List of Speakers
Date : 19/08/2020, Time: 3:00 pm
Date : 19/08/2020, Time: 3:00 pm
Title : Approximation of eigenvalue problems arising from partial differential equations: examples and counterexamples
Title : Approximation of eigenvalue problems arising from partial differential equations: examples and counterexamples
Abstract : We discuss the finite element approximation of eigenvalue problems arising from elliptic partial differential equations. We present various examples of non-standard schemes, including mixed finite elements, approximation of operators related to the least-squares finite element method, parameter dependent formulations such as those produced by the virtual element method. Each example is studied theoretically; advantages and disadvantages of each approach are pointed out.
Abstract : We discuss the finite element approximation of eigenvalue problems arising from elliptic partial differential equations. We present various examples of non-standard schemes, including mixed finite elements, approximation of operators related to the least-squares finite element method, parameter dependent formulations such as those produced by the virtual element method. Each example is studied theoretically; advantages and disadvantages of each approach are pointed out.
boffi.pdf
Date : 10/09/2020, Time : 5:30 pm
Date : 10/09/2020, Time : 5:30 pm
Title : Random ordinary differential equations and their numerical approximation
Title : Random ordinary differential equations and their numerical approximation
Abstract : Random ordinary differential equations (RODEs) are pathwise ordinary differential equations that contain a stochastic process in their vector field functions. They have been used for many years in a wide range of applications, but have been very much overshadowed by stochastic ordinary differential equations (SODEs). The stochastic process could be a fractional Brownian motion, but when it is a diffusion process there is a close connection between RODEs and SODEs through the Doss-Sussmann transformation and its generalisations, which relate a RODE and an SODE with the same (transformed) solutions. RODEs play an important role in the theory of random dynamical systems and random attractors.
Abstract : Random ordinary differential equations (RODEs) are pathwise ordinary differential equations that contain a stochastic process in their vector field functions. They have been used for many years in a wide range of applications, but have been very much overshadowed by stochastic ordinary differential equations (SODEs). The stochastic process could be a fractional Brownian motion, but when it is a diffusion process there is a close connection between RODEs and SODEs through the Doss-Sussmann transformation and its generalisations, which relate a RODE and an SODE with the same (transformed) solutions. RODEs play an important role in the theory of random dynamical systems and random attractors.
Classical numerical schemes such as Runge-Kutta schemes can be used for RODEs but do not achieve their usual high order since the vector field does not inherit enough smoothness in time from the driving process. It will be shown how, nevertheless, Taylor expansions of the solutions of RODES can be obtained when the stochastic process has H\"older continuous sample paths and then used to derive pathwise convergent numerical schemes of arbitrarily high order. Some other methods will be indicated for more specific kinds of noise.
Classical numerical schemes such as Runge-Kutta schemes can be used for RODEs but do not achieve their usual high order since the vector field does not inherit enough smoothness in time from the driving process. It will be shown how, nevertheless, Taylor expansions of the solutions of RODES can be obtained when the stochastic process has H\"older continuous sample paths and then used to derive pathwise convergent numerical schemes of arbitrarily high order. Some other methods will be indicated for more specific kinds of noise.
RODEsTalk.pdf
Date : 17/09/2020, Time : 6:30 pm
Date : 17/09/2020, Time : 6:30 pm
Title : Physics-Informed Neural Networks (PINNs): Algorithms & Applications
Title : Physics-Informed Neural Networks (PINNs): Algorithms & Applications
Abstract : We will present a new approach to develop a data-driven, learning-based framework for predicting outcomes of physical and biological systems and for discovering hidden physics from noisy data. We will introduce a deep learning approach based on neural networks (NNs) and generative adversarial networks (GANs). We also introduce new NNs that learn functionals and nonlinear operators from functions and corresponding responses for system identification. Unlike other approaches that rely on big data, here we "learn" from small data by exploiting the information provided by the physical conservation laws, which are used to obtain informative priors or regularize the neural networks. We will also make connections between Gauss Process Regression and NNs and discuss the new powerful concept of meta-learning. We will demonstrate the power of PINNs for several inverse problems in fluid mechanics, solid mechanics and biomedicine including wake flows, shock tube problems, material characterization, brain aneurysms, etc, where traditional methods fail due to lack of boundary and initial conditions or material properties.
Abstract : We will present a new approach to develop a data-driven, learning-based framework for predicting outcomes of physical and biological systems and for discovering hidden physics from noisy data. We will introduce a deep learning approach based on neural networks (NNs) and generative adversarial networks (GANs). We also introduce new NNs that learn functionals and nonlinear operators from functions and corresponding responses for system identification. Unlike other approaches that rely on big data, here we "learn" from small data by exploiting the information provided by the physical conservation laws, which are used to obtain informative priors or regularize the neural networks. We will also make connections between Gauss Process Regression and NNs and discuss the new powerful concept of meta-learning. We will demonstrate the power of PINNs for several inverse problems in fluid mechanics, solid mechanics and biomedicine including wake flows, shock tube problems, material characterization, brain aneurysms, etc, where traditional methods fail due to lack of boundary and initial conditions or material properties.
