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Numerical methods for wave interaction with marine structures

Speaker:R. Gayathri

Abstract
The main objective of this talk is to employ the numerical methods for analyzing a broad class of boundary value problems, originating in the field of ocean engineering, that model wave interactions with permeable and impermeable structures. The impact of an array of porous breakwaters for reducing wave forces on a floating dock based on the small-amplitude water wave theory in finite water depth will be presented. The first part explains about breakwaters and their applications. The second part will be on the boundary element method followed by a problem on wave interaction with multiple porous structures. The subsequent study explores the wave interaction problem and illustrates various hydrodynamic coefficients. This talk establishes a physical- mathematical connection between physical oceanography and ocean engineering.

Title of the talk: Mathematical modeling and simulation of Mtb dynamics based on
experimental data
Speaker: Dr. Saikat Batabyal
Abstract:


Tuberculosis (TB), a disease caused by bacteria Mycobacterium tuberculosis (Mtb), remains
the major infectious disease in humans worldwide responsible for 10 million TB cases and 1.5
million deaths due to TB in 2018. In recent years, the increase of available data has led to
greater use of more complex phenomenological and mechanistic models, which increasingly go
by names such as machine learning or deep learning approaches. Currently, I am focusing on
building mathematical model of within-host dynamics of pathogen i.e. TB and the kinetics of
the immune response to the pathogen. A strong connection between mathematical models and
data is at the core of my research. The model formulation is typically driven by the quality and
type of the data available to parameterize these models. Fitting models to data is one of the
main aspects of my work. Another important aspect of my research is to formulate alternative
models and use data as an arbiter to discriminate between the alternatives. I also investigate the
kinetics of the T cell and incorporate it into the model. Hopefully, this experience will intersect
the industrial sector (company) where mathematical model-based understanding of the
phenomena in question and make predictions of how a specific intervention would impact the
dynamics of the system.

Approximation of Bayesian  inverse problems of  determining initial data from Eulerian observations in convective Brinkman-Forchheimer equations
 
Speaker: Manil T Mohan
Abstract:
 
The Bayesian approach to an  inverse problem  for two and three dimensional convective Brinkman-Forchheimer (CBF) equations  in periodic domains is considered in this work. Given noisy Eulerian observations of the velocity field, the inverse problem is to determine  the initial data whose prior is known in terms of a prior probability measure. The well-posedness of this inverse problem is carried out by a regularization strategy using the Bayesian formulation of the problem at the level of probability measures. We prove a stability property by estimating the distance between the true and approximate posterior distributions, in the Hellinger metric, in terms of error estimates for approximation of the underlying forward problem.

Numerical Identification of Initial Temperatures in Heat Equation with Dynamic Boundary Conditions

Walid Zouhair

Abstract:

In this work, we investigate the inverse problem of numerically identifying unknown initial temperatures in a heat equation with dynamic boundary conditions whenever some overdetermination data are provided after a final time. This is a backward parabolic problem which is severely ill-posed. As a first step, the inverse problem is reformulated as a minimization problem for an associated Tikhonov functional. Using the weak solution approach, an explicit formula for the Fréchet gradient of the cost functional is derived from the corresponding sensitivity and adjoint problems. Then, the Lipschitz continuity of the gradient is proved. Next, further spectral properties of the input–output operator are established. Finally, the numerical results for noisy measured data are performed using the regularization framework and the conjugate gradient method. We consider both one- and two-dimensional numerical experiments using finite difference discretization to illustrate the efficiency of the designed algorithm. Aside from dealing with a time derivative on the boundary, the presence of a boundary diffusion makes the analysis more complicated. This issue is handled in the 2-D case by considering the polar coordinate system. The presented method implies fast numerical results.

Keywords: Inverse problem, backward parabolic problem, dynamic boundary condition, adjoint problem, conjugate gradient.

Utilization of AI algorithms in Atmospheric science problems
Speaker :Bipin Kumar
Abstract:
Within the field of earth science, particularly in atmospheric science, the majority of forecasting challenges are traditionally addressed through the application of dynamical models. Predictions regarding weather and climate primarily rely on numerical models that solve sets of probability density functions (PDFs). However, recent progress in artificial intelligence algorithms and computational capabilities has led to a shift in focus towards the development of data-driven models for weather and climate forecasting, among other applications. This presentation will delve into several instances of weather forecasting and statistical data downscaling problems, showcasing the utilization of deep learning algorithms in these contexts.


 






Deep learning is a rapidly evolving field with a wide range of applications. This presentation provides a concise introduction to deep learning and its underlying mathematical principles. Special emphasis is placed on Physics-Informed Neural Networks (PINN) and their utility in solving differential equations. The discussion encompasses various research areas and highlights the ongoing improvements required in the domain of PINN. Additionally, the presentation delves into the research conducted by our group, focusing on the challenges posed by discontinuous problems and offering strategies to stabilize PINN solvers, particularly in the context of the gas dynamics Euler equation. An abbreviated exploration of deep learning's application in reduced order modeling is also presented, along with an introduction to its use in numerical stability analysis, including associated challenges. The presentation concludes by emphasizing the significance of neural networks and introduces activation functions developed within our research laboratory.

