BIT'65 - INVITED SPEAKERS
January 14-16, 2025, Uppsala, Sweden
Gitta Kutyniok
Reliable and Sustainable AI: Towards Next Generation Scientific Computing
The new wave of artificial intelligence is impacting industry, public life, and the sciences in an unprecedented manner. It has also by now already led to paradigm changes in scientific computing. However, one current major drawback is the lack of reliability as well as the enormous energy problem of AI.
In this lecture we will first provide an introduction into the vibrant research area of reliable and sustainable AI. We will then touch upon the necessity to reconsider the classical von Neumann computing hardware and embrace novel approaches in analog computing such as neuromorphic computing, which are intrinsically coupled with spiking neural networks. Finally, we will discuss chances and challenges of scientific computing, in particular at the intersection of AI and PDEs.
Anders Hansen
The consistent reasoning paradox, hallucinations and fallibility of super AI: The power of 'I don't know'
We introduce the Consistent Reasoning Paradox (CRP), which applies to any artificial super intelligence (ASI) (surpassing human intelligence). Consistent reasoning, at the core of logical reasoning, is the ability to handle questions that are equivalent, yet described by different sentences ('Is 1 > 0?' and 'Is one greater than 0?'). The CRP asserts that any ASI, because it must attempt to consistently reason, will always be fallible — like a human. Specifically, the CRP states that there are problems, e.g. in basic arithmetic, where any ASI that always answers and strives to reason consistently will hallucinate (produce wrong, yet plausible answers) infinitely often. The paradox is that there exists a non-consistently reasoning AI — which is not on the level of human intelligence — that will be correct on the same set of problems. The CRP also shows that detecting these hallucinations, even in a probabilistic sense, is strictly harder than solving the original problems, and that there are problems that an ASI may answer correctly, but it cannot provide a correct logical explanation for the answer. Therefore, the CRP implies that any trustworthy AI (i.e., an AI that never answers incorrectly) that also reasons consistently must be able to say 'I don't know'. Moreover, this can only be done by implicitly computing a new concept that we introduce, termed the 'I don't know' function — something currently lacking in modern AI. In view of these insights, the CRP provides a glimpse into the behaviour of ASI. An ASI cannot be 'almost sure', nor can it always explain itself, and therefore to be trustworthy it must be able to say 'I don't know’.
Ralf Zimmermann
The Stiefel manifold of orthogonal frames revisited: Recent algorithmic and theoretical advances
Column-orthogonal matrices are vital in a multitude of numerical applications. They feature at prominent positions in machine learning and data science, signal and image processing, and numerical optimization. In fact, the goal of many classical matrix decompositions is to orthogonalize input data or to find an orthogonal frame that is best suited for a target application.
The set of all column-orthogonal matrices forms the Stiefel manifold, which makes it possible to use Riemannian geometry techniques to solve matrix problems. Starting with a gentle introduction, in this talk we will present the most important tools for numerical algorithms with Stiefel matrices. Then we discuss new theoretical insights on its geometric properties and their implications on numerical algorithms. In particular, we investigate how strongly the manifold is curved, how long shortest closed curves can be, and how far it is from a base point to the edge of normal coordinate domains. The latter is rendered more precisely by the so-called injectivity radius. The injectivity radius acts as a computational barrier, e.g., in Riemannian interpolation algorithms, and in determining Riemannian centers of mass. We will also report a new way to compute Riemannian normal coordinates under a family of Riemannian metrics.
The theoretical concepts will be illustrated and demonstrated throughout by numerical experiments.
Charles-Edouard Bréhier
Asymptotic preserving schemes for stochastic multiscale systems
It is well-known that numerical approximation of multiscale systems is a challenging issue. I will consider this question for several classes of slow-fast systems of stochastic (partial) differential equations. When the time-scale separation parameter goes to 0, the behavior of the slow component is described by averaged or homogenized equations. I will describe how to construct and analyze some temporal discretization schemes which are asymptotic preserving: a similar asymptotic behavior is satisfied for the numerical scheme, and the limiting scheme is consistent with the associated averaged or homogenized equation. I will also describe how uniform error estimates can be obtained for such methods. This will be illustrated by numerical experiments.
Martin Kronbichler
Efficient iterative linear solvers for high-order finite element
discretizations
My talk will discuss the iterative solution of linear systems arising
from discretizing partial differential equations with finite element
methods in fluid dynamics. High-order discretizations have attractive
properties in terms of accuracy-per-unknown also in barely resolved
flows such as turbulence, but come at the price of a denser coupling
between the unknowns. In consequence, classical linear solvers and
preconditioners based on sparse matrices would get more expensive per unknown. Matrix-free methods, which compute the action of the underlying linear operator on a vector, without explicitly building the system matrix, are an attractive alternative for modern computer hardware. Here, the additional computations due to high-order basis functions happen on cached data, without an increase in the global memory transfer. The main challenge for matrix-free methods is the construction of suitable preconditioners. My talk will present developments on block-Jacobi methods, where the blocks are formed by element-wise unknowns of high-order discontinuous Galerkin finite element methods and usually involve between 100 and 500 local unknowns when applied to the context of the incompressible Navier-Stokes equations. For computational efficiency, these block-wise cannot be formed by a locally dense LU decomposition, and approximations with the fast diagonalization method and the Sherman-Morrison-Woodbury formula are proposed instead.
Stefano De Marchi
Multivariate approximation of functions and data: from Padua points to "fake" nodes and beyond
In this talk, we start by recalling the well-known problem of multivariate approximation by polynomials of total degree, concerning finding good points for interpolation. This allowed the discovery of the Padua Points, the first set of unisolvent and explicitly known points on the square whose Lebesgue constant has optimal growth. i.e. O(log^2(n)), with n the polynomial degree. A more general set is the points obtained by sampling on Lissajous curves, which have the advantage of being defined in all space dimensions and have found application in Magnetic Particle Imaging, an emerging medical imaging technique alternative to MRI. Different approaches when approximating functions (even discontinuous). We describe the Variably Scaled (Discontinuos) Kernels and their suitability in approximate multivariate functions. Finally, a general approach called the "fake" nodes approach, which is essentially a mapping basis technique, will be described. If time allows, we can discuss how to use these techniques in Support Vector Machine.
Elisabeth Larsson
Efficient kernel-based approximation of PDEs: An odyssey
In this talk, I will discuss the development of kernel-based methods for solving partial differential equations (PDE) into the current state-of-the-art methods. The basic properties that made these methods attractive to the numerical PDE community is that they are meshfree and quite flexible in terms of the smoothness of the kernel and the resulting approximation orders. The shape parameter that can be applied, especially to infinitely smooth kernels, to modify their properties can be seen as an opportunity or a problem. Choosing the optimal shape parameter can improve results by orders of magnitude, but in the case that the optimal value is small, the corresponding discretization matrices are highly ill-conditioned. This led to the development of stable evaluation methods, which will be discussed here. Initially global approximations were used, but this quickly leads to intractable computational problems in realistic settings and more than two dimensions. There are currently three main strategies for localization, using compactly supported kernels, using stencil-based approximations (RBF-FD), or using a partition of unity method (RBF-PUM). In the talk, the focus will be on the two latter methods. Techniques for theoretical analysis have been developed in parallel with the methods and we will discuss the latest results in this area. In a scattered node setting, the quality of node sets becomes important, and we take a look at the interplay between nodes and methods, and how to easily generate good quality nodes for approximation. Finally, I will talk a bit about optimal shape parameters and recent results for stable evaluation of kernel approximations, and show some results for a challenging application problem.
