Reduced order modeling of the cardiac function across the scales

Providing fast and reliable numerical simulations becomes extremely important when dealing with the complex applications arising in computational medicine. Indeed, even more strikingly than in other branches of computational mechanics and applied science, being able to perform accurate numerical simulations in a short amount of time—second or minutes, rather than hours or even days—is crucial for problems arising from the life sciences, e.g., in the simulation of the behavior of living organs. Indeed, in order to be representative, integrative and predictive, in-silico (or numerical) models must be able to (i) account for sources of variability carried by subject-specific features and (ii) run on deployed platforms rather than on supercomputers. The virtual description of the heart, to which this chapter is devoted, involves additional sources of complexities, such as the need to deal with several coupled models across different spatial and temporal scales. Reduced order modeling techniques, and the more recent algorithms for machine learning and deep learning, represent key strategies to meet these challenges, making the numerical modeling of the heart an extraordinary and fascinating test bed for these methods.