Thu, 08 Feb 2024
14:00
Lecture Room 3

From Chebfun3 to RTSMS: A journey into deterministic and randomized Tucker decompositions

Behnam Hashemi
(Leicester University)
Abstract
The Tucker decomposition is a family of representations that break up a given d-dimensional tensor into the multilinear product of a core tensor and a factor matrix along each of the d-modes. It is a useful tool in extracting meaningful insights from complex datasets and has found applications in various fields, including scientific computing, signal processing and machine learning. 
 In this talk we will first focus on the continuous framework and revisit how Tucker decomposition forms the foundation of Chebfun3 for numerical computing with 3D functions and the deterministic algorithm behind Chebfun3. The key insight is that separation of variables achieved via low-rank Tucker decomposition simplifies and speeds up lots of subsequent computations.
 We will then switch to the discrete framework and discuss a new algorithm called RTSMS (randomized Tucker with single-mode sketching). The single-mode sketching aspect of RTSMS allows utilizing simple sketch matrices which are substantially smaller than alternative methods leading to considerable performance gains. Within its least-squares strategy, RTSMS incorporates leverage scores for efficiency with Tikhonov regularization and iterative refinement for stability. RTSMS is demonstrated to be competitive with existing methods, sometimes outperforming them by a large margin.
We illustrate the benefits of Tucker decomposition via MATLAB demos solving problems from global optimization to video compression. RTSMS is joint work with Yuji Nakatsukasa.
Thu, 19 Nov 2015

14:00 - 15:00
L5

Adaptivity and blow-up detection for nonlinear evolution PDEs

Dr. Emmanuil Georgoulis
(Leicester University)
Abstract

I will review some recent work on the problem of reliable automatic detection of blow-up behaviour for nonlinear parabolic PDEs. The adaptive algorithms developed are based on rigorous conditional a posteriori error bounds. The use of space-time adaptivity is crucial in making the problem computationally tractable. The results presented are applicable to quite general spatial operators, rendering the approach potentially useful in informing respective PDE theory. The new adaptive algorithm is shown to accurately estimate the blow-up time of a number of problems, including ones exhibiting regional blow-up. 

Thu, 04 Jun 2015

14:00 - 15:00
L5

Polytopic Finite Element Methods

Dr Andrea Cangiani
(Leicester University)
Abstract

Can we extend the FEM to general polytopic, i.e. polygonal and polyhedral, meshes while retaining 
the ease of implementation and computational cost comparable to that of standard FEM? Within this talk, I present two approaches that achieve just  that (and much more): the Virtual Element Method (VEM) and an hp-version discontinuous Galerkin (dG) method.

The Virtual Element spaces are like the usual (polynomial) finite element spaces with the addition of suitable non-polynomial functions. This is far from being a novel idea. The novelty of the VEM approach is that it avoids expensive evaluations of the non-polynomial "virtual" functions by basing all 
computations solely on the method's carefully chosen degrees of freedom. This way we can easily deal 
with complicated element geometries and/or higher continuity requirements (like C1, C2, etc.), while 
maintaining the computational complexity comparable to that of standard finite element computations.

As you might expect, the choice and number of the degrees of freedom depends on such continuity 
requirements. If mesh flexibility is the goal, while one is ready to  give up on global regularity, other approaches can be considered. For instance, dG approaches are naturally suited to deal with polytopic meshes. Here I present an extension of the classical Interior Penalty dG method which achieves optimal rates of convergence on polytopic meshes even under elemental edge/face degeneration. 

The next step is to exploit mesh flexibility in the efficient resolution of problems characterised by 
complicated geometries and solution features, for instance within the framework of automatic FEM 
adaptivity. I shall finally introduce ongoing work in this direction.

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