Forthcoming events in this series


Thu, 18 Feb 2021
14:00
Virtual

The reference map technique for simulating complex materials and multi-body interactions

Chris Rycroft
(Harvard University)
Abstract

Conventional computational methods often create a dilemma for fluid-structure interaction problems. Typically, solids are simulated using a Lagrangian approach with grid that moves with the material, whereas fluids are simulated using an Eulerian approach with a fixed spatial grid, requiring some type of interfacial coupling between the two different perspectives. Here, a fully Eulerian method for simulating structures immersed in a fluid will be presented. By introducing a reference map variable to model finite-deformation constitutive relations in the structures on the same grid as the fluid, the interfacial coupling problem is highly simplified. The method is particularly well suited for simulating soft, highly-deformable materials and many-body contact problems, and several examples will be presented.

 

This is joint work with Ken Kamrin (MIT).

 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Thu, 11 Feb 2021

14:00 - 15:00
Virtual

From design to numerical analysis of partial differential equations: a unified mathematical framework

Annalisa Buffa
(École Polytechnique Fédérale de Lausanne (EPFL))
Abstract

Computer-based simulation of partial differential equations (PDEs) involves approximating the unknowns and relies on suitable description of geometrical entities such as the computational domain and its properties. The Finite Element Method (FEM) is by large the most popular technique for the computer-based simulation of PDEs and hinges on the assumption that discretized domain and unknown fields are both represented by piecewise polynomials, on tetrahedral or hexahedral partitions. In reality, the simulation of PDEs is a brick within a workflow where, at the beginning, the geometrical entities are created, described and manipulated with a geometry processor, often through Computer-Aided Design systems (CAD), and then used for the simulation of the mechanical behaviour of the designed object. This workflow is often repeated many times as part of a shape optimisation loop. Within this loop, the use of FEM on CAD geometries (which are mainly represented through their boundaries) calls then for (re-) meshing and re-interpolation techniques that often require human intervention and result in inaccurate solutions and lack of robustness of the whole process. In my talk, I will present the mathematical counterpart of this problem, I will discuss the mismatch in the mathematical representations of geometries and PDEs unknowns and introduce a promising framework where geometric objects and PDEs unknowns are represented in a compatible way. Within this framework, the challenges to be addressed in order to construct robust PDE solvers are many and I will discuss some of them. Mathematical results will besupported by numerical validation.

Thu, 04 Feb 2021
14:00
Virtual

Modeling composite structures with defects

Anne Reinarz
(University of Durham)
Abstract

Composite materials make up over 50% of recent aircraft constructions. They are manufactured from very thin fibrous layers  (~10^-4 m) and even  thinner resin interfaces (~10^-5 m). To achieve the required strength, a particular layup sequence of orientations of the anisotropic fibrous layers is used. During manufacturing, small localised defects in the form of misaligned fibrous layers can occur in composite materials, adding an additional level of complexity. After FE discretisation the model exhibits multiple scales and large spatial variations in model parameters. Thus the resultant linear system of equations can be very ill-conditioned and extremely large. The limitations of commercially available modelling tools for solving these problems has led us to the implementation of a robust and scalable preconditioner called GenEO for parallel Krylov solvers. I will discuss using the GenEO coarse space as an effective multiscale model for the fine-scale displacement and stress fields. For the coarse space construction, GenEO computes generalised eigenvectors of the local stiffness matrices on the overlapping subdomains and builds an approximate coarse space by combining the smallest energy eigenvectors on each subdomain via a partition of unity.

 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Thu, 28 Jan 2021
14:00
Virtual

Spontaneous periodic orbits in the Navier-Stokes flow via computer-assisted proofs

Jean-Philippe Lessard
(McGill University)
Abstract
In this talk, we introduce a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier-Stokes equations on the three-torus. After introducing a zero finding problem posed on a Banach space of geometrically decaying Fourier coefficients, a Newton-Kantorovich theorem is applied to obtain the (computer-assisted) proofs of existence. As applications, we present proofs of existence of spontaneous periodic orbits in the Navier-Stokes equations with Taylor-Green forcing.