Date : 07/10/2020, Time : 8:00 pm
Date : 07/10/2020, Time : 8:00 pm
Title : Structure preserving methods for approximating fluid stresses
Title : Structure preserving methods for approximating fluid stresses
Abstract : This talk will focus on finite element techniques for simulating incompressible fluid flow. An age-old subject of discussion in computational fluid dynamics is the proper treatment of the incompressibility constraint on the fluid velocity u, namely div(u)=0. A new twist in this topic arising from a series of recent developments by multiple authors is the treatment of the incompressibility constraint using the Sobolev space H(div), the space of vector fields whose components and whose divergence are square integrable. Instead of using the standard Lagrange finite element spaces, the use of H(div)-conforming finite elements for velocity approximation brings new tools into play. A natural question to ask in this context is this: what is a natural Sobolev space for viscous fluid stresses to pair with an H(div) velocity? We report on results obtained in our search for a mixed formulation with a stress space that pairs well with H(div)-spaces for velocity. The main new insight is that stresses should lie in a nonstandard Sobolev space H(curl div). The need to study this space and develop finite elements for it will be amply evident in this presentation. We will show that structure-preservation properties like mass conservation and pressure robustness are immediate in the newly introduced framework.
Abstract : This talk will focus on finite element techniques for simulating incompressible fluid flow. An age-old subject of discussion in computational fluid dynamics is the proper treatment of the incompressibility constraint on the fluid velocity u, namely div(u)=0. A new twist in this topic arising from a series of recent developments by multiple authors is the treatment of the incompressibility constraint using the Sobolev space H(div), the space of vector fields whose components and whose divergence are square integrable. Instead of using the standard Lagrange finite element spaces, the use of H(div)-conforming finite elements for velocity approximation brings new tools into play. A natural question to ask in this context is this: what is a natural Sobolev space for viscous fluid stresses to pair with an H(div) velocity? We report on results obtained in our search for a mixed formulation with a stress space that pairs well with H(div)-spaces for velocity. The main new insight is that stresses should lie in a nonstandard Sobolev space H(curl div). The need to study this space and develop finite elements for it will be amply evident in this presentation. We will show that structure-preservation properties like mass conservation and pressure robustness are immediate in the newly introduced framework.
- Prof. Emmanuel Gobet (Ecole Polytechinque, France)
Date : 22/10/2020, Time: 5:30 pm
Date : 22/10/2020, Time: 5:30 pm
Title : WEAK APPROXIMATIONS AND VIX OPTION PRICES EXPANSIONS IN ROUGH FORWARD VARIANCES MODELS
Title : WEAK APPROXIMATIONS AND VIX OPTION PRICES EXPANSIONS IN ROUGH FORWARD VARIANCES MODELS
Abstract : We provide approximation formulas for VIX derivatives in forward variance models, with particular emphasis on the family of so-called Bergomi models: the rough Bergomi model of Bayer, Friz, and Gatheral (2016), and an enhanced version that is able to generate realistic positive skew for VIX smiles – introduced simultaneously by De Marco (2018) and Guyon (2018) mimicking the ideas of Bergomi (2008), that we refer to as “mixed rough Bergomi model”. Following the methodology of Gobet and Miri (2014), we derive weak approximations for the VIX random variable, leading to option price approximations under the form of explicit combinations of Black–Scholes prices and greeks. We stress that our approach does not rely on small-time asymptotics and can, therefore, be applied to any option maturity. Our results are illustrated by several numerical experiments and calibrations to VIX market data.
Abstract : We provide approximation formulas for VIX derivatives in forward variance models, with particular emphasis on the family of so-called Bergomi models: the rough Bergomi model of Bayer, Friz, and Gatheral (2016), and an enhanced version that is able to generate realistic positive skew for VIX smiles – introduced simultaneously by De Marco (2018) and Guyon (2018) mimicking the ideas of Bergomi (2008), that we refer to as “mixed rough Bergomi model”. Following the methodology of Gobet and Miri (2014), we derive weak approximations for the VIX random variable, leading to option price approximations under the form of explicit combinations of Black–Scholes prices and greeks. We stress that our approach does not rely on small-time asymptotics and can, therefore, be applied to any option maturity. Our results are illustrated by several numerical experiments and calibrations to VIX market data.
(Joint work with F. Bourgey and S. De Marco.)
(Joint work with F. Bourgey and S. De Marco.)
- Prof. Alan Demlov (Texas A&M University, TX, USA)
Date : 28/10/2020, Time: 7:30 pm
Date : 28/10/2020, Time: 7:30 pm
Title : Geometric errors in surface finite element methods
Title : Geometric errors in surface finite element methods
Abstract : Surface finite element methods (SFEM) are widely used to approximately solve partial differential equations posed on surfaces. Such PDE arise in a range of applications, from image processing to fluid dynamics. Typical SFEM involve first approximating the underlying surface and then formulating the finite element method on the approximate surface. In this talk we discuss how approximation of the underlying surface affects the overall quality of the finite element approximation. The talk includes discussion of the effects of surface smoothness on geometric errors and some surprising recent results on approximation of surface eigenvalue problems.
