Abstract

Session 1, Nov 3, Wednesday Afternoon

Tony F. Chan (KAUST)
1:30 PM
The Jinchao I know

Mary F. Wheeler
2:10 PM
Bayesian Optimization for Field Scale Geological Carbon Sequestration

We present a framework of the application of Bayesian Optimization (BO) to well management for geological carbon sequestration. The coupled compositional flow and poroelasticity simulator, IPARS, is utilized to accurately capture the underlying physical processes during CO2 sequestration. We further couple the IBM Bayesian Optimization (IBO) with IPARS for parallel optimization of well injection strategies during field-scale CO2 sequestration. Bayesian optimization builds a surrogate for the objective and quantifies the uncertainty in that surrogate using a Bayesian machine learning technique, Gaussian process regression, and then uses an acquisition function defined from this surrogate to decide where to sample. IBO addresses the three weak points of standard BO in that it supports parallel (batch) execution, scales better for higher-dimensional problems, and is more robust to initializations. An application to optimize the CO2 injection schedule in the Cranfield site using field data benchmarked with genetic algorithm (GA) and covariance matrix adaptation evolution strategy (CMA-ES) shows that IBO achieves competitive objective function value with significantly less number of forward model evaluations. Furthermore, the Bayesian framework that IBO builds upon allows uncertainty quantification and naturally extends to optimization under uncertainty.

L. Ridgway Scott
3:10 PM
Multigrid and Pressure Robust Elements

Pressure-robust elements pose special challenges for multigrid methods. Jinchao Xu and co-workers made one of the fundamental observations about such methods.
We review the issues and present some new results. We make two conjectures. First, we conjecture that in three dimensions, the Scott-Vogelius method on a structured tetrahedral mesh is inf-sup stable for velocities of degree 4 and higher. We introduce a new method for computing the inf-sup constant and use it to support the conjecture.
The second conjecture on such a mesh relates to optimal multigrid performance for degree 5 and higher. This conjecture is connected to the existence of a local basis for divergence-free functions.

Randolph E. Bank
3:50 PM
An Analysis of the Saturation Assumption with Application to Adaptive Finite Elements

The saturation assumption plays a central role in much of the analysis of a posteriori error estimates and refinement algorithms for adaptive finite element methods. In this work we provide an analysis of this assumption in the simple setting of interpolation. We then construct a useful computational framework to evaluate the efficiency and reliability of adaptive feedback loops.

Michael Holst
4:30 PM
Some Research Problems in Mathematical and Numerical Relativity

The 2017 Nobel Prize in Physics was awarded to three of the key scientists involved in the development of LIGO and its eventual successful first detections of gravitational waves. These detections are made possible through the exploration of a multi-dimensional model parameter space, using detailed numerical simulations of one of the most complicated PDE systems known: the Einstein equations.
In the first part of the talk we start with the mathematical formulation of general relativity, and then focus in on the Einstein constraint equations. This nonlinear PDE system has been studied extensively since 1944, with a complex solution theory starting to take shape, until analysts reached an impasse in 1996. A mathematical breakthrough occurred in 2009, allowing the full complexity of the solution theory to emerge. I will give an overview of the problem and results prior to 2009, and then summarize the new developments since 2009. An interesting footnote to this story is that the 2009 developments in the theory came from the numerical analysis community, and were driven largely by difficulties encountered in designing reliable numerical methods for the PDE system.
In the second part of the talk I will shift over to the quest for developing good numerical methods for these types of geometric PDE systems posed on 2- and 3-manifolds. We will work primarily within the finite element exterior calculus framework, and our journey will take us through a prior error estimates, a posteriori estimates, adaptive algorithms, fast solvers, and parallel computing. This lecture will touch on several joint projects that span more than a decade, involving a number of collaborators.

Session 2, Nov 4, Thursday Morning

Franco Brezzi and L. Donatella Marini
8:30 AM
Virtual Elements and curved boundaries

We present a way of constructing conforming Virtual Element spaces on polygons with curved edges. Unlike previous VEM approaches for curvilinear elements, the present construction ensures that the local VEM spaces contain all the polynomials of a given degree, thus providing the full satisfaction of the patch test.

Wolfgang Hackbusch
9:10 AM
On nonclosed tensor formats

Since tensor spaces may have a huge dimension, it is often not possible to store tensors by all their entries. Instead one uses certain representations (also called ‘formats’), which describe a subset of tensors. For some formats used in practice the set of representable tensors is not closed. This leads to an instability comparable with the cancellation effect in the case of numerical differentiation. Under general conditions we proof for the finite-dimensional case that there is some minimal strenghth of the instability. For the special case of the 2-term format a quantitative result can be proved.
In the infinite-dimensional case with a tensor norm not weaker than the injective crossnorm, the same instability behaviour can be proved. Even the constants in the estimates are under control. As a result, it is sufficient to study the instability behaviour for finite-dimensional model spaces.

