Prof. Akash Anand

Indian Institute of Technology Kanpur

Title of the talk : A numerical integration scheme for fast and high-order approximation of convolutions with weakly singular kernels

Abstract : Computationally efficient numerical methods for high-order approximations of convolution integrals involving weakly singular kernels find many practical applications including those in the development of fast quadrature methods for numerical solution of integral equations. Most fast techniques in this direction utilize uniform grid discretizations of the integral that facilitate the use of FFT for 𝑂(𝑛log𝑛) computations on a grid of size 𝑛. In general, however, the resulting error converges slowly with increasing 𝑛 when the integrand does not have a smooth periodic extension. Such extensions, in fact, are often discontinuous, and, therefore, their approximations by truncated Fourier series suffer from Gibbs oscillations. In this talk, we present an 𝑂(𝑛log𝑛) scheme, based on a Fourier extension approach for removing such unwanted oscillations, that not only converges with high order but is also relatively simple to implement.

Prof. K. Murugesan

National Institute of Technology Tiruchirappalli

Title of the talk : Image Processing using Numerical Techniques

Abstract : This talk concentrates in image compression, image edge detection, image segmentation, digital image inpainting, image fusion and their applications. Further, applications of numerical techniques in image processing will be discussed.

Prof. M.P. Rajan

Indian Institute of Science Education And Research Thiruvananthapuram

Title of the talk : Inverse Problems and its applications

Abstract : Many science and engineering problems can be modelled as an operator equation of the for Tx=y. In this talk, we will discuss how one could find a stable approximate solution to such problems and some of the applications of the inverse problem.

Prof. Sashikumar Ganesan

Indian Institute of Science Bangalore

Title of the talk : Artificial Intelligence-augmented Stabilized Finite Element Scheme

Abstract : Singularly perturbed partial differential equations are challenging to solve with standard numerical methods such as the Galerkin finite element method due to the presence of boundary and interior layers. The solution obtained with the standard numerical methods has spurious oscillations in the vicinity of these layers, and often stabilization techniques are employed to eliminate these oscillations. The accuracy of these stabilization techniques depends on a user-chosen stabilization parameter, of which an optimal value is challenging to find.

This talk will present an artificial intelligence-augmented Streamline Upwind Petrov-Galerkin finite element scheme for solving singularly perturbed partial differential equations. In particular, an artificial neural network framework is proposed to predict an optimal value for the stabilization parameter. The neural network is trained by minimizing a physics-informed cost function, where the equation’s mesh and physical parameters are used as input features. Further, the predicted stabilization parameter is normalized with the gradient of the Galerkin solution to treat the boundary and/or interior layer region adequately. The proposed approach suppresses the undershoots and overshoots in the stabilized finite element solution and outperforms the existing neural network-based partial differential equation solvers, such as Physics-Informed Neural Networks and the standard stabilized finite element scheme.