 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Thu, 21 Jan 2021
14:00
Virtual

Domain specific languages for convex optimization

Stephen Boyd
(Stanford University)
Abstract

Specialized languages for describing convex optimization problems, and associated parsers that automatically transform them to canonical form, have greatly increased the use of convex optimization in applications. These systems allow users to rapidly prototype applications based on solving convex optimization problems, as well as generate code suitable for embedded applications. In this talk I will describe the general methods used in such systems.

 

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A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Thu, 03 Dec 2020
14:00
Virtual

Reconstructing Signals with Simple Fourier Transforms

Haim Avron
(Tel Aviv University)
Abstract

Reconstructing continuous signals based on a small number of discrete samples is a fundamental problem across science and engineering. In practice, we are often interested in signals with ``simple'' Fourier structure -- e.g., those involving frequencies within a bounded range, a small number of frequencies, or a few blocks of frequencies. More broadly, any prior knowledge about a signal's Fourier power spectrum can constrain its complexity.  Intuitively, signals with more highly constrained Fourier structure require fewer samples to reconstruct.

We formalize this intuition by showing that, roughly speaking, a continuous signal from a given class can be approximately reconstructed using a number of samples equal to the statistical dimension of the allowed power spectrum of that class. We prove that, in nearly all settings, this natural measure tightly characterizes the sample complexity of signal reconstruction.

Surprisingly, we also show that, up to logarithmic factors, a universal non-uniform sampling strategy can achieve this optimal complexity for any class of signals. We present a simple, efficient, and general algorithm for recovering a signal from the samples taken. For bandlimited and sparse signals, our method matches the state-of-the-art. At the same time, it gives the first computationally and sample efficient solution to a broad range of problems, including multiband signal reconstruction and common kriging and Gaussian process regression tasks.

Our work is based on a novel connection between randomized linear algebra and the problem of reconstructing signals with constrained Fourier structure. We extend tools based on statistical leverage score sampling and column-based matrix reconstruction to the approximation of continuous linear operators that arise in the signal fitting problem. We believe that these extensions are of independent interest and serve as a foundation for tackling a broad range of continuous time problems using randomized methods.

This is joint work with Michael Kapralov, Cameron Musco, Christopher Musco, Ameya Velingker and Amir Zandieh

 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

Thu, 26 Nov 2020
14:00
Virtual

Why should we care about Steklov eigenproblems?

Nilima Nigam
(Simon Fraser University)
Abstract

Steklov eigenproblems and their variants (where the spectral parameter appears in the boundary condition) arise in a range of useful applications. For instance, understanding some properties of the mixed Steklov-Neumann eigenfunctions tells us why we shouldn't use coffee cups for expensive brandy. 

In this talk I'll present a high-accuracy discretization strategy for computing Steklov eigenpairs. The strategy can be used to study questions in spectral geometry, spectral optimization and to the solution of elliptic boundary value problems with Robin boundary conditions.

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A link for the talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

 

Thu, 19 Nov 2020
14:00
Virtual

A foundation for automated high performance scientific machine learning

Chris Rackauckas
(MIT)
Abstract

Scientific machine learning is a burgeoning discipline for mixing machine learning into scientific simulation. Use cases of this field include automated discovery of physical equations and accelerating physical simulators. However, making the analyses of this field automated will require building a set of tools that handle stiff and ill-conditioned models without requiring user tuning. The purpose of this talk is to demonstrate how the methods and tools of scientific machine learning can be consolidated to give a single high performance and robust software stack. We will start by describing universal differential equations, a flexible mathematical object which is able to represent methodologies for equation discovery, 100-dimensional differential equation solvers, and discretizations of physics-informed neural networks. Then we will showcase how adjoint sensitivity analysis on the universal differential equation solving process gives rise to efficient and stiffly robust training methodologies for a large variety of scientific machine learning problems. With this understanding of differentiable programming we will describe how the Julia SciML Software Organization is utilizing this foundation to provide high performance tools for deploying battery powered airplanes, improving the energy efficiency of buildings, allow for navigation via the Earth's magnetic field, and more.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

