Optimization & Goal-Oriented Computation
PDE-constrained optimization, state and control constraints, goal-oriented error estimation, and adaptive computation for decision-relevant quantities.
Overview
My optimization research studies how PDE models can be used not only for simulation, but also for design, control, and decision-making. The emphasis is on optimization problems governed by heterogeneous or multiphysics PDEs, where constraints, discretization error, and solver cost interact strongly.
A complementary direction is goal-oriented computation: rather than resolving the full solution uniformly, the numerical method targets the quantities of interest that matter for the final objective, control, or decision.
PDE-Constrained Optimization in Heterogeneous Media
A central model problem is optimal control of Darcy flow in heterogeneous porous media, where the pressure is the state and source or sink terms act as controls. Heterogeneous permeability introduces nonuniform regularity and strongly varying sensitivity, so the discretization and optimization method must remain robust with respect to coefficient contrast.
- Heterogeneous coefficients. Optimization under strongly varying permeability fields.
- Finite element discretization. Interior-penalty formulations designed for the reduced optimization problem.
- Analysis. Stability and a priori error estimates for the discrete control problem.
- Applications. Flow control and source/sink placement in porous-media systems.
State and Control Constraints
State-constrained optimization introduces additional mathematical and computational difficulty because admissibility must be enforced on the PDE solution itself. Recent work develops formulations for Darcy control with pointwise state or control constraints using discontinuous Galerkin and C0 interior-penalty techniques.
These problems naturally lead to active-set or variational-inequality structures and require efficient solvers for linear systems that change as the active and inactive sets evolve.
Goal-Oriented Error Estimation and Adaptivity
In many applications, the objective depends on a localized or derived quantity rather than on the global PDE solution. Goal-oriented error estimation provides a principled way to identify which discretization errors most strongly affect that quantity of interest.
Current work develops goal-oriented a posteriori error estimators based on equilibrated stress reconstruction for elasticity and connects these ideas with adaptive computation for optimization and decision-oriented simulation.
- quantity-of-interest error representation;
- equilibrated reconstruction and computable estimators;
- adaptive refinement targeted to downstream objectives;
- interaction between discretization error and optimization error.
Efficient Solvers for Repeated Optimization
Optimization requires repeated PDE solves, often with closely related linear systems. My current work therefore also studies reusable solver components for changing active-set systems, including spectral-deflation strategies and other reduced representations that accelerate repeated solves across optimization iterations or parameter values.
Optimization with Reduced and Learned Models
Reduced-order and learned surrogates can substantially lower the cost of repeated simulation inside an optimization loop, but only if their errors are controlled relative to the objective. This motivates a task-aware perspective in which model fidelity is chosen according to sensitivity of the quantity of interest rather than solution accuracy alone.
Current Directions
- State-constrained control. Robust discretization and solvers for pointwise state and control constraints.
- Goal-oriented adaptivity. Mesh refinement driven by objective or quantity-of-interest error.
- Reusable linear algebra. Deflation and reduced representations for changing active-set systems.
- Optimization under uncertainty. Incorporating uncertain parameters and data into PDE-constrained decisions.
Selected Publications
- A Goal-Oriented A Posteriori Error Estimator Based on Equilibrated Stress Reconstruction for Linear Elasticity. Submitted.
- A Monolithic Discontinuous Galerkin Framework for Darcy Optimal Control with Radon-Measure Tracking and Pointwise Control Constraints. Submitted. arXiv
- A Proximal C0 Interior Penalty Method for State-Constrained Darcy Flow Control. Submitted.
- Reference-Space Reuse for Steady and Transient State-Constrained Conjugate Heat Transfer. Submitted. arXiv
- Optimal Control for Darcy's Flow in a Heterogeneous Porous Media. Applied Numerical Mathematics, 207, 303-322, 2025. Journal
- Optimal Design of Hydraulic Fracturing in Porous Media Using the Phase-Field Fracture Model Coupled with Genetic Algorithm. Computational Geosciences, 22(3), 833-849, 2018. Journal
Related Research Connections
- Structure-Preserving Numerical PDEs: discretization and error estimation provide the numerical foundation for optimization.
- Inverse Problems & Data Assimilation: inference provides parameter and state estimates that feed downstream decisions.
- Scientific Machine Learning: reduced and learned models can accelerate repeated simulation when their error is controlled relative to the objective.