies multicomponent reaction-diffusion systems in which the diffusion structure carries physical or geometric meaning rather than serving as a generic smoothing term.
A recurring theme is the use of Dirichlet concentration parameters to encode both composition and uncertainty, especially in heterogeneous tissue remodeling scenarios where sharp categorical labels are too crude.
Key questions
Which invariant regions and positivity bounds remain available in anisotropic multicomponent models?
How can uncertainty-aware state variables be introduced without losing analytical control?
What numerical regimes preserve the qualitative interpretation of tissue transitions?
Cross-diffusion, entropy, and Markov kinetics
Entropy-structured analysis of coupled diffusion-reaction systems beyond the reach of classical maximum principles.
We study reaction-diffusion systems with cross-diffusion and Markovian reaction structure, where nonsymmetry and degeneracy require tools that go beyond standard parabolic arguments.
The emphasis is on existence, entropy dissipation, convergence to equilibrium, and analytical decompositions that separate spatial transport from kinetic relaxation.
Key questions
Which structural assumptions turn entropy into a workable global control mechanism?
How do spatial diffusion and finite-state Markov kinetics split the dissipation landscape?
When can weak solutions be complemented by uniqueness or enhanced regularity statements?
Dirichlet uncertainty models for imaging
Distributional representations of predictions and uncertainty for segmentation problems with spatial regularization.
This line of work uses Dirichlet fields to represent predictive uncertainty in dense image segmentation, so that calibration, confidence structure, and spatial smoothness are handled within a unified variational framework.
Instead of relying only on repeated stochastic inference, the goal is to make uncertainty an explicit part of the state description and of the optimization problem itself.
Key questions
How can Dirichlet parameter fields deliver useful uncertainty maps in a single forward pass?
Which regularizers improve calibration without destroying boundary sensitivity?
How should epistemic and data-driven uncertainty be separated in structured segmentation models?
Scale-consistent variational regularization
Discrete-to-continuum analysis of edge-aware energies and structurally consistent anisotropic discretizations.
The group also studies what happens to weighted variational energies under mesh refinement, with a particular focus on the loss of anisotropy caused by scale-inconsistent edge-aware weights.
This direction links practical regularization schemes to rigorous questions of well-posedness, Gamma-convergence, and resolution-independent parameter tuning.
Key questions
Which weight constructions remain nontrivial in the continuum limit?
How can edge-aware discretizations preserve dimensional consistency across resolutions?
What convergence guarantees are needed for inverse problems and imaging applications?