Thu, 12 Nov 2020
14:00
Virtual

High order Whitney forms on simplices

Francesca Rapetti
(University of Nice Sophia-Antipolis)
Abstract

Whitney elements on simplices are perhaps the most widely used finite elements in computational electromagnetics. They offer the simplest construction of polynomial discrete differential forms on simplicial complexes. Their associated degrees of freedom (dofs) have a very clear physical meaning and give a recipe for discretizing physical balance laws, e.g., Maxwell’s equations. As interest grew for the use of high order schemes, such as hp-finite element or spectral element methods, higher-order extensions of Whitney forms have become an important computational tool, appreciated for their better convergence and accuracy properties. However, it has remained unclear what kind of cochains such elements should be associated with: Can the corresponding dofs be assigned to precise geometrical elements of the mesh, just as, for instance, a degree of freedom for the space of Whitney 1-forms belongs to a specific edge? We address this localization issue. Why is this an issue? The existing constructions of high order extensions of Whitney elements follow the traditional FEM path of using higher and higher “moments” to define the needed dofs. As a result, such high order finite k-elements in d dimensions include dofs associated to q-simplices, with k < q ≤ d, whose physical interpretation is obscure. The present paper offers an approach based on the so-called “small simplices”, a set of subsimplices obtained by homothetic contractions of the original mesh simplices, centered at mesh nodes (or more generally, when going up in degree, at points of the principal lattice of each original simplex). Degrees of freedom of the high-order Whitney k-forms are then associated with small simplices of dimension k only.  We provide an explicit  basis for these elements on simplices and we justify this approach from a geometric point of view (in the spirit of Hassler Whitney's approach, still successful 30 years after his death).   

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

Thu, 05 Nov 2020
14:00
Virtual

Modeling and simulation of fluidic surfaces

Maxim Olshanskii
(University of Houston)
Abstract

We briefly review mathematical models of viscous deformable interfaces (such as plasma membranes) leading to fluid equations posed on (evolving) 2D surfaces embedded in $R^3$. We further report on some recent advances in understanding and numerical simulation of the resulting fluid systems using an unfitted finite element method.

 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

 

Thu, 29 Oct 2020
14:00
Virtual

An algorithm for constructing efficient discretizations for integral equations near corners

Kirill Serkh
(University of Toronto)
Abstract

It has long been known that many elliptic partial differential equations can be reformulated as Fredholm integral equations of the second kind on the boundaries of their domains. The kernels of the resulting integral equations are weakly singular, which has historically made their numerical solution somewhat onerous, requiring the construction of detailed and typically sub-optimal quadrature formulas. Recently, a numerical algorithm for constructing generalized Gaussian quadratures was discovered which, given 2n essentially arbitrary functions, constructs a unique n-point quadrature that integrates them to machine precision, solving the longstanding problem posed by singular kernels.

When the domains have corners, the solutions themselves are also singular. In fact, they are known to be representable, to order n, by a linear combination (expansion) of n known singular functions. In order to solve the integral equation accurately, it is necessary to construct a discretization such that the mapping (in the L^2-sense) from the values at the discretization points to the corresponding n expansion coefficients is well-conditioned. In this talk, we present exactly such an algorithm, which is optimal in the sense that, given n essentially arbitrary functions, it produces n discretization points, and for which the resulting interpolation formulas have condition numbers extremely close to one.

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A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

Thu, 22 Oct 2020
14:00
Virtual

A new block preconditioner for implicit Runge-Kutta methods for parabolic PDE

Victoria Howle
(Texas Tech University)
Abstract

In this talk, we introduce a new preconditioner for the large, structured systems appearing in implicit Runge–Kutta time integration of parabolic partial differential equations. This preconditioner is based on a block LDU factorization with algebraic multigrid subsolves for scalability.

We compare our preconditioner in condition number and eigenvalue distribution, and through numerical experiments, with others in the literature. In experiments run with implicit Runge–Kutta stages up to s = 7, we find that the new preconditioner outperforms the others, with the improvement becoming more pronounced as the spatial discretization is refined and as temporal order is increased.