Abstract : Surface finite element methods (SFEM) are widely used to approximately solve partial differential equations posed on surfaces. Such PDE arise in a range of applications, from image processing to fluid dynamics. Typical SFEM involve first approximating the underlying surface and then formulating the finite element method on the approximate surface. In this talk we discuss how approximation of the underlying surface affects the overall quality of the finite element approximation. The talk includes discussion of the effects of surface smoothness on geometric errors and some surprising recent results on approximation of surface eigenvalue problems.
IITR_seminar.pdf
- Prof. J Hesthaven (EPFL Lausanne, Switzerland)
Date : 26/11/2020, Time: 3:00 pm
Date : 26/11/2020, Time: 3:00 pm
Title : Nonintrusive reduced order models using physics informed neural networks
Title : Nonintrusive reduced order models using physics informed neural networks
Abstract : The development of reduced order models for complex applications, offering the promise for rapid and accurate evaluation of the output of complex models under parameterized variation, remains a very active research area. Applications are found in problems which require many evaluations, sampled over a potentially large parameter space, such as in optimization, control, uncertainty quantification, and in applications where a near real-time response is needed.
Abstract : The development of reduced order models for complex applications, offering the promise for rapid and accurate evaluation of the output of complex models under parameterized variation, remains a very active research area. Applications are found in problems which require many evaluations, sampled over a potentially large parameter space, such as in optimization, control, uncertainty quantification, and in applications where a near real-time response is needed.
However, many challenges remain unresolved to secure the flexibility, robustness, and efficiency needed for general large-scale applications, in particular for nonlinear and/or time-dependent problems.
However, many challenges remain unresolved to secure the flexibility, robustness, and efficiency needed for general large-scale applications, in particular for nonlinear and/or time-dependent problems.
After giving a brief general introduction to projection based reduced order models, we discuss the use of artificial feedforward neural networks to enable the development of fast and accurate nonintrusive models for complex problems. We demonstrate that this approach offers substantial flexibility and robustness for general nonlinear problems and enables the development of fast reduced order models for complex applications.
After giving a brief general introduction to projection based reduced order models, we discuss the use of artificial feedforward neural networks to enable the development of fast and accurate nonintrusive models for complex problems. We demonstrate that this approach offers substantial flexibility and robustness for general nonlinear problems and enables the development of fast reduced order models for complex applications.
In the second part of the talk, we discuss how to use residual based neural networks in which knowledge of the governing equations is built into the network and show that this has advantages both for training and for the overall accuracy of the model.
In the second part of the talk, we discuss how to use residual based neural networks in which knowledge of the governing equations is built into the network and show that this has advantages both for training and for the overall accuracy of the model.
Time permitting, we finally discuss the use of reduced order models in the context of prediction, i.e. to estimate solutions in regions of the parameter beyond that of the initial training. With an emphasis on the Mori-Zwansig formulation for time-dependent problems, we discuss how to accurately account for the effect of the unresolved and truncated scales on the long term dynamics and show that accounting for these through a memory term significantly improves the predictive accuracy of the reduced order model.
Time permitting, we finally discuss the use of reduced order models in the context of prediction, i.e. to estimate solutions in regions of the parameter beyond that of the initial training. With an emphasis on the Mori-Zwansig formulation for time-dependent problems, we discuss how to accurately account for the effect of the unresolved and truncated scales on the long term dynamics and show that accounting for these through a memory term significantly improves the predictive accuracy of the reduced order model.
- Prof. Bernardo Cockburn (University of Minnesota, USA)
Date : 10/02/2021, Time: 7:00 pm
Date : 10/02/2021, Time: 7:00 pm
Title : Static condensation, hybridization and the devising of the HDG methods
Title : Static condensation, hybridization and the devising of the HDG methods
Abstract : The hybridizable discontinuous Galerkin (HDG) methods were introduced in the framework of second-order diffusion problems by hybridization and static condensation. We show that the exact solution can be characterized as the solution of local Dirichlet problems (hybridization) which can then be patched together by the transmission conditions (static condensation). Our goal is to show that the HDG methods are nothing but a discrete version of this characterization. To do so, we show that this is also the case for the well known continuous Galerkin and the mixed methods. We end by sketching how to define HDG methods for general PDEs.
Abstract : The hybridizable discontinuous Galerkin (HDG) methods were introduced in the framework of second-order diffusion problems by hybridization and static condensation. We show that the exact solution can be characterized as the solution of local Dirichlet problems (hybridization) which can then be patched together by the transmission conditions (static condensation). Our goal is to show that the HDG methods are nothing but a discrete version of this characterization. To do so, we show that this is also the case for the well known continuous Galerkin and the mixed methods. We end by sketching how to define HDG methods for general PDEs.
- Prof. Michael Rockner (Universität Bielefeld, Germany)
Date : 31/03/2021, Time: 5:30 pm
Date : 31/03/2021, Time: 5:30 pm
Title : Nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations.
Title : Nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations.
