• Brigitte Forster, Frame Recycling

Joint work with: Peter Massopust (TU Munchen), Ole Christensen (DTU Lyngby) and Florian Heinrich (University of Passau).

Abstract: Grafakos and Sansing have shown how to obtain directionally sensitive time-frequency decompositions in L^2(R^n) based on Gabor systems in L^2(R); the key tool is the "ridge idea", which lifts a function of one variable to a function of several variables. We extend their idea by showing that similar results hold starting with general frames for L^2(R), both in the setting of discrete frames and continuous frames. This allows to apply the theory for several other classes of frames, e.g., wavelet frames and shift-invariant systems. We will consider applications to the Meyer wavelet and the B-spline family. We will close with a short discussion of partial ridges.

forster_osna_2.pdf

• Filomena Di Tommaso, Multinode Shepard methods and Applications

Abstract: In this talk we give an overview on the Multinode Shepard methods for scattered data interpolation and apply them to the numerical solution of PDEs by collocation. These methods derive from the Little’s observation about the possibility to improve the precision and the behaviour of the classical Shepard interpolants only by fixing triangulations of the scattered nodes and by blending local linear interpolants on the vertices of triangles with Shepard like basis functions based on those triangles. The main advantage of the Little interpolants is their explicit expressions that rely only on function values without using any derivative data (exact or approximated). Moreover, it is possible to provide algorithms for their fast computation based on a criterion of choice of triangles (which may overlap or being disjoint) and on a searching technique to detect and select the nearest neighbour points. Further improvements of such interpolants require the solution of two main problems: the partitioning of the node set in ordered subsets that guarantees the existence and accuracy of approximation of local interpolation polynomials of fixed total degree and the possibility to compute them in a stable way. In line with the Kansa idea, the multinode Shepard operators can be applied in the numerical solution of elliptic PDEs with Dirichlet conditions via collocation. Numerical experiments show accuracy of approximation comparable with that one of the Kansa method but a lower condition number of the collocation matrix.

ditommaso_osna_2.pdf

• Elisabeth Larsson, The least-squares RBF-FD method

Joint work with: Igor Tominec​ (Divison of Scientific Computing, Department of Information Technology, Uppsala University, Sweden) and Alfa Heryudono (Department of Mathematics, University of Massachusetts Dartmouth, MA, USA.)

Abstract: The radial basis function-generated finite difference method (RBF-FD) aims to combine the simplicity of finite difference methods and the flexibility of finite element methods. The approximations are based on scattered data, and do not require a grid or a mesh. Stencil weights for approximation of function values and derivatives are computed based on interpolation of local data. The RBF-FD method was first introduced as a collocation method, where partial differential equations (PDEs) and boundary conditions are enforced at the stencil points. Here, we introduce the least-squares RBF-FD method, where the PDE and boundary conditions are enforced at ​ M>N ​ points, where ​ N ​ is the number of stencil points. The introduction of oversampling improves the stability of the approximation as well as its behaviour close to boundaries with Neumann conditions. This new formulation of the method also allowed us to prove theoretical convergence results for a Poisson problem with mixed boundary conditions. In the talk, we discuss the principles behind the convergence results, we show numerical experiments that support the theory, as well as results from some non-trivial application problems.

Larsson_osna_2.pdf

• Sourav Dutta, Kernel-based approximation methods for reduced order modeling applications in hydrology

Abstract: The depth-averaged two-dimensional Shallow Water Equations (SWE) are a well-studied system of hyperbolic or near-hyperbolic partial differential equations (PDEs) that are widely adopted to study various flow scenarios which arise in engineering applications that are critical for achieving the U.S. Army Corps of Engineer’s mission of delivering vital public and military engineering services.

However, for multi-query, real-time and fast-replay applications arising in optimal design, risk assessment, or ensemble forecasting problems, fully resolved numerical simulations of a parametrized SWE model pose a significant computational challenge. Reduced order modeling (ROM) encompasses a wide array of approaches that aim to drastically reduce the computational burden versus reference, high-fidelity simulations. These techniques introduce a linear or non-linear mapping (or projection) from the space of high-fidelity (full order) solutions to a lower-dimensional latent space in which the model dynamics can be efficiently approximated using various data-driven methods.

In this talk, radial basis functions (RBF) will be adopted as a classical kernel-based regression method to devise non-intrusive latent space representations of the high-fidelity model dynamics. A novel greedy strategy will be presented to generate an optimal selection of RBF centers. Moreover, higher-order discretization techniques will be employed to achieve improved stability in the long-term forecasting of the system coefficients in the latent space. The performance of this framework will be demonstrated using real-world, large-scale riverine and coastal flow problems. Quantitative comparison results with existing data-driven strategies like Gaussian process regression (GPR), dynamic mode decomposition (DMD), and machine learning (ML)-based time series learning techniques will be presented to assess predictive accuracy, efficiency, and computational stability of the different reduced order models.

Sourav_osna21.pdf