ABSTRACT. Matrix Riccati equations have numerous mathematical applications, especially in control engineering. In this talk we discuss a derivation of analytical formulas for solutions of matrix Riccati equations in terms of matrix transfer functions for the original linear control systems. For many practical control systems such transfer functions can be evaluated by measuring response of the system to the input oscillating signal without the need to identify control system's parameters.

Thus, these analytical representations for solutions of matrix Riccati equations can be used for the design of stabilizing feedback control in the framework of some data-driven control procedure.

ABSTRACT. A derivation of so-called ``soft-margin Support Vector Machines with kernel'' is presented which does not rely on concepts from functional analysis such as Mercer's theorem that is frequently cited in this context, and that leads to a new analysis of the continuity properties of the kernel functions such as a new self-concordance condition for the kernel. The derivations are intended for a general audience, requiring some knowledge of calculus and linear algebra, while more advanced results used from optimization theory are being introduced in a self-contained form.

ABSTRACT. Consensus-based optimization (CBO) is a versatile multi-particle optimization method for performing non-convex and non-smooth global optimizations in high dimensions. Proofs of global convergence in the large-particle regime have been achieved for a broad class of objective functions in unconstrained optimizations. In this work, we adapt the algorithm for solving constrained optimizations on compact and unbounded domains with boundaries by introducing reflective boundary conditions. We also provide proof of global convergence for the large-particle regime. On the one hand, to minimize computational costs, it is desirable to keep the number of particles low. On the other hand, reducing the number of particles implies a diminished capability of the algorithm to explore. Hence, numerical heuristics are needed to ensure convergence of CBO in the low-particle regime. In this work, we significantly improve the convergence and complexity of CBO by utilizing an adaptive region control mechanism and choosing geometry-specific random noise. In particular, combining a hierarchical noise structure with the finite element method allows us to compute global minimizers for a high-dimensional, constrained p-Allen-Cahn problem with obstacles.

ABSTRACT. Curvature plays an important role in the function of biological membranes, and is therefore a readout of interest in microscopy data. The PyCurv library established itself as a valuable tool for curvature estimation in 3D microscopy images. However, in noisy images, the method exhibits visible instabilities, which are not captured by the standard error measures. In this article, we investigate the source of these instabilities, provide ade- quate measures to detect them, and introduce a novel post-processing step which corrects the errors. We illustrate the robustness of our enhanced method over various noise regimes and demonstrate that with our orientation correcting post-processing step, the PyCurv library becomes a truly stable curvature estimation tool for curvature quantification.

ABSTRACT. This presentation, based on the publication "Duswald, T., Lima, E. A. B. F., Oden, J. T., & Wohlmuth, B. (2024). Bridging scales: A hybrid model to simulate vascular tumor growth and treatment response," introduces an advanced computational framework for simulating vascular tumor growth and treatment response in a three-dimensional environment. Our hybrid model integrates two agent-based models to represent tumor cells and the vasculature alongside partial differential equations governing the diffusive dynamics of essential nutrients, vascular endothelial growth factors, and cancer drugs.

Focused on breast cancer cells with over-expressed HER2 receptors, the model simulates treatment with a combination of Doxorubicin and Trastuzumab, capturing this therapy's complex interactions and effects. The model's adaptability allows its application to other cancer scenarios beyond breast cancer. We validate the model by comparing simulation outcomes with existing pre-clinical data, demonstrating its qualitative accuracy in capturing treatment effects.

A significant feature of this work is its scalability, illustrated by simulating a vascular tumor with a 400 mm³ volume involving approximately 92.4 million agents. This showcases the model's robustness and the efficiency of the associated C++ code in handling large-scale simulations, which is crucial for detailed and accurate analysis. This work exemplifies the convergence of mathematical modeling and scientific computing in addressing complex biomedical problems.

ABSTRACT. An optimal control problem for nonlinear Fisher-Kolmogorov model of tumor growth is studied. In a given subdomain, it is required to minimize the density of tumor cells, while the drug concentration in tissue is limited by given minimal and maximal values. Based on derived estimates of the solution of the controlled system, the solvability of the control problem is proved. The problem is reduced to an optimal control problem with a penalty. An algorithm for solving the optimal control problem with a penalty is constructed and implemented. The efficiency of the algorithm is illustrated by a numerical example.

ABSTRACT. For patients suffering metastatic cancerous diseases, successful management of metastases is of utmost importance. About 90% of all cancer-related deaths are due to the presence of metastases. A common therapeutic approach is radiation: more than 50% of all cancer patients receive a form of radiotherapy during their treatment course [1,2,3,4]. But there are several dynamics that feature both challenges and opportunities as well as pose fundamental questions on current knowledge and understanding of radiation biology [5].

Clinical case reports of observed regression or progression of metastatic lesions distant from the irradiation site following local radiation on the primary tumor have been increasing in recent years. This phenomenon is sometimes referred to as the abscopal effect and has been claimed to stem from immune activation as a potential result of radiation therapy [6,7]. However, these effects are to date rather unpredictable events, as they have been classified with fundamentally different outcomes for patients featuring similar genetic profiles. The beneficial outcome of these systemic effects of local radiotherapy reliably triggered in clinical application could lead to the shrinkage of both treated tumor and distant untreated tumors.

In this presentation we propose a mathematical model based on a McKendrick-von Foerster transport equation with a global carrying capacity shared by all tumors. We calibrate the model with experimental data of 4T1 tumors in BALB/c mice and Py230 tumors in C57BL/6 mice. This allows us to examine the interaction dynamics of primary tumor and metastases in an untreated setting. We then simulate the local and global effects of radiotherapy to the primary tumor, and dive into different outcome scenarios that the assumed dynamics may exhibit. The outlook features insights into possible clinical integration.

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ABSTRACT. Bacterial biofilms are microbial depositions on inert surfaces. In the initial stages of biofilm formation bacteria attach to the surface, proliferate and start the production of extracellular polymeric substances that hold them together. EPS production is controlled by a quorum sensing mechanism. Biomass (cells and EPS) is detached into the aqueous environment by erosion or sloughing. Utilising the classical Wanner-Gujer 1D biofilm modeling concept one arrives at a model that consists of a system of ODEs for the reactor, a nonlocal hyperbolic system of balance laws for the biofilm proper, and a system of two point boundary value problems for dissolved susbtances such as nutrients and quorum sensing signal in the biofilm. We report and discuss numerical simulations that show the system can, depending on parameters, attain an upregulated steady state, a down-regulated steady state, and in the transition between these two passes through an oscillatory regime. This is joint work with Maryam Ghasemi (Waterloo/Queensland UT) and Firaz Khan (Guelph).