Abstract : In the first part of the talk we shall explain how to construct a solution to the above PDE based on classical nonlinear operator semigroup theory on L^1 (R^d ) and new results on L^1−L^∞ regularization of the solution semigroups in our case. In the second part of the talk we shall present a general result about the correspondence of nonlinear Fokker-Planck equations (FPEs) and McKean-Vlasov type SDEs. In particualr, it is shown that if one can solve the nonlinear FPE, then one can always construct a weak solution to the corresponding McKean-Vlasov SDE. We would like to emphasize that this, in particular, applies to the singular case, where the coefficients depend “Nemytski-type” on the time-marginal law of the solution process, hence the coefficients are not continuous in the measure-variable with respect to the weak topology on probability measures. This is in contrast to the literature in which the latter is standardly assumed. Hence we can cover nonlinear FPEs as the ones above, which are PDEs for the marginal law densities, realizing an old vision of McKean.
Abstract : In the first part of the talk we shall explain how to construct a solution to the above PDE based on classical nonlinear operator semigroup theory on L^1 (R^d ) and new results on L^1−L^∞ regularization of the solution semigroups in our case. In the second part of the talk we shall present a general result about the correspondence of nonlinear Fokker-Planck equations (FPEs) and McKean-Vlasov type SDEs. In particualr, it is shown that if one can solve the nonlinear FPE, then one can always construct a weak solution to the corresponding McKean-Vlasov SDE. We would like to emphasize that this, in particular, applies to the singular case, where the coefficients depend “Nemytski-type” on the time-marginal law of the solution process, hence the coefficients are not continuous in the measure-variable with respect to the weak topology on probability measures. This is in contrast to the literature in which the latter is standardly assumed. Hence we can cover nonlinear FPEs as the ones above, which are PDEs for the marginal law densities, realizing an old vision of McKean.
- Prof. Robert Pego (Carnegie Mellon University, USA)
Date : 16/04/2021, Time: 7:00 pm
Date : 16/04/2021, Time: 7:00 pm
Title : Dynamics and oscillations in models of coagulation and fragmentation
Title : Dynamics and oscillations in models of coagulation and fragmentation
Abstract : Coaglation-fragmentation equations are rather simple-looking kinetic models for evolution of the size distribution of clusters, appearing widely in science and technology. But few general analytical results characterize their dynamics. Solutions can exhibit self-similar growth, singular mass transport, and weak or slow approach to equilibrium. I will review some recent results in this vein for models that lack detailed balance: For a data-driven model of fish school size, we derive results on equilibration and self-similar spreading from the use of Bernstein transforms and complex function theory for Pick or Herglotz functions. Formal modeling of physical Becker-Doering models of nucleation and condensation with small-scale source and large-scale sinks identifies a scaling regime yielding oscillations in first-order phase transitions.
Abstract : Coaglation-fragmentation equations are rather simple-looking kinetic models for evolution of the size distribution of clusters, appearing widely in science and technology. But few general analytical results characterize their dynamics. Solutions can exhibit self-similar growth, singular mass transport, and weak or slow approach to equilibrium. I will review some recent results in this vein for models that lack detailed balance: For a data-driven model of fish school size, we derive results on equilibration and self-similar spreading from the use of Bernstein transforms and complex function theory for Pick or Herglotz functions. Formal modeling of physical Becker-Doering models of nucleation and condensation with small-scale source and large-scale sinks identifies a scaling regime yielding oscillations in first-order phase transitions.
- Prof. Sivaguru S. Sritharan
Date : 03/06/2021, Time: 5:30 pm
Date : 03/06/2021, Time: 5:30 pm
Title : ML/QC: AN INVITATION TO MACHINE LEARNING AND ITS QUANTUM COUNTERPART
Title : ML/QC: AN INVITATION TO MACHINE LEARNING AND ITS QUANTUM COUNTERPART
Abstract : Artificial Intelligence (AI) and Quantum Computing are occupying the center stage in the national conversation on critical technologies. Many countries across the globe are heavily investing in these two technologies. These subjects are not only powerful and timely, with wide range of applications in engineering, science, medicine and social sciences, but also the underlying mathematics is of immense elegance and of broad scope. In this talk we will introduce the key concepts in machine learning such as neural networks, VC dimension and entropy, kernel techniques and support vector machines, and then discuss the key ingredients such as the hidden subgroup problem for quantum computing and quantum machine learning.
Abstract : Artificial Intelligence (AI) and Quantum Computing are occupying the center stage in the national conversation on critical technologies. Many countries across the globe are heavily investing in these two technologies. These subjects are not only powerful and timely, with wide range of applications in engineering, science, medicine and social sciences, but also the underlying mathematics is of immense elegance and of broad scope. In this talk we will introduce the key concepts in machine learning such as neural networks, VC dimension and entropy, kernel techniques and support vector machines, and then discuss the key ingredients such as the hidden subgroup problem for quantum computing and quantum machine learning.
