SEAMSSchool2025 Baguio
Inverse Problems and Optimal Control
UP Baguio, Philippines
December 3 - 11, 2025
UP Baguio, Philippines
December 3 - 11, 2025
| Home | Registration | Program | Directions | Acknowledgments |
| Date DD/MM/YY |
08:00–09:00 | 09:00–10:30 | 10:30–11:00 | 11:00–12:30 | 12:30–13:30 | 13:30–15:00 | 15:00–15:30 | 15:30–17:00 |
|---|---|---|---|---|---|---|---|---|
| 2/12/25 | Arrival in Baguio City, Philippines | |||||||
| 3/12/25 | Opening | Mohammad | Break | Biswas | Lunch | Simon | Break | Cueva |
| 4/12/25 | Real | Break | Biswas | Lunch | Simon | Break | Cueva | |
| 5/12/25 | Mohammad | Break | Mohammad | Lunch | Biswas | Break | Cueva | |
| 6/12/25 | Excursion | |||||||
| 7/12/25 | Break | |||||||
| 8/12/25 | Cueva | Break | Simon | Lunch | Mendoza | Break | Biswas | |
| 9/12/25 | Real | Break | Simon | Lunch | Real | Break | Mendoza | |
| 10/12/25 | Mendoza | Break | Real | Lunch | Mendoza | Break | Mini-Project Discussion |
|
| 11/12/25 | Mini-Project Discussion |
Break | Project presentation |
Lunch | Project presentation |
Break | Closing | |
| 12/12/25 | Departure from Baguio City, Philippines | |||||||
This course introduces numerical methods and applications of infinite-dimensional optimization problems. Here, the solution can be a function, leading to problems that appear in the calculus of variations, inverse problem, and optimal control; or can be a shape, leading to problems that appear in shape and topology optimization. We start with the basics using tools from functional analysis. Then we present some numerical methods for both unconstrained and constrained problems, including their discretization.
Inverse problems are fundamental in creating precise models across natural sciences. Traditional approaches often struggle with the complexity and high dimensionality of typical datasets in these fields. To solve these problems, this lecture will explore intelligent systems, including machine learning algorithms and metaheuristic optimization. We will explore how these advanced methods can enhance the accuracy and reliability of model predictions and reveal patterns that conventional techniques may miss. We present real-world examples.
Inverse problems aim to solve the parameters given an observed or desired effect. These problems often lead to mathematical models that are inherently ill-posed. In many inverse problems, exact data are not available, instead, only noisy data are available. Using the linear deconvolution problem as a model, we will demonstrate ill-posedness of the inverse problems in finite-dimensional space, and how regularization overcomes this ill-posedness. We will focus on Tikhonov regularization for linear inverse problems, with a special attention on choosing the regularization parameter. Finally we will examine the total variation regularization to solve solutions with specific properties.
This course will use MATLAB to demonstrate the numerics of the course.
This lecture presents the reconstruction of medical images as an ill-posed inverse problem, focusing on the mathematical models and computational methods that enable recovery from incomplete data. Emphasis is placed on magnetic resonance imaging (MRI), described through the Fourier model, and on strategies to address accelerated acquisitions and data undersampling. The session reviews key approaches from Parallel Imaging and Compressed Sensing to deep learning–based reconstructions, highlighting the role of regularization and physics-informed priors. Participants will gain an integrated view of how modern mathematical modeling connects theory and computation in medical imaging.
The hands-on labs will be conducted in Python using Jupyter Notebooks, prepared to run directly on Google Colab. No local installation will be required.
The optimal control problem can be viewed as a cost functional to be minimized subject to a partial differential equation describing the motion, a control function which can be taken in a boundary condition, an initial condition, a coefficient in a partial differential equation, or any parameter in the equation. The control may be chosen, while the state is uniquely determined by the solution of the differential equations. Our aim is to choose a control in such a way that the cost functional is minimized. Such controls are called optimal control and the corresponding state is called optimal state. In these lectures, I will introduce the basic tools to prove the existence of solutions to optimal control problems and its characterization by deriving first order optimality conditions. Moreover, I will also discuss some applications of optimal control problems in real life.
This lecture addresses a box constrained optimal control problem with a linear objective functional in relation to the control, often leading to bang-bang solutions that lack regularity. To address this, Tikhonov regularization is employed. The lecture investigates the stability of optimal controls for the regularized problem by introducing strong Holder subregularity (sHs). Conditions for sHs are established and applied to problems governed by semilinear and quasilinear parabolic equations. The sHs for the optimality map is proved, demonstrating stability not only under Tikhonov regularization but also for perturbations in initial data and source functions.