Harry Yserentant
10:20 AM
A class of variational multiscale methods based on subspace decomposition

Numerical homogenization tries to approximate the solutions of elliptic partial differential equations with strongly oscillating coefficients by functions from modified finite element spaces. I will present in this talk a class of such methods that are very closely related to a method that has years ago been developed by Malqvist and Peterseim. Like the method of Malqvist and Peterseim, the presented methods do not make explicit or implicit use of a scale separation and rest upon the localization and smoothing properties of elliptic equations. Their analysis, and not only that, is based on the theory of additive subspace decomposition methods and in the end thus on Jinchao Xu’s seminal work.

Gabriel Wittum
11:00 AM
Double enriched finite volume spaces for the DNS of fluid-particle interaction

We present double enriched finite volume spaces for the simulation of free particles in a fluid. This involves forces beeing exchanged between the particles and the fluid at the interface. In an earlier work we derived a monotithic scheme which includes the interaction forces into the Navier-Stokes equations by direct coupling. In multiphase flows oscillations and spurious velocities are a common issue. The surface force term yields a jump in the pressure and therefore the oscillations are usually resolved by extending the spaces on cut elements in order to resolve the discontinuity. For the construction of the enriched spaces proposed in this paper we exploit the Petrov-Galerkin formulation of the vertex-centered finite volume method (PG-FVM). From the perspective of the finite volume scheme we argue that wrong discrete normal directions at the interface are the origin of the oscillations. The new perspective of normal vectors suggests to look at gradients rather than values of the enriching shape functions. The crucial parameter of the enrichement functions therefore is the gradient of the shape functions and especially the one of the test space. The distinguishing feature of our construction therefore is an enrichment that is based on the choice of shape functions with consistent gradients. These derivations finally yield a fitted scheme for the immersed interface. We further propose a strategy ensuring a well-conditioned system independent of the location of the interface. Numerical tests were conducted using the PG-FVM. We demonstrate that the enriched spaces are able to eliminate the oscillations.

Qing Nie
11:40 am
Multiscale inference and modeling of cell fate via single-cell data

Cells make fate decisions in response to dynamic environmental and pathological stimuli as well as cell-to-cell communications. Recent technological breakthroughs have enabled gathering data in previously unthinkable quantities at single cell level, starting to suggest that cell fate decision is much more complex, dynamic, and stochastic than previously recognized. Multiscale interactions, sometimes through cell-cell communications, play a critical role in cell decision-making. Dissecting cellular dynamics emerging from molecular and genomic scale in single-cell demands novel computational tools and multiscale models. In this talk, through multiple biological examples we will present our recent effort to use single-cell RNA-seq data and spatial imaging data to uncover new insights in development, regeneration, and cancers. We will also present several new computational tools and mathematical modeling methods that are required to study the complex and dynamic cell fate process through the lens of single cells.

Session 3, Nov 4, Thursday Afternoon

Sherry Li
1:30 PM
Interplay of linear algebra, machine learning, and high performance computing

In recent years, we have seen a large body of research using hierarchical matrix algebra to construct low complexity linear solvers and preconditioners. Not only can these fast solvers significantly accelerate the speed of large scale PDE based simulations, but also they can speed up many AI and machine learning algorithms which are often matrix-computation-bound. On the other hand, statistical and machine learning methods can be used to help select best solvers or solvers configurations for specific problems and computer platforms. In both of these fields, high performance computing becomes an indispensable tool to achieve real-time solutions for big data problems. In this talk, we will show our recent developments in the intersection of these areas.

Chun Liu
2:10 PM
Energetic Variatioanl Approaches for Phase Field Models: Boundary Conditions and Temperature Effects

Active/reactive fluids convert and transduce energy from their surrounding into a motion and other mechanical activities. These systems are usually out of mechanical or even thermodynamic equilibrium. One can find such examples in almost all biological systems. In this talk I will develop a general theory for active fluids which convert chemical energy into various types of mechanical energy and laws of thermodynamics. This is the extension of the classical energetic variational approaches for mechanical systems and nonequilibrium thermodynamics. In this talk I will focus on the applications to the phase field models. This is a joint work with Yiwei Wang.

Constantin Bacuta
3:10 PM
Least squares discretization for singularly perturbed problems

We consider a least squares method for discretizing singularly perturbed elliptic problems. Choices for discrete stable spaces are considered for the mixed formulation and a preconditioned conjugate gradient iterative process for solving the saddle point reformulation is proposed. We provide a preconditioning strategy that works for a large range of the perturbation parameter. Using the concepts of optimal test norm and saddle point reformulation we provide an efficient discretization strategy that works for uniform and non-uniform refinements for Convection- Reaction-Diffusion Problems.

Youngju Lee
3:50 PM
Helicity-conservative finite element discretization for incompressible MHD systems

We construct finite element methods for the incompressible magnetohydrodynamics (MHD) system that precisely preserve magnetic and cross helicity, the energy law and the magnetic Gauss law at the discrete level. The variables are discretized as discrete differential forms in a de Rham complex. We present a couple of numerical tests to show the performance of the algorithm. This is a joint work with Kaibo Hu and Jinchao Xu.

Jonathan Siegel
4:30 PM
Training Asymptotically Optimal Shallow Neural Networks using Greedy Algorithms

A shallow neural network is a linear combination of ridge functions whose profile is determined by a fixed activation function. We will introduce spaces of functions which can be efficiently approximated by shallow neural networks for a wide variety of activation functions and analyze their approximation properties. Specifically, we will compute their metric entropy and n-widths, which are fundamental quantities in approximation theory that control the limits of linear and non-linear approximation and statistical estimation for a given class of functions. Consequences of these results include: the optimal approximation rates which can be attained for shallow neural networks, that shallow neural networks dramatically outperform linear methods of approximation, and even that shallow neural networks are optimal among all non-linear methods on these spaces, if stability or continuity of the non-linear method is required. Next, we discuss algorithmic aspects of approximation by shallow networks. Specifically, we show that the optimal approximation rates can be attained algorithmically using the orthogonal greedy algorithm, assuming that an optimization over single neurons can be solved. Finally, we show some applications to solving PDEs numerically.

Session 4, Nov 5, Friday Morning

Ralf Hiptmair
8:30 AM
Regular Decompositions: Continuous and Discrete

Regular decompositions first rose to prominence in the context of function spaces related to Maxwell’s equations and in the analysis of associated variational (boundary integral)equations. The classical version refers to stable splittings of the space of vector fields with square-integrable curl of the form \textbf{H}(\textbf{curl},\Omega) = (H^1(\Omega))^3 + \textbf{grad} H^1(\Omega). (1)
This turns out to be a special case of a more general result for the De Rahm-Sobolev complex of differential forms on bounded Lipschitz subsets of Rn. It can also be extended to homogeneous boundary conditions imposed on parts of the boundary. Harnessing abstract theory of Hilbert complexes, regular decompositions then pave the way for establishing compact embeddings, closed range theorems, Poincare/Friedrichs-type estimates and finite-dimensional co-homology. They have also been extended to Hilbert complexes derived from the de Rham complex like the elasticity complex or div Div/Grad grad complexes.In finite-element exterior calculus (FEEC)discrete regular decompositions assert the existence of “uniformly stable” splittings of (piecewise) polynomial spaces of discrete differential forms into (i) fully continuous piecewise polynomial forms, (ii) closed discrete differential forms, and, slightly extending (1), (iii) an “oscillatory” remainder. They have been established for both the h version and p version of FEEC, also taking into account(partial) homogeneous boundary conditions. They form the theoretical foundation for the numerical analysis of nodal auxiliary space preconditioning for discrete variational problems in Sobolev spaces of differential forms.

Ralf Kornhuber
9:10 AM
Towards Numerical Simulation of Multiscale Fault Networks

We consider a scalar elliptic model problem with jump conditions on a fractal network of interfaces inspired by frictional fault systems in the geosciences, and derive an associated asymptotic limit problem. The resulting solution space is characterized in terms of generalized jumps and gradients, and we prove continuous embeddings into L^2 and H^s, s < \frac{1}{2}. Based on novel projection operators with suitable stability and approximation properties, we also derive finite element methods allowing for scale-independent error estimates together with successive subspace correction methods with scale-independent convergence rates. Our theoretical findings are illustrated by numerical computations.
Based on these we briefly present a mathematical model for the deformation of a geological structure with non-intersecting faults that can be represented by a layered system of viscoelastic bodies satisfying rate- and state-depending friction conditions along the common interfaces and undergoing large tangential displacements and sketch a numerical solution algorithm based on a Newmark discretization in time, decoupling of rate and state by a fixed point iteration, a mortar method in space and non-smooth multigrid. Numerical experiments illustrate the behavior of the model together with the efficiency and reliability of our solver.

John Urschel
10:20 AM
Learning Determinantal Point Processes

Determinantal point processes (DPPs) are a class of random point processes where subset probabilities correspond to principal minors of a matrix. DPPs appear in a wide variety of settings, including random matrix theory, combinatorics, quantum mechanics, and more recently machine learning. In this talk, I will give an introduction to DPPs, a few examples and interesting properties, and treat the problems of 1. recognizing when a point process is a DPP, and 2. recovering the kernel of a DPP given samples. These two questions are closely related to the principal minor assignment problem.

Long Chen
11:00 AM
Efficient and Flexible Transformer via Decomposed Near-field and Far-field Attention

We propose FMMformers, a class of efficient and flexible transformers inspired by the celebrated fast multipole method (FMM) for accelerating interacting particle simulation. FMM decomposes particle-particle interaction into near-field and far-field components and then performs direct and coarse-grained computation, respectively. Similarly, FMMformers decompose the attention into near-field and far-field attention, modeling the near-field attention by a banded matrix and the far-field attention by a low-rank matrix. Computing the attention matrix for FMMformers requires linear complexity in computational time and memory footprint with respect to the sequence length. In contrast, standard and linear transformers suffer from quadratic complexity. We analyze and validate the advantage of FMMformers over the standard transformer on the Long Range Arena and language modeling benchmarks. FMMformers can even outperform the standard transformer in terms of accuracy by a significant margin.
This is a joint work with Tan Minh Nguyen, Vai Suliafu, Stanley Osher, and Bao Wang.