- Prof. Roger Temam (Indiana University, USA)
Date : 23/06/2021, Time: 2:30 pm
Date : 23/06/2021, Time: 2:30 pm
Title : Two-Fluid Flows: The Diffuse Interface / Phase Field Method
Title : Two-Fluid Flows: The Diffuse Interface / Phase Field Method
Abstract : This lecture is devoted to the study of some two-fluid flows in the context of the Diffuse Interface Method also called the Phase Field Method. The traditional approach to two-fluids flows leads to some free boundary value problems which are very nice to study, but the theory is limited by the possible appearance of singularities. The Diffuse Interface method which developed recently tends to replace the sharp interface by a thin layer interface making the problems more regular. These models are fully physically grounded since their derivation is based on the principles of thermodynamics and statistical mechanics. The theory has been very effective for the modeling and numerical simulation of classical examples such as transition in alloys, droplet interactions in fluids and solidification, as well as novel applications in biology and engineering, such as tumor growth, lithium battery, architecture of nanomaterials and image processing. This lecture will focus on the Navier-Stokes-Cahn-Hilliard equations. Work done in collaboration with Andrea Giorgini (Indiana University and Imperial College London).
Abstract : This lecture is devoted to the study of some two-fluid flows in the context of the Diffuse Interface Method also called the Phase Field Method. The traditional approach to two-fluids flows leads to some free boundary value problems which are very nice to study, but the theory is limited by the possible appearance of singularities. The Diffuse Interface method which developed recently tends to replace the sharp interface by a thin layer interface making the problems more regular. These models are fully physically grounded since their derivation is based on the principles of thermodynamics and statistical mechanics. The theory has been very effective for the modeling and numerical simulation of classical examples such as transition in alloys, droplet interactions in fluids and solidification, as well as novel applications in biology and engineering, such as tumor growth, lithium battery, architecture of nanomaterials and image processing. This lecture will focus on the Navier-Stokes-Cahn-Hilliard equations. Work done in collaboration with Andrea Giorgini (Indiana University and Imperial College London).
- Prof. Endre Suli (University of Oxford, UK)
Date : 01/07/2021, Time: 4:00 pm
Date : 01/07/2021, Time: 4:00 pm
Title : Discrete De Giorgi--Nash--Moser theory and the finite element approximation of chemically reacting fluids
Title : Discrete De Giorgi--Nash--Moser theory and the finite element approximation of chemically reacting fluids
Abstract : The talk is concerned with the convergence analysis of finite element methods for the approximate solution of a system of nonlinear elliptic partial differential equations that arise in models of chemically reacting viscous incompressible fluids. The shear-stress appearing in the model involves a power-law type nonlinearity, where, instead of being a fixed constant, the power law-exponent is a function of a spatially varying nonnegative concentration function, which, in turn, solves a nonlinear convection-diffusion equation. In order to prove the convergence of the sequence of finite element approximations to a solution of this coupled system of nonlinear PDEs, a uniform H\"{o}lder norm bound needs to be derived for the sequence of finite element approximations to the concentration in a setting, where the diffusion coefficient in the convection-diffusion equation satisfied by the concentration is merely an L^\infty function. This necessitates the development of a finite element counterpart of the De Giorgi--Nash--Moser theory. Motivated by an early paper by Aguilera and Caffarelli (1986) in the simpler setting of Laplace's equation, we derive such uniform H\"older norm bounds on the sequence of continuous piecewise linear finite element approximations to the concentration. We then use results from the theory of variable-exponent Sobolev spaces equipped with a Luxembourg norm, Minty's method for monotone operators and an extension to variable-exponent Sobolev spaces of the finite element version of the Acerbi--Fusco Lipschitz-truncation method, originally developed in classical Sobolev spaces in collaboration with Lars Diening and Christian Kreuzer (SIAM J. Numer. Anal. 51(2): 984--1015 (2014)), to pass to the limit in the coupled system of nonlinear PDEs under consideration.
Abstract : The talk is concerned with the convergence analysis of finite element methods for the approximate solution of a system of nonlinear elliptic partial differential equations that arise in models of chemically reacting viscous incompressible fluids. The shear-stress appearing in the model involves a power-law type nonlinearity, where, instead of being a fixed constant, the power law-exponent is a function of a spatially varying nonnegative concentration function, which, in turn, solves a nonlinear convection-diffusion equation. In order to prove the convergence of the sequence of finite element approximations to a solution of this coupled system of nonlinear PDEs, a uniform H\"{o}lder norm bound needs to be derived for the sequence of finite element approximations to the concentration in a setting, where the diffusion coefficient in the convection-diffusion equation satisfied by the concentration is merely an L^\infty function. This necessitates the development of a finite element counterpart of the De Giorgi--Nash--Moser theory. Motivated by an early paper by Aguilera and Caffarelli (1986) in the simpler setting of Laplace's equation, we derive such uniform H\"older norm bounds on the sequence of continuous piecewise linear finite element approximations to the concentration. We then use results from the theory of variable-exponent Sobolev spaces equipped with a Luxembourg norm, Minty's method for monotone operators and an extension to variable-exponent Sobolev spaces of the finite element version of the Acerbi--Fusco Lipschitz-truncation method, originally developed in classical Sobolev spaces in collaboration with Lars Diening and Christian Kreuzer (SIAM J. Numer. Anal. 51(2): 984--1015 (2014)), to pass to the limit in the coupled system of nonlinear PDEs under consideration.