 

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

Thu, 15 Oct 2020
14:00
Virtual

Generalized Gaussian quadrature as a tool for discretizing singular integral equations

Jim Bremer
(UC Davis)
Abstract

 

One of the standard methods for the solution of elliptic boundary value problems calls for reformulating them as systems of integral equations.  The integral operators that arise in this fashion typically have singular kernels, and, in many cases of interest, the solutions of these equations are themselves singular.  This makes the accurate discretization of the systems of integral equations arising from elliptic boundary value problems challenging.

Over the last decade, Generalized Gaussian quadrature rules, which are n-point quadrature rules that are exact for a collection of 2n functions, have emerged as one of the most effective tools for discretizing singular integral equations. Among other things, they have been used to accelerate the discretization of singular integral operators on curves, to enable the accurate discretization of singular integral operators on complex surfaces and to greatly reduce the cost of representing the (singular) solutions of integral equations given on planar domains with corners.

We will first briefly outline a standard method for the discretization of integral operators given on curves which is highly amenable to acceleration through generalized Gaussian quadratures. We will then describe a numerical procedure for the construction of Generalized Gaussian quadrature rules.

Much of this is joint work with Zydrunas Gimbutas (NIST Boulder) and Vladimir Rokhlin (Yale University).

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please send email to @email.

Thu, 11 Jun 2020

14:00 - 15:00

Dense networks that do not synchronize and sparse ones that do.

Alex Townsend
(Cornell)
Abstract

Consider a network of identical phase oscillators with sinusoidal coupling. How likely are the oscillators to globally synchronize, starting from random initial phases? One expects that dense networks have a strong tendency to synchronize and the basin of attraction for the synchronous state to be the whole phase space. But, how dense is dense enough? In this (hopefully) entertaining Zoom talk, we use techniques from numerical linear algebra and computational Algebraic geometry to derive the densest known networks that do not synchronize and the sparsest networks that do. This is joint work with Steven Strogatz and Mike Stillman.


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Thu, 04 Jun 2020

14:00 - 15:00

Do Galerkin methods converge for the classical 2nd kind boundary integral equations in polyhedra and Lipschitz domains?

Simon Chandler-Wilde
(Reading University)
Abstract

The boundary integral equation method is a popular method for solving elliptic PDEs with constant coefficients, and systems of such PDEs, in bounded and unbounded domains. An attraction of the method is that it reduces solution of the PDE in the domain to solution of a boundary integral equation on the boundary of the domain, reducing the dimensionality of the problem. Second kind integral equations, featuring the double-layer potential operator, have a long history in analysis and numerical analysis. They provided, through C. Neumann, the first existence proof to the Laplace Dirichlet problem in 3D, have been an important analysis tool for PDEs through the 20th century, and are popular computationally because of their excellent conditioning and convergence properties for large classes of domains. A standard numerical method, in particular for boundary integral equations, is the Galerkin method, and the standard convergence analysis starts with a proof that the relevant operator is coercive, or a compact perturbation of a coercive operator, in the relevant function space. A long-standing open problem is whether this property holds for classical second kind boundary integral equations on general non-smooth domains. In this talk we give an overview of the various concepts and methods involved, reformulating the problem as a question about numerical ranges. We solve this open problem through counterexamples, presenting examples of 2D Lipschitz domains and 3D Lipschitz polyhedra for which coercivity does not hold. This is joint work with Prof Euan Spence, Bath.

 

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Thu, 28 May 2020

14:00 - 15:00

Robust preconditioners for non-Newtonian fluids and magnetohydrodynamics

Patrick Farrell
(Oxford University)
Abstract

We discuss two recent extensions of work on Reynolds-robust preconditioners for the Navier-Stokes equations, to non-Newtonian fluids and to the equations of magnetohydrodynamics.  We model non-Newtonian fluids by means of an implicit constitutive relation between stress and strain. This framework is broadly applicable and allows for proofs of convergence under quite general assumptions. Since the stress cannot in general be solved for in terms of the strain, a three-field stress-velocity-pressure formulation is adopted. By combining the augmented Lagrangian approach with a kernel-capturing space decomposition, we derive a preconditioner that is observed to be robust to variations in rheological parameters in both two and three dimensions.  In the case of magnetohydrodynamics, we consider the stationary incompressible resistive Newtonian equations, and solve a four-field formulation for the velocity, pressure, magnetic field and electric field. A structure-preserving discretisation is employed that enforces both div(u) = 0 and div(B) = 0 pointwise. The basic idea of the solver is to split the fluid and electromagnetic parts and to employ our existing Navier-Stokes solver in the Schur complement. We present results in two dimensions that exhibit robustness with respect to both the fluids and magnetic Reynolds numbers, and describe ongoing work to extend the solver to three dimensions.

