Unlocking the power of unstructured Isogeometric Analysis: some recent mathematical advances and a more unified framework for the numerical analysis of PDEs

B-spline and NURBS (Non-Uniform Rational B-Spline) functions have enjoyed a long and prosperous career in Computer-Aided Design (CAD), and have recently been successfully employed in the numerical analysis of partial differential equations (PDEs), spawning a new discipline called Isogeometric Analysis (IGA). Advantages include an almost perfect representation of the problem geometry, less numerical noise and dispersion, longer simulation timesteps and better stability. However, in the multivariate case, the usual functions obtained via tensor products tend to limit the topology and complexity of the problem geometry, perhaps explaining its limited adoption outside engineering, where CAD models are not generally available. In this talk, we present some new mathematical advances which have allowed us to formulate an isogeometric analysis scheme based on general, unstructured multivariate splines built on a simple point cloud. We show how these advances allow us to construct bases in all dimensions, for all polynomial degrees and in the presence of affinely dependent and repeated points, an essential feature needed for the definition of boundary conditions and domain decomposition. Based on this approach, we have developed accurate and fast evaluation algorithms for unstructured spline bases, unlocking their use in the numerical simulation of PDEs. We show how this method naturally incorporates both the Finite Elements and Discontinous Galerkin(DG) bases in a single unified framework, thus allowing the natural blending of DG and IGA domains in the same simulation, drawing advantages from both methods. We showcase our method and its favorable numerical properties in the context of 2D and 3D acoustic wave propagation, with particular attention on the formulation of the inverse problem through Full Waveform Inversion (FWI) and how adding the position of points in the point cloud as degrees of freedom can allow the automatic reconstruction of the position and shape of sharp discontinuities in the velocity model.