Computational Mathematics and Applications Seminar

In this talk, we consider the problem of approximating failure regions. More specifically, given a costly computational model with random parameters and a failure condition, our objective is to determine the parameter region in which the failure condition is likely to not be satisfied. In mathematical terms, this problem can be cast as approximating the level set of a probability density function. We solve this problem by dividing it into two: 1) The design of an efficient Monte Carlo strategy for probability estimation. 2) The construction of an efficient algorithm for level-set approximation. Following this structure, this talk is comprised of two parts:

In the first part, we present a new multi-output multilevel best linear unbiased estimator (MLBLUE) for approximating expectations. The advantage of this estimator is in its convenience and optimality: Given any set of computational models with known covariance structure, MLBLUE automatically constructs a provenly optimal estimator for any (finite) number of quantities of interest. Nevertheless, the optimality of MLBLUE is tied to its optimal set-up, which requires the solution of a nonlinear optimization problem. We show how the latter can be reformulated as a semi-definite program and thus be solved reliably and efficiently.

In the second part, we construct an adaptive level-set approximation algorithm for smooth functions corrupted by noise in $\mathbb{R}^d$. This algorithm only requires point value data and is thus compatible with Monte Carlo estimators. The algorithm is comprised of a criterion for level-set adaptivity combined with an a posteriori error estimator. Under suitable assumptions, we can prove that our algorithm will correctly capture the target level set at the same cost complexity of uniformly approximating a $(d-1)$-dimensional function.

The classical finite element method uses piecewise-polynomial function spaces satisfying continuity and boundary conditions. Hybrid finite element methods, by contrast, drop these continuity and boundary conditions from the function spaces and instead enforce them weakly using Lagrange multipliers. The hybrid approach has several numerical and implementational advantages, which have been studied over the last few decades.

In this talk, we show how the hybrid perspective has yielded new insights—and new methods—in structure-preserving numerical PDEs. These include multisymplectic methods for Hamiltonian PDEs, charge-conserving methods for the Maxwell and Yang-Mills equations, and hybrid methods in finite element exterior calculus.

Joint work Marta Betcke and Bolin Pan

In photoacoustic tomography (PAT) with flat sensor, we routinely encounter two types of limited data. The first is due to using a finite sensor and is especially perceptible if the region of interest is large relatively to the sensor or located farther away from the sensor. In this talk we focus on the second type caused by a varying sensitivity of the sensor to the incoming wavefront direction which can be modelled as binary i.e. by a cone of sensitivity. Such visibility conditions result, in Fourier domain, in a restriction of the data to a bowtie, akin to the one corresponding to the range of the forward operator but further narrowed according to the angle of sensitivity. 

We show how we can separate the visible and invisible wavefront directions in PAT image and data using a directional frame like Curvelets, and how such decomposition allows for decoupling of the reconstruction involving application of expensive forward/adjoint solvers from the training problem. We present fast and stable approximate Fourier domain forward and adjoint operators for reconstruction of the visible coefficients for such limited angle problem and a tailored UNet matching both the multi-scale Curvelet decomposition and the partition into the visible/invisible directions for learning the invisible coefficients from a training set of similar data.

As machine learning algorithms get integrated into the decision-making process of companies and organizations, insurance products are being developed to protect their providers from liability risk. Algorithmic liability differs from human liability since it is based on data-driven models compared to multiple heterogeneous decision-makers and its performance is known a priori for a given set of data. Traditional actuarial tools for human liability do not consider these properties, primarily focusing on the distribution of historical claims. We propose, for the first time, a quantitative framework to estimate the risk exposure of insurance contracts for machine-driven liability, introducing the concept of algorithmic insurance. Our work provides ML model developers and insurance providers with a comprehensive risk evaluation approach for this new class of products. Thus, we set the foundations of a niche area of research at the intersection of the literature in operations, risk management, and actuarial science. Specifically, we present an optimization formulation to estimate the risk exposure of a binary classification model given a pre-defined range of premiums. Our approach outlines how properties of the model, such as discrimination performance, interpretability, and generalizability, can influence the insurance contract evaluation. To showcase a practical implementation of the proposed framework, we present a case study of medical malpractice in the context of breast cancer detection. Our analysis focuses on measuring the effect of the model parameters on the expected financial loss and identifying the aspects of algorithmic performance that predominantly affect the risk of the contract.

Paper Reference: Bertsimas, D. and Orfanoudaki, A., 2021. Pricing algorithmic insurance. arXiv preprint arXiv:2106.00839.

Paper link: https://arxiv.org/pdf/2106.00839.pdf

Bilevel optimization has been part of machine learning for over 4 decades now, although perhaps not always in an obvious way. The interconnection between the two topics started appearing more clearly in publications since about 20 years now, and in the last 10 years, the number of machine learning applications of bilevel optimization has literally exploded. This rise of bilevel optimization in machine learning has been highly positive, as it has come with many innovations in the theoretical and numerical perspectives in understanding and solving the problem, especially with the rebirth of the implicit function approach, which seemed to have been abandoned at some point.
Overall, machine learning has set the bar very high for the whole field of bilevel optimization with regards to the development of numerical methods and the associated convergence analysis theory, as well as the introduction of efficient tools to speed up components such as derivative calculations among other things. However, it remains unclear how the techniques from the machine learning—based bilevel optimization literature can be extended to other applications of bilevel programming. 
For instance, many machine learning loss functions and the special problem structures enable the fulfillment of some qualification conditions that will fail for multiple other applications of bilevel optimization. In this talk, we will provide an overview of machine learning applications of bilevel optimization while giving a flavour of corresponding solution algorithms and their limitations. 
Furthermore, we will discuss possible paths for algorithms that can tackle more complicated machine learning applications of bilevel optimization, while also highlighting lessons that can be learned for more general bilevel programs.

In this talk, we discuss continuous in time dynamics for the problem of approaching the set of zeros of a single-valued monotone and continuous operator V . Such problems are motivated by minimax convexconcave and, in particular, by convex optimization problems with linear constraints. The central role is played by a second-order dynamical system that combines a vanishing damping term with the time derivative of V along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. We show that these methods exhibit fast convergence rates for kV (z(t))k as t ! +1, where z( ) denotes the generated trajectory, and for the restricted gap function, and that z( ) converges to a zero of the operator V . For the corresponding implicit and explicit discrete time models with Nesterov’s momentum, we prove that they share the asymptotic features of the continuous dynamics.

Extensions to variational inequalities and fixed-point problems are also addressed. The theoretical results are illustrated by numerical experiments on bilinear games and the training of generative adversarial networks.

This talk explores recent advancements in stress and flux-based finite element methods. It focuses on addressing the limitations of traditional finite elements, in order to describe complex material behavior and engineer new metamaterials.

Stress and flux-based finite element methods are particularly useful in error estimation, laying the groundwork for adaptive refinement strategies. This concept builds upon the hypercircle theorem [1], which states that in a specific energy space, both the exact solution and any admissible stress field lie on a hypercircle. However, the construction of finite element spaces that satisfy admissible states for complex material behavior is not straightforward. It often requires a relaxation of specific properties, especially when dealing with non-symmetric stress tensors [2] or hyperelastic materials.

Alternatively, methods that directly approximate stresses can be employed, offering high accuracy of the stress fields and