An Overview of Deep Learning for Engineers and Mathematicians
Speaker: Arun Govind
abstract:

FRACTIONAL MATHEMATICAL MODELING AND BEYOND
 

Speaker: BONGSOO JANG

Abstract

 The theory of derivatives of non-integer order goes back to the Leibniz’s note
in his list to L’Hospital, Sep 30, 1695, in which the meaning of the derivative of order
one half is discussed (Fractional-order). Fractional derivatives provide an excellent tool for
the description of memory and hereditary properties of various materials and processes.
Due to this reason, Fractional Mathematical Modeling(FMM) or Fractional-order (partial)
differential equations(FPDEs) have been successfully applied in physics, biology, applied
sciences, and engineering.

In this talk, I discuss several difficulties in finding numerical approximations for Frac-
tional Mathematical Modeling, such as an expensive computational cost. Also, I introduce

recent research improvements to overcome these difficulties and new engineering applica-
tions in nanofluids. In addition, I introduce Fractional Physics-informed neural networks

(fPINNs), an extended variant of PINNs that utilize standard feedforward neural networks
(NN) while explicitly incorporating partial differential equations (PDEs) into the neural
network architecture via automatic differentiation.

Title: Machine Learning for Fluid Flow Problems
Speaker: H.Thameem Basha
Abstract:

Fluid dynamics challenges are common in various scientific and engineering fields, such as energy-related, aerospace, automotive, environmental science, and biomedical research. Traditional methods for solving fluid flow equations can be computationally intensive, especially for complex large-scale systems. Recently, the integration of data-driven approaches, particularly machine learning techniques like deep learning and supervised machine learning, has brought about significant advancements in computational fluid dynamics. In this invited talk, we explore this convergence of fluid dynamics and data-driven methodologies, with a specific focus on the role of neural networks. We aim to demonstrate how these data-driven techniques enhance our understanding, predictive capabilities, and optimization strategies for fluid flow phenomena. Through a discussion of various data-driven ML methods, our presentation highlights the transformative potential of these approaches and their applicability to complex fluid flow problems.

Discontinuous Galerkin finite element approximation for elasticity/poroelasticity coupling

Arbaz Khan
Department of Mathematics,
IIT Roorkee Roorkee,
India. arbaz@ma.iitr.ac.in
Abstract
It is the aim of this talk to discuss the Hdiv-conforming discontinuous Galerkin formulation for elasticity/poroelasticity coupling. We show that our a priori error analysis is robust w. r. t. the model parameters. Moreover, a posteriori error analysis for interfacial elasticity-poroelasticity is discussed. The proposed residual-based estimator is also shown to be reliable, efficient and robust. Numerical results are presented that validate the theoretical es



In this presentation, we will introduce numerical methods that effectively deal with time-fractional partial differential equations (TFPDEs) with Caputo sense. For the Caputo differential operator with fractional order ‘nu’ less than 1, we rewrite the TFPDEs as an equivalent integral form with respect to time, circumventing the derivative of the solution by the integral by parts. We obtain a discrete formulation directly from the integral form using the Lagrange interpolating polynomials. It turns out that the explicit scheme has the order 2 - nu and 3 - nu of the truncation error for the linear and quadratic interpolation, respectively. To navigate challenges arising as 'nu' approaches zero, characterized by a strong singularity, we put forth an explicit predictor-corrector scheme fortified with a perturbation technique. Conversely, when 'nu' gravitates toward one, the truncation errors diminish by O(h) for linear interpolation and O(h^2) for quadratic interpolation. To further bolster the convergence rate, we dissect the fractional order, transmute the TFPDEs into a system of fractional differential equations, and harness the predictor-corrector approach.

Leveraging the benefits of the proposed explicit scheme, we integrate multiple numerical techniques in the spatial domain. We favor the Isogeometric collocation method over the Galerkin approach, primarily due to the computational complexity imposed by the memory term in the fractional differential operator and the enhanced geometric representation provided by Isogeometric analysis. The efficacy of our proposed scheme is illustrated through the lens of fractional phase-field models and time-fractional convection-diffusion-reaction models.

In our concluding remarks, we synergize the explicit scheme with a geometrical mapping strategy. This approach utilizes a push-forward transformation from the parameter space to the physical space, producing point singularity functions anchored on the parametrization of the circular arc and NURBS (non-uniform rational B-spline). This innovative combination offers a robust approximate solution for TFPDEs containing crack singularities.

An Explicit Scheme Coupled with the Isogeometric Collocation Method for Time-Fractional PDEs
Speaker: Hyunju Kim
Abstract

Title: Advanced reservoir computing and its applications
Speaker:Pilwon kim
Abstract:
 

Reservoir computing(RC) is a relatively new computational framework that harnesses the power of complex systems for performing computations. It is based on the dynamics of nonlinear system called a reservoir, through which input signals are projected into higher dimensional computational spaces. Then a simple linear readout function layer can turn the projected features into output signals to perform tasks like classification and time-series analysis.I will introduce a new class of RCs based on networks of coupled oscillators and chaotic systems. These RCs are easy to parameterize to maximize their performance and provide a starting point for building RCs with real physical systems. In the last part of the talk, I will present several applications of RCs, including image/signal denoising and detecting early warning precursors from the systems that undergo a critical transition.

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