The talk is based on joint work with Seungchan Ko and Petra Pustejovska, and recent results obtained in collaboration with Lars Diening and Toni Scharle (IMA Journal of Numerical Analysis (2021): https://doi.org/10.1093/imanum/drab029 )
The talk is based on joint work with Seungchan Ko and Petra Pustejovska, and recent results obtained in collaboration with Lars Diening and Toni Scharle (IMA Journal of Numerical Analysis (2021): https://doi.org/10.1093/imanum/drab029 )
- Prof. Thomas Ritchter (Otto-von-Guericke University, Magdeburg, Germany)
Date : 27/10/2021, Time: 5:00 pm
Date : 27/10/2021, Time: 5:00 pm
Title : Deep neural networks for accelerating fluid-dynamics simulations
Title : Deep neural networks for accelerating fluid-dynamics simulations
Abstract : In this talk we discuss the use of deep neural networks for augmenting classical finite element simulations in fluid-dynamics.
Abstract : In this talk we discuss the use of deep neural networks for augmenting classical finite element simulations in fluid-dynamics.
Classical simulation methods often reach their limits. Even if the finite element method is highly efficient and established for the discretization of the Navier-Stokes equations, fundamental problems, such as the resolution of fine structures or a correct information transport between scales, are still not sufficiently solved. A similar problem arises in the description of fluid-structure interactions, where it is often not possible to accurately represent both the flow field and the detailed interaction with solids simultaneously. For example, consider the interaction of blood plasma with the small, often very irregularly shaped, solid components of the blood. These are too small, too diverse, and too unstructured for coupling models to be known.
Classical simulation methods often reach their limits. Even if the finite element method is highly efficient and established for the discretization of the Navier-Stokes equations, fundamental problems, such as the resolution of fine structures or a correct information transport between scales, are still not sufficiently solved. A similar problem arises in the description of fluid-structure interactions, where it is often not possible to accurately represent both the flow field and the detailed interaction with solids simultaneously. For example, consider the interaction of blood plasma with the small, often very irregularly shaped, solid components of the blood. These are too small, too diverse, and too unstructured for coupling models to be known.
We discuss approaches to connect the finite element method with neural networks to overcome these obstacles. The paradigm is to use classical simulation techniques where their strengths are eminent, such as in the very efficient representation of a coarse, large-scale flow field. Neural networks are used where a full resolution of the effects does not seem possible or efficient. In addition to the desired increase in efficiency, the focus is particularly on issues of stability, generalizability and error accuracy.
We discuss approaches to connect the finite element method with neural networks to overcome these obstacles. The paradigm is to use classical simulation techniques where their strengths are eminent, such as in the very efficient representation of a coarse, large-scale flow field. Neural networks are used where a full resolution of the effects does not seem possible or efficient. In addition to the desired increase in efficiency, the focus is particularly on issues of stability, generalizability and error accuracy.
- Prof. Rolf Hiptmair (ETH Zurich, Switzerland)
Date : 08/11/2021, Time: 5:30 pm
Date : 08/11/2021, Time: 5:30 pm
Title : Nodal Auxiliary Space Preconditioning for Edge Elements (The HX-Preconditioner)
Title : Nodal Auxiliary Space Preconditioning for Edge Elements (The HX-Preconditioner)
Abstract : I am going to present a general approach to preconditioning large sparse linear systems of equations arising from conforming edge finite element discretizations of H(curl, Ω)-elliptic variational problems. Like geometric multigrid, the methods are asymptotically optimal in the sense that their performance does not deteriorate on arbitrarily fine meshes. Unlike geometric multigrid, no hierarchy of nested meshes is required, only fast solvers for discrete 2nd-order elliptic problems have to be available, which are provided, e.g., by standard algebraic multigrid codes. In a sense, the method described in this talk makes it possible to construct optimal algebraic preconditioners for discrete curl curl-equations.
Abstract : I am going to present a general approach to preconditioning large sparse linear systems of equations arising from conforming edge finite element discretizations of H(curl, Ω)-elliptic variational problems. Like geometric multigrid, the methods are asymptotically optimal in the sense that their performance does not deteriorate on arbitrarily fine meshes. Unlike geometric multigrid, no hierarchy of nested meshes is required, only fast solvers for discrete 2nd-order elliptic problems have to be available, which are provided, e.g., by standard algebraic multigrid codes. In a sense, the method described in this talk makes it possible to construct optimal algebraic preconditioners for discrete curl curl-equations.
This class of auxiliary space preconditioners is motivated by the possibility to decompose any vector field in the energy space H(curl, Ω), Ω a 3D computational domain, into a more regular vector field componentwise belonging to H^1(Ω), and a gradient: in a sense, gradients fill the gap between H(curl, Ω) and (H1(Ω))^3. This hints at the efficacy of additive subspace correction based on variational problems in H^1(Ω) (“Poisson problems”) to deliver good preconditioners for discrete H(curl, Ω)-elliptic variational problems. However, the splitting does not immediately carry over to edge finite element subspaces of H(curl, Ω). Oscillatory remainder functions have to be introduced. Fortunately, they can effectively be dealt with by local relaxation sweeps.