[To be added to our seminars mailing list, or to receive a Zoom invitation for a particular seminar, please contact @email.]

Thu, 21 May 2020

14:00 - 15:00

System Interpolation with Loewner Pencils: Background, Pseudospectra, and Nonlinear Eigenvalue Problems

Mark Embree
(Virginia Tech)
Abstract

In 2007, Andrew Mayo and Thanos Antoulas proposed a rational interpolation algorithm to solve a basic problem in control theory: given samples of the transfer function of a dynamical system, construct a linear time-invariant system that realizes these samples.  The resulting theory enables a wide range of data-driven modeling, and has seen diverse applications and extensions.  We will introduce these ideas from a numerical analyst's perspective, show how the selection of interpolation points can be guided by a Sylvester equation and pseudospectra of matrix pencils, and mention an application of these ideas to a contour algorithm for the nonlinear eigenvalue problem. (This talk involves collaborations with Michael Brennan (MIT), Serkan Gugercin (Virginia Tech), and Cosmin Ionita (MathWorks).)

[To be added to our seminars mailing list, or to receive a Zoom invitation for a particular seminar, please contact @email.]

Thu, 12 Mar 2020

14:00 - 15:00
L4

The Statistical Finite Element Method

Mark Girolami
(University of Cambridge)
Abstract

The finite element method (FEM) is one of the great triumphs of applied mathematics, numerical analysis and software development. Recent developments in sensor and signalling technologies enable the phenomenological study of systems. The connection between sensor data and FEM is restricted to solving inverse problems placing unwarranted faith in the fidelity of the mathematical description of the system. If one concedes mis-specification between generative reality and the FEM then a framework to systematically characterise this uncertainty is required. This talk will present a statistical construction of the FEM which systematically blends mathematical description with observations.

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Thu, 27 Feb 2020

14:00 - 15:00
L4

Randomised algorithms for solving systems of linear equations

Gunnar Martinsson
(University of Texas at Austin)
Abstract

The task of solving large scale linear algebraic problems such as factorising matrices or solving linear systems is of central importance in many areas of scientific computing, as well as in data analysis and computational statistics. The talk will describe how randomisation can be used to design algorithms that in many environments have both better asymptotic complexities and better practical speed than standard deterministic methods.

The talk will in particular focus on randomised algorithms for solving large systems of linear equations. Both direct solution techniques based on fast factorisations of the coefficient matrix, and techniques based on randomised preconditioners, will be covered.

Note: There is a related talk in the Random Matrix Seminar on Tuesday Feb 25, at 15:30 in L4. That talk describes randomised methods for computing low rank approximations to matrices. The two talks are independent, but the Tuesday one introduces some of the analytical framework that supports the methods described here.

Thu, 20 Feb 2020

14:00 - 15:00
Rutherford Appleton Laboratory, nr Didcot

Learning with nonlinear Perron eigenvectors

Francesco Tudisco
(Gran Sasso Science Institute GSSI)
Abstract

In this talk I will present a Perron-Frobenius type result for nonlinear eigenvector problems which allows us to compute the global maximum of a class of constrained nonconvex optimization problems involving multihomogeneous functions.

I will structure the talk into three main parts:

First, I will motivate the optimization of homogeneous functions from a graph partitioning point of view, showing an intriguing generalization of the famous Cheeger inequality.

Second, I will define the concept of multihomogeneous function and I will state our main Perron-Frobenious theorem. This theorem exploits the connection between optimization of multihomogeneous functions and nonlinear eigenvectors to provide an optimization scheme that has global convergence guarantees.

Third, I will discuss a few example applications in network science and machine learning that require the optimization of multihomogeneous functions and that can be solved using nonlinear Perron eigenvectors.