SEAMSSchool2025 Baguio
Inverse Problems and Optimal Control
UP Baguio, Philippines
December 3 - 11, 2025
UP Baguio, Philippines
December 3 - 11, 2025
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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, optimal design, 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.
Electrical Impedance Tomography (EIT) is a non-invasive, portable, low-cost technology developed to image the distribution of electrical properties, conductivity and/or permittivity, within an object from measurements of electric currents and voltages on its surface. The lectures will offer first an overview of the technique highlighting its theoretical challenges ill-posedness, nonlinearity and poor spatial resolution, as well as its practical applications including medical ones such as breast cancer screening and lung function monitoring, and then will be focused on non-iterative reconstruction algorithms for EIT which are based on integral equation formulations and could be used to produce images in real-time. Regularization and numerical methods for solving Fredholm integral equations will also be presented.
This course offers an in-depth exploration of the mathematical foundations and techniques essential to computed tomography (CT), a critical tool in biomedical imaging. We will begin by examining the physics underlying the Radon transform, a key concept in the projection of multi-dimensional data. The course will then explore the slice Fourier theorem, which connects the Radon transform with the Fourier transform, providing the theoretical basis for image reconstruction. Participants will gain insights into the role of image discretization in transforming continuous models into practical algorithms. We will also cover the formation and interpretation of sinograms, which represent the CT measurement data. The course will culminate with a discussion on reconstructing images using these sinograms, starting with the classical filtered back-projection formula. Designed for researchers and practitioners in mathematical modeling, inverse problems, and biomedical engineering, this course offers a comprehensive understanding of how advanced mathematical tools are applied to real-world challenges in medical imaging.
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.