Speakers
Description
The mechanical models describe many functions of populations, organisms and cells including their motility, proliferation, and morphogenesis. The mathematical modeling is crucial for description of biological cells, soft tissues, and multifunctional materials in various applications stemming from mechanical and environmental engineering, biological and medical sciences. For example, we refer to tumor growth and invasion, nanoindentation and creep relaxation tests, computed tomography scanning and other nondestructive methodologies used for imaging in the applications. From the mathematical point of view, such models are characterized by nonlinear evolutionary PDE and ODE systems with hysteresis, hereditary integrals with memories, fractional derivatives, and nonlocal effects such as adhesion. From a mechanical viewpoint, the most of biological tissues are viscoelastic. Viscoelastic response of cells and tissues represents a class of implicit constitutive relations between the histories of the stress and the relative deformation gradient. Their analysis needs non-standard theoretical methods like hemivariational inequalities, nonlinear inclusions, sweeping processes, maximal monotone graphs for implicit relations, and measure-theoretical tools. It should be supported by advanced numerical techniques, multiscale modelling at different length scales and AI-based computer simulations. Contributions to the fields in a broad scope are highly welcome.
Bibliography
@book{efendiev_evolution_2013,
address = {Basel},
series = {International {Series} of {Numerical} {Mathematics}},
title = {Evolution {Equations} {Arising} in the {Modelling} of {Life} {Sciences}},
volume = {163},
copyright = {http://www.springer.com/tdm},
isbn = {9783034806145 9783034806152},
url = {http://link.springer.com/10.1007/978-3-0348-0615-2},
language = {en},
urldate = {2025-11-11},
publisher = {Springer Basel},
author = {Efendiev, Messoud},
year = {2013},
doi = {10.1007/978-3-0348-0615-2},
}
@article{hauptmann_convergent_2025,
title = {Convergent {Regularization} in {Inverse} {Problems} and {Linear} {Plug}-and-{Play} {Denoisers}},
volume = {25},
issn = {1615-3375, 1615-3383},
url = {https://link.springer.com/10.1007/s10208-024-09654-x},
doi = {10.1007/s10208-024-09654-x},
abstract = {Abstract
Regularization is necessary when solving inverse problems to ensure the well-posedness of the solution map. Additionally, it is desired that the chosen regularization strategy is convergent in the sense that the solution map converges to a solution of the noise-free operator equation. This provides an important guarantee that stable solutions can be computed for all noise levels and that solutions satisfy the operator equation in the limit of vanishing noise. In recent years, reconstructions in inverse problems are increasingly approached from a data-driven perspective. Despite empirical success, the majority of data-driven approaches do not provide a convergent regularization strategy. One such popular example is given by iterative plug-and-play (PnP) denoising using off-the-shelf image denoisers. These usually provide only convergence of the PnP iterates to a fixed point, under suitable regularity assumptions on the denoiser, rather than convergence of the method as a regularization technique, that is under vanishing noise and regularization strength. This paper serves two purposes: first, we provide an overview of the classical regularization theory in inverse problems and survey a few notable recent data-driven methods that are provably convergent regularization schemes. We then continue to discuss PnP algorithms and their established convergence guarantees. Subsequently, we consider PnP algorithms with learned linear denoisers and propose a novel spectral filtering technique of the denoiser to control the strength of regularization. Further, by relating the implicit regularization of the denoiser to an explicit regularization functional, we are the first to rigorously show that PnP with a learned linear denoiser leads to a convergent regularization scheme. The theoretical analysis is corroborated by numerical experiments for the classical inverse problem of tomographic image reconstruction.},
language = {en},
number = {4},
urldate = {2025-11-11},
journal = {Foundations of Computational Mathematics},
author = {Hauptmann, Andreas and Mukherjee, Subhadip and Schönlieb, Carola-Bibiane and Sherry, Ferdia},
month = aug,
year = {2025},
pages = {1087--1120},
}
@article{itou_lagrange_2021,
title = {Lagrange multiplier approach to unilateral indentation problems: {Well}-posedness and application to linearized viscoelasticity with non-invertible constitutive response},
volume = {31},
issn = {0218-2025, 1793-6314},
shorttitle = {Lagrange multiplier approach to unilateral indentation problems},
url = {https://www.worldscientific.com/doi/10.1142/S0218202521500159},
doi = {10.1142/S0218202521500159},
abstract = {The Boussinesq problem describing indentation of a rigid punch of arbitrary shape into a deformable solid body is studied within the context of a linear viscoelastic model. Due to the presence of a non-local integral constraint prescribing the total contact force, the unilateral indentation problem is formulated in the general form as a quasi-variational inequality with unknown indentation depth, and the Lagrange multiplier approach is applied to establish its well-posedness. The linear viscoelastic model that is considered assumes that the linearized strain is expressed by a material response function of the stress involving a Volterra convolution operator, thus the constitutive relation is not invertible. Since viscoelastic indentation problems may not be solvable in general, under the assumption of monotonically non-increasing contact area, the solution for linear viscoelasticity is constructed using the convolution for an increment of solutions from linearized elasticity. For the axisymmetric indentation of the viscoelastic half-space by a cone, based on the Papkovich–Neuber representation and Fourier–Bessel transform, a closed form analytical solution is constructed, which describes indentation testing within the holding-unloading phase.},
language = {en},
number = {03},
urldate = {2025-11-11},
journal = {Mathematical Models and Methods in Applied Sciences},
author = {Itou, Hiromichi and Kovtunenko, Victor A. and Rajagopal, Kumbakonam R.},
month = mar,
year = {2021},
pages = {649--674},
}
@article{krejci_analysis_2022,
title = {Analysis of a tumor model as a multicomponent deformable porous medium},
volume = {24},
issn = {1463-9963, 1463-9971},
url = {https://ems.press/doi/10.4171/ifb/472},
doi = {10.4171/ifb/472},
abstract = {We propose a diffuse interface model to describe a tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn–Hilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reaction-diffusion type equation for the nutrient proportion, and a quasistatic momentum balance.We prove here that the corresponding initial-boundary value problem has a solution in appropriate function spaces.},
number = {2},
urldate = {2025-11-28},
journal = {Interfaces and Free Boundaries, Mathematical Analysis, Computation and Applications},
author = {Krejčí, Pavel and Rocca, Elisabetta and Sprekels, Jürgen},
month = mar,
year = {2022},
pages = {235--262},
}