This class of auxiliary space preconditioners is motivated by the possibility to decompose any vector field in the energy space H(curl, Ω), Ω a 3D computational domain, into a more regular vector field componentwise belonging to H^1(Ω), and a gradient: in a sense, gradients fill the gap between H(curl, Ω) and (H1(Ω))^3. This hints at the efficacy of additive subspace correction based on variational problems in H^1(Ω) (“Poisson problems”) to deliver good preconditioners for discrete H(curl, Ω)-elliptic variational problems. However, the splitting does not immediately carry over to edge finite element subspaces of H(curl, Ω). Oscillatory remainder functions have to be introduced. Fortunately, they can effectively be dealt with by local relaxation sweeps.
- Prof. James C. Robinson (University of Warwick, UK)
Date : 29/11/2021, Time: 5:15 pm
Date : 29/11/2021, Time: 5:15 pm
Title : Approximating solutions of the Navier-Stokes equations on the whole space using large periodic domains
Title : Approximating solutions of the Navier-Stokes equations on the whole space using large periodic domains
Abstract : I will show that the numerical heuristic of using a large period domain to approximate solutions of the 3D Navier-Stokes equations on "the whole space" can be placed on a rigorous footing. As a consequence, I will show that if, for given compactly-supported initial data, a regular solution of the equations is assumed to exist on R^3 on some time interval [0,T], then a regular solution will also exist on a "large enough" periodic domain for the same initial data and on the same time interval.
Abstract : I will show that the numerical heuristic of using a large period domain to approximate solutions of the 3D Navier-Stokes equations on "the whole space" can be placed on a rigorous footing. As a consequence, I will show that if, for given compactly-supported initial data, a regular solution of the equations is assumed to exist on R^3 on some time interval [0,T], then a regular solution will also exist on a "large enough" periodic domain for the same initial data and on the same time interval.
- Prof. Martin Hairer (Imperial College London, UK)
Date : 12/01/2022, Time: 04:30 pm
Date : 12/01/2022, Time: 04:30 pm
Title : Two-scale systems with fractional noise.
Title : Two-scale systems with fractional noise.
Abstract : We consider fast-slow systems with the slow variables driven by fractional noise. On suitable timescales, we show that the limiting dynamic of the slow variables is Markovian with a generator involving a fractional power of the generator of the slow dynamic. Interestingly, this interpolates between the classical averaging and homogenisation results, which correspond to Hurst parameters 1 and 1/2 respectively. This is joint work with Xue-Mei Li.
Abstract : We consider fast-slow systems with the slow variables driven by fractional noise. On suitable timescales, we show that the limiting dynamic of the slow variables is Markovian with a generator involving a fractional power of the generator of the slow dynamic. Interestingly, this interpolates between the classical averaging and homogenisation results, which correspond to Hurst parameters 1 and 1/2 respectively. This is joint work with Xue-Mei Li.
- Prof. Arnulf Jentzen (University of Munster, Germany)
Date : 24/01/2022, Time: 1:00 pm
Date : 24/01/2022, Time: 1:00 pm
Title : Convergence analysis for gradient descent optimization methods in the training of artificial neural networks.
Title : Convergence analysis for gradient descent optimization methods in the training of artificial neural networks.
Abstract : Gradient descent (GD) type optimization methods are the standard instrument to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Despite the great success of GD type optimization methods in numerical simulations for the training of ANNs with ReLU activation, it remains -- even in the simplest situation of the plain vanilla GD optimization method with random initializations -- an open problem to prove (or disprove) the conjecture that the true risk of the GD optimization method converges in the training of ANNs with ReLU activation to zero as the width/depth of the ANNs, the number of independent random initializations, and the number of GD steps increase to infinity. In this talk we establish this conjecture in the situation where the probability distribution of the input data is equivalent to the continuous uniform distribution on a compact interval, where the probability distributions for the random initializations of the ANN parameters are standard normal distributions, and where the target function under consideration is continuous and piecewise affine linear.
Abstract : Gradient descent (GD) type optimization methods are the standard instrument to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Despite the great success of GD type optimization methods in numerical simulations for the training of ANNs with ReLU activation, it remains -- even in the simplest situation of the plain vanilla GD optimization method with random initializations -- an open problem to prove (or disprove) the conjecture that the true risk of the GD optimization method converges in the training of ANNs with ReLU activation to zero as the width/depth of the ANNs, the number of independent random initializations, and the number of GD steps increase to infinity. In this talk we establish this conjecture in the situation where the probability distribution of the input data is equivalent to the continuous uniform distribution on a compact interval, where the probability distributions for the random initializations of the ANN parameters are standard normal distributions, and where the target function under consideration is continuous and piecewise affine linear.
Date : 11/03/2022, Time: 6:00 pm
Date : 11/03/2022, Time: 6:00 pm
Title : Local and L^2 bounded projections in FEEC
Title : Local and L^2 bounded projections in FEEC
Abstract : One of the chief tools of Finite element exterior calculus (FEEC) are bounded commuting projections. They provided stability and convergence results for FEEC applied to the Hodge Laplacian. Early in the development of FEEC such projections were constructed by Schoberl and Christiansnen and Winther. More recently Falk and Winther have developed projections that are also local. Inspired by these results we discuss new projections that are local and bounded in L^2. This is joint work with Douglas Arnold.
Abstract : One of the chief tools of Finite element exterior calculus (FEEC) are bounded commuting projections. They provided stability and convergence results for FEEC applied to the Hodge Laplacian. Early in the development of FEEC such projections were constructed by Schoberl and Christiansnen and Winther. More recently Falk and Winther have developed projections that are also local. Inspired by these results we discuss new projections that are local and bounded in L^2. This is joint work with Douglas Arnold.
Date : 29/04/2022, Time: 6:00 pm
Date : 29/04/2022, Time: 6:00 pm
Title : Error estimation and adaptivity for stochastic collocation finite elements
Title : Error estimation and adaptivity for stochastic collocation finite elements
Abstract : Partial differential equations (PDEs) with uncertain inputs have provided engineers and scientists with enhanced fidelity in the modelling of real-life phenomena, especially within the last decade. Sparse grid stochastic collocation representations of parametric uncertainty, in combination with finite element discretization of physical space, have emerged as an efficient alternative to Monte-Carlo strategies over this period, especially in the context of nonlinear PDE models or linear PDE problems that are nonlinear in the parameterization of the uncertainty.
Abstract : Partial differential equations (PDEs) with uncertain inputs have provided engineers and scientists with enhanced fidelity in the modelling of real-life phenomena, especially within the last decade. Sparse grid stochastic collocation representations of parametric uncertainty, in combination with finite element discretization of physical space, have emerged as an efficient alternative to Monte-Carlo strategies over this period, especially in the context of nonlinear PDE models or linear PDE problems that are nonlinear in the parameterization of the uncertainty.
A multilevel adaptive refinement strategy for solving linear elliptic partial differential equations with random data is developed in this talk. The strategy extends the a posteriori error estimation framework introduced by Guignard & Nobile in 2018 to cover problems with a nonaffine parametric coefficient dependence. A suboptimal, but nonetheless reliable and convenient implementation of the strategy involves approximation of the decoupled PDE problems with a common finite element approximation space. Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, will also be discussed in detail.
A multilevel adaptive refinement strategy for solving linear elliptic partial differential equations with random data is developed in this talk. The strategy extends the a posteriori error estimation framework introduced by Guignard & Nobile in 2018 to cover problems with a nonaffine parametric coefficient dependence. A suboptimal, but nonetheless reliable and convenient implementation of the strategy involves approximation of the decoupled PDE problems with a common finite element approximation space. Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, will also be discussed in detail.
This is joint work with Alex Bespalov (University of Birmingham, UK) and Feng Xu.
This is joint work with Alex Bespalov (University of Birmingham, UK) and Feng Xu.
Date : 09/12/2022, Time: 4:00 pm
Date : 09/12/2022, Time: 4:00 pm
Title : Staggered discontinuous Galerkin methods on polygonal meshes
Title : Staggered discontinuous Galerkin methods on polygonal meshes
Abstract : Recently, polygonal finite element methods have received considerable attention. It is because general meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements, and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and non-matching interfaces. This talk presents a new computational paradigm for discretizing PDEs via the staggered Galerkin method on general meshes. First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal meshes for elliptic problems is proposed. The method can be flexibly applied to rough grids such as highly distorted meshes. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes. We derive a simple residual-type error estimator. Numerical results indicate that optimal convergence can be achieved for both the potential and vector variables, and the singularity can be well-captured by the proposed error estimator. Then, some applications to diffusion equations, Stokes equations, and linear elasticity equations are considered. Finally, we extend this approach to high-order polynomial approximations on general meshes.
Abstract : Recently, polygonal finite element methods have received considerable attention. It is because general meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements, and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and non-matching interfaces. This talk presents a new computational paradigm for discretizing PDEs via the staggered Galerkin method on general meshes. First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal meshes for elliptic problems is proposed. The method can be flexibly applied to rough grids such as highly distorted meshes. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes. We derive a simple residual-type error estimator. Numerical results indicate that optimal convergence can be achieved for both the potential and vector variables, and the singularity can be well-captured by the proposed error estimator. Then, some applications to diffusion equations, Stokes equations, and linear elasticity equations are considered. Finally, we extend this approach to high-order polynomial approximations on general meshes.
Date : 30/03/2023, Time: 4:00 pm
Date : 30/03/2023, Time: 4:00 pm
Title : Hybrid high-order methods for the wave equation
Title : Hybrid high-order methods for the wave equation
Abstract : We first review the devising principles of hybrid-high-order methods for static problems and outline the main analysis results. Then we present the development of HHO methods for wave propagation problems, by considering the second-order and the first-order formulations in time of the wave equation. We also briefly discuss the use of unfitted meshes for heterogeneous problems. Various numerical results are presented to illustrate the method.
Abstract : We first review the devising principles of hybrid-high-order methods for static problems and outline the main analysis results. Then we present the development of HHO methods for wave propagation problems, by considering the second-order and the first-order formulations in time of the wave equation. We also briefly discuss the use of unfitted meshes for heterogeneous problems. Various numerical results are presented to illustrate the method.