• Tomas Sauer, Continued Fractions: Music, Moments and Hurwitz

Abstract: Continued fractions are a classical concept in mathematics that works not only for numbers but in fact in any Euclidean ring, allowing us to extend them to univariate polynomials as well. Their extraordinary rate approximation is reflected in music (why does the octave have 12 semitones) as well as in the study of moment sequences, leading to Gaussian quadrature formulas. And, moreover, they allow for classification and enumeration of Hurwitz polynomials via positive pairs. The talk presents a short overview over the main concepts.

• Wolfgang Erb, Approximation and classification of graph signals with positive definite graph basis functions

Abstract: For the approximation of graph signals with generalized shifts of a graph basis function (GBF), we study how the concept of positive definite functions can be translated to graphs. This concept merges kernel-based approximation with spectral theory on graphs and can be regarded as a graph analog of radial basis function methods in euclidean spaces or on the sphere. We provide several descriptions of positive definite functions on graphs, the most relevant one is a Bochner- type characterization in terms of positive Fourier coefficients. These descriptions allow us to design GBF’s and to study GBF approximation in more detail: we are able to characterize the native spaces of the interpolants, we give explicit estimates for the approximation error and provide ways on how to calculate the approximants in an efficient manner. As a final application, we show how GBFs can be used for classification tasks on graphs.

• Anna Maria Massone, De-saturation of EUV images from the Solar Dynamics Observatory

Abstract: Launched on February 2010 the Atmospheric Imaging Assembly in the Solar Dynamics Observatory (SDO/AIA) is still providing us with full Sun images in 7 different wavelengths (mainly EUV) with 12-second cadence. However, for a significant amount of these images, saturation affects their most intense core, inhibiting their full exploitation. All information on the radiation flux which is lost due to primary saturation is actually present in the diffraction pattern associated to the telescope observations and therefore the signal in the primary saturation region can be restored by solving an inverse diffraction problem. This talk describes the mathematics at the basis of an automatic procedure for the recovery of information in the primary saturation region, relying on an inversion analysis of the diffraction pattern.

• Slides, TBA

• Varun Shankar, An Efficient High-Order Meshless Method for Advection-Diffusion Equations on Time-Varying Irregular Domains.

Abstract: We present a high-order radial basis function finite difference (RBF-FD) framework for the solution of advection-diffusion equations on time-varying domains. Our framework is based on a generalization of the recently-developed Overlapped RBF-FD method that utilizes a novel automatic procedure for computing RBF-FD weights on stencils in variable-sized regions around stencil centers. This procedure eliminates the overlap parameter, thereby enabling tuning-free assembly of RBF-FD differentiation matrices on moving domains. In addition, our framework utilizes a simple and efficient procedure for updating differentiation matrices on moving domains tiled by node sets of time-varying cardinality. Finally, advection-diffusion in time-varying domains is handled through a combination of rapid node set modification, a new high-order semi-Lagrangian method that utilizes the new tuning-free overlapped RBF-FD method, and a high-order time-integration method. The resulting framework has no tuning parameters, has O(N) time complexity, and allows for high orders of convergence across a range of Peclet numbers on time-varying irregular domains.

• Abstract, Joint work with Pakshal Bohra, Joaquim Campos, Harshit Gupta, Shayan Aziznejad

We present a unifying functional framework for the implementation and training of deep neural networks (DNN) with free-form activation functions. To make the problem well posed, we constrain the shape of the trainable activations (neurons) by penalizing their second-order total-variations. We prove that the optimal activations are adaptive piecewise-linear splines, which allows us to recast the problem as a parametric optimization. We then specify some corresponding trainable B-spline-based activation units. These modules can be inserted in deep neural architectures and optimized efficiently using standard tools. We provide experimental results that demonstrate the benefit of our approach.

• Emiliano Cirillo, Handling heterogeneous structures and materials using interpolatory blending schemes in V-reps.

Abstract: Additive manufacturing has recently enabled the creation of heterogeneous objects in which different materials can be specified at different locations. This creates new exciting perspectives in additive crafting but also presents several challenges, including the problem of specifying rules for varying materials at particular locations as well as encoding user specified properties (such as stress tensors) inside a volumetric model. The aim of this work is to provide an answer to these problems, with the definition of local feature preserving blending schemes that allow both the encodement and the blend of property fields over volumetric representation models, while preserving the original values of such properties where possible. A distance-based blending scheme satisfying this characteristic is described both theoretically and practically, with examples of both homogeneous and heterogeneous primitives in volumetric models.