| layout | title | subtitle |
|---|---|---|
page |
Projects |
What have I done in the last few years |
In this research area, mathematicians and researchers applying mathematics form research teams to attack social issues that have been difficult to be solved. Through the problem-attacking process, we expect also the development of mathematics itself. More specifically, targeting the phenomena for which the governing principles and rules have been unclear, by using the abstractness and universality of mathematics as well as knowledge obtained through application, we advance research to derive innovative models based on mathematical ideas to find the hidden essence of them and to develop new mathematical methods to approach them. We include research for proving the validity of the description on the essence of phenomena by old and new mathematical models, as well as building mathematical theories and techniques for evaluating these models. The targets may be found in social, natural, or life phenomena. This may include others if the research aims to create new fields and give solutions to social issues.
Our center was created by faculty members of the Mathematical Institute in the Graduate School of Science, Research Center for Pure and Applied Mathematics in the Graduate School of Information Sciences, Advanced Institute for Materials Research Mathematical Science Group, Graduate School of Economics and Management, and the Institute for Fluid Science in April 2017. It was created to promote the fundamental research of mathematical sciences through collaboration between mathematics and various fields. Our center was created as an international hub for academic research for interdisciplinary fields that use mathematical science as a base. We promote the creation of new fields and basic constructions of mathematical science that work towards solving social problems. We also work to train people with knowledge of mathematical sciences and a global world-view.
This project aims at the design and the analysis of innovative methods for reducing the computational complexity of mathematical models relevant for medical applications, by appropriate surrogating techniques. Although it is now evident the potential role of mathematical modeling in medical applications for many proofs of concept, the real penetration of clinical practice is prevented by many factors. One of those is the high computational costs that the solution of complex problems like medical ones may require and the need for remote high-performance computing architectures. These requirements may impair the real impact of scientific computing in clinics. As an example, we mention the data assimilation techniques required to personalize numerical models to patient-specific settings. These problems almost invariably lead to the solution of inverse problems, with constrained minimization - computationally intensive - procedures. More specifically, we mention the identification of arterial compliance based on imaging, the solution of inverse fluid-structure interaction problems, the identification of cardiac conductivities/fibers from ventricular potential measures or the identification of sources in electroencephalography and magnetoencephalography. More in general, all the numerical models developed for medical applications - as for other fields - need to be equipped by Uncertainty Quantification procedures. These require additional computational costs. On the other hand, the concept of accuracy in clinics is different than in mathematics, as it relies on the correct stratification of patients into operative clusters more than on the number of decimal digits of a numerical simulation. Surrogate models, even if not completely correct from the physical point of view, maybe an appropriate trade-off between accuracy and efficiency, leading to clinically correct conclusions in reasonable timelines.
Hierarchical Model Reduction Techniques for Incompressible Fluid-Dynamics and Fluid-Structure Interaction Problems
Networks perfused by fluids are found in several engineering applications, ranging from hydrogeology, oil distribution, and internal combustion engines to hemodynamics. A quantitative analysis of these problems is of utmost interest for understanding fluid dynamics in the network, for predicting effects of local changes on the network (for instance, the effects of a surgical operation over the fluid dynamics in the arterial tree), and for optimizing flow distribution. Mathematical description and numerical approximation of these problems are challenging when coupling the accurate description of local dynamics with the large scale of the network. This proposal investigates a novel numerical method to undertake the quantitative analysis of fluid dynamics in complex networks called HiMod (Hierarchical Model Reduction). The primary (but not exclusive) application is the physiopathology of the arterial system, including in the mathematical model up to almost 2000 segments of the network. Several specific properties of this method need to be investigated for its development and engineering. The research provides a graduate student the opportunity of working on advanced mathematical and numerical techniques - including theoretical as well as practical aspects - in a truly interdisciplinary framework with frequent contacts with engineers and doctors expected to be the end-users of these methodologies.
Advanced Numerical Methods Combined with Computational Reduction Techniques for Parameterised PDEs and Applications
The objective of this project has been the design and analysis of innovative numerical methods for the approximation of partial differential equations (PDEs) in computational science and engineering. The increasing complexity of realistic models and the evolution of computational platforms and architectures has led the numerical analysis community to develop more efficient, effective, and innovative methods. We have shared consolidated expertise on advanced discretization schemes based on variational approaches, such as h-type finite elements (FEM), isogeometric analysis (IGA), spectral, and boundary elements (BEM). In several applied fields and scenarios a successful resolution of PDEs with different values of control or design variables or different physical/geometric quantities has been required, thus demanding high computational efficiency. We have developed suitable Reduced Order Methods (ROM), such as reduced basis methods (RBM), proper orthogonal decomposition (POD) and hierarchical model (HiMod) reduction, representing effective strategies to contain the overall computational costs. We would expect that our results are opening new scenarios, making possible the solution of more complex problems with significantly reduced computational effort. Applications are playing a leading role in developing demonstrative proofs of concept related to methodological advances, as well as important software developments, in order to enhance efficient scientific computing on modern platforms/technological devices.