Computational Mathematics and Applications Seminar

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.

Data scientists are often faced with the challenge of understanding a high dimensional data set organized as a table. These tables may have columns of different (sometimes, non-numeric) types, and often have many missing entries. In this talk, we discuss how to use low rank models to analyze these big messy data sets. Low rank models perform well --- indeed, suspiciously well — across a wide range of data science applications, including applications in social science, medicine, and machine learning. In this talk, we introduce the mathematics of low rank models, demonstrate a few surprising applications of low rank models in data science, and present a simple mathematical explanation for their effectiveness.

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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).

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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.

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This talk concerns applications of differential geometry in numerical optimization. They arise when the optimization problem can be formulated as finding an optimum of a real-valued cost function defined on a smooth nonlinear search space. Oftentimes, the search space is a "matrix manifold", in the sense that its points admit natural representations in the form of matrices. In most cases, the matrix manifold structure is due either to the presence of certain nonlinear constraints (such as orthogonality or rank constraints), or to invariance properties in the cost function that need to be factored out in order to obtain a nondegenerate optimization problem. Manifolds that come up in practical applications include the rotation group SO(3) (generation of rigid body motions from sample points), the set of fixed-rank matrices (low-rank models, e.g., in collaborative filtering), the set of 3x3 symmetric positive-definite matrices (interpolation of diffusion tensors), and the shape manifold (morphing).

In the recent years, the practical importance of optimization problems on manifolds has stimulated the development of geometric optimization algorithms that exploit the differential structure of the manifold search space. In this talk, we give an overview of geometric optimization algorithms and their applications, with an emphasis on the underlying geometric concepts and on the numerical efficiency of the algorithm implementations.

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.

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.

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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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.

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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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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.

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