Special Session 157: Advances in PDE-Based and Data-Driven Approaches for Applied Sciences

Inverse and constrained optimization problems with applications to personalized medicine

Abramo Agosti University of Pavia Italy Co-Author(s):    Elena Beretta, Cecilia Cavaterra, Matteo Fornoni, Elisabetta Rocca

Abstract: In this talk I will present different types of inverse and PDE-constrained optimization problems for multiphase tumour growth models which are of some interest for clinical applications. In particular, I will present the inverse identification problem of the initial condition from a single-point measurement at a final time for a Cahn-Hilliard type model describing tumour evolution in the a-vascular stage, and the parameters optimization problem for a multiphase model describing tumour growth in the vascular stage. I will show both analytical and numerical results, and present numerical simulations for both academic and patient-specific test cases.

On the reconstruction of early stages in tumour growth models

Elena Beretta New York University Abu Dhabi United Arab Emirates Co-Author(s):    Abramo Agosti Cecilia Cavaterra Matteo Fornoni Masahiro Yamamoto

Abstract: We study backward inverse problems for coupled Cahn--Hilliard--reaction--diffusion systems. Our main result is a Carleman estimate for a fourth-order/second-order parabolic system with cross-diffusion terms. As a consequence, we derive conditional stability estimates for the reconstruction of past states from a single final-time observation: H\older stability at positive times and logarithmic stability for the initial datum. We then apply the abstract theory to a phase-field tumour growth model coupling the tumour volume fraction with the nutrient concentration. In this setting, we obtain backward uniqueness and quantitative stability for the recovery of early tumour states, extending previous results and removing additional restrictions on the chemotactic coupling. We also discuss how these results support Lipschitz stability on finite-dimensional admissible sets, which is relevant for reconstruction algorithms.

Complete stickiness for nonlocal minimal graphs with obstacles in highly nonlocal regimes

Claudia D Bucur University of Milan Italy Co-Author(s):

Abstract: We study the geometric and functional framework for a nonlocal Plateau problem with obstacles. In particular, we formulate the minimization of the fractional perimeter in cylinders with respect to graphical exterior data, as well as the equivalent variational problem for the nonlocal area functional. We then show that, when the prescribed exterior data is not too large at infinity and the fractional parameter is sufficiently small, minimizers exhibit complete stickiness: they adhere entirely to the obstacle and leave the remainder of the domain asymptotically empty.

PDE-constrained optimal control of viscous Cahn--Hilliard systems with hyperbolic relaxation

Pierluigi Colli University of Pavia Italy Co-Author(s):

Abstract: Phase-field models such as the Cahn--Hilliard system play a central role in the PDE-based modeling of phase separation phenomena in applied sciences. In this talk, we investigate an optimal control problem for a viscous Cahn--Hilliard system in which the chemical potential is subject to a hyperbolic relaxation. We present recent analytical results for the associated state system, including well-posedness, regularity properties, and the Fr\`echet differentiability of the control-to-state mapping in suitable Banach spaces. Building on this framework, we study a sparse optimal control problem and derive first-order necessary conditions for local optimality. Finally, we analyze the asymptotic behavior of the system as the relaxation parameter tends to zero, providing convergence results for the state variables, optimal controls, and adjoint system.

A mathematical model of phase separation on biomembranes

Maurizio Grasselli Politecnico di Milano Italy Co-Author(s):    D. Caetano, C.M. Elliott, A. Poiatti

Abstract: We present a model for phase separation of multiple components on a biological membrane, allowing for small deformations of a spherical shape. The model couples a multi-component Cahn--Hilliard equation with an equation for the membrane geometry, derived using variational principles. We show that the system is well posed and that solutions become regular over time, staying away from pure phases. Moreover, every solution eventually converges to a single equilibrium configuration.

An inverse problem for the Monge-Ampere equation

Yi-Hsuan Lin National Yang Ming Chiao Tung University Taiwan Co-Author(s):    Tony Liimatainen

Abstract: We extend the study of inverse boundary value problems to the setting of fully nonlinear PDEs by considering an inverse source problem for the Monge--Amp\`ere equation \[ \det D^2 u = F. \] We prove that, on a convex Euclidean domain in the plane, the associated Dirichlet-to-Neumann (DN) map uniquely determines a positive source function $F$. The proof relies on recovering the Hessian of a solution to the equation, which is interpreted as a Riemannian metric $g$. Interestingly, although the equation is posed on a Euclidean domain, the inverse problem becomes anisotropic since the metric $g$ appears as a coefficient matrix in the linearized equation. As an intermediate step, we prove that the DN map of the non-divergence form equation \[ g^{ab} \partial_{ab} v = 0 \] uniquely determines the conformal class of the metric $g$ on a simply connected planar domain, without the usual diffeomorphism invariance. To address the challenges of full nonlinearity, we develop asymptotic expansions for complex geometric optics solutions in the planar setting and solve a resulting nonlocal $\op$-equation by proving a unique continuation principle for it. These techniques are expected to be applicable to a wide range of inverse problems for nonlinear equations.

Enhanced dissipation and applications to non-linear PDEs

Anna L Mazzucato Penn State University USA Co-Author(s):

Abstract: I will discuss recent results concerning enhanced dissipation in drift-diffusion equations by the action of the drift. If time permits I will then give an application to global existence for the 2D Kuramoto-Sivashinsky equation, a model of front propagation in combustion. Enhanced dissipation has applications in speeding up Langevin dynamics and MCMC methods.

Convergence to equilibrium of weak solutions to the Cahn--Hilliard equation with non-degenerate mobility: a novel perspective

Andrea Poiatti University of Parma Italy Co-Author(s):    Maurizio Grasselli

Abstract: We consider the initial and boundary value problem for the Cahn--Hilliard equation with non-degenerate mobility and singular potential. We show that any weak solution converges to a single equilibrium using only minimal assumptions, i.e., the existence of a global weak solution satisfying an energy inequality. This result also holds in the three-dimensional case, which was an open problem so far due to the lack of regularity of solutions, especially when the mobility is just a continuous function. This novel method is robust and can be used also for other models like, for instance, Cahn--Hilliard-Navier--Stokes type systems with unmatched densities and viscosities as the one proposed by Abels, Garcke, and Gr\{u}n (Math. Models Methods Appl. Sci. 22, 2012).

The Calder\`on problem with random measurements

Simone Sanna University of Genoa Italy Co-Author(s):    Giovanni S. Alberti, Damiano Poletti, Matteo Santacesaria

Abstract: The Calder\`on problem consists in recovering an unknown parameter of a partial differential equation from boundary measurements of its solution. While global uniqueness is well-established for the full Dirichlet-to-Neumann operator, corresponding to infinitely many boundary measurements, practical applications, such as EIT, operate with finite and discrete data. In this realistic setting, the unknown parameter is assumed to lie in a finite-dimensional space, and the boundary data are restricted to a finite family of Dirichlet-Neumann pairs. This reduction leads to a fundamental question regarding sample complexity: what is the minimal number of measurements M required to reconstruct a d-dimensional unknown? Due to the non-linearity and severe ill-posedness of the problem, existing deterministic results often yield sample complexity estimates that are exponential in d. In this work, we address this question using randomized boundary data. Instead of deterministically choosing a finite number of boundary data from a fixed orthonormal basis on the boundary of the domain, we take random linear combinations of such basis elements. We prove that this approach ensures almost sure uniqueness with a number of measurements M that is merely proportional to the dimension of the parameter space d, significantly improving upon current deterministic bounds.

Neural network parametrized level sets for image segmentation

Cong Shi University of Vienna Austria Co-Author(s):    Otmar Scherzer and Thi Lan Nhi Vu

Abstract: The Chan-Vese functionals have proven to by a first-class method for segmentation and classification. Previously they have been implemented with level-set methods based on a pixel-wise representation of the level-sets. Later parametrized level-set approximations, such as splines, have been studied. In this talk we consider neural networks as parametrized approximations of level-set functions. We show in particular, that parametrized two-layer networks are most efficient to approximate polyhedral segments and classes. We also prove the efficiency for segmentation and classification.

Recovery of an Anisotropic Conductivity from the Neumann-to-Dirichlet Map in a Semilinear Elliptic Equation

Eva Sincich University of Trieste Italy Co-Author(s):    Elena Beretta, Elisa Francini, Dario Pierotti

Abstract: We study the inverse boundary value problem of detecting a non-uniform conductivity motivated by pacing-guided ablation in cardiac electrophysiology. At the stationary level, the transmembrane potential $u$ in a region \(\Omega\subset\mathbb{R}^3\) of cardiac tissue satisfies \[ -\nabla\!\cdot(\gamma\nabla u)+\alpha u^3=0 \quad \text{in }\Omega,\qquad \gamma\nabla u\cdot\nu=g \quad \text{on }\partial\Omega, \] where $\gamma$ is an anisotropic conductivity tensor and $\alpha$ a nonlinear ionic response coefficient. The Neumann data $g$ represent pacing currents, and the boundary values $u|_{\partial\Omega}$ correspond to invasive voltage measurements. Ischemic regions are modeled by a subdomain $D\subset\Omega$ where $\gamma$ is piecewise constant. We address the inverse problem of determining $\gamma$ from the Neumann-to-Dirichlet (NtD) map, assuming that $\alpha$ and $D$ are known.

Curvature-driven pattern formation in biomembranes: A gradient flow approach

Dennis Trautwein University of Regensburg Germany Co-Author(s):    Patrik Knopf, Anastasija Pe\v{s}i\`{c}

Abstract: In this talk, we study a phase-field model for curvature-driven pattern formation in biomembranes. The model is derived as a gradient flow of an energy functional that approximates the two-phase Canham--Helfrich energy. This leads to a Cahn--Hilliard-type equation with cross diffusion for the relative chemical concentration of one lipid phase, coupled to a fourth-order reaction-diffusion equation describing the height profile of the membrane. We first prove the existence of weak solutions for the case of regular double-well potentials, using a minimizing movement scheme to construct approximate solutions. The analysis is then extended to singular potentials, e.g., the Flory--Huggins potential, by approximating them with a Moreau--Yosida regularization. For both cases, we establish higher regularity, continuous dependence on the initial data, and consequently the uniqueness of weak solutions. Finally, we propose a well-posed finite element discretization of the model and present numerical experiments illustrating the effect of different physical parameters on the resulting membrane patterns. Depending on the parameter regime, we observe purely striped, dotted, or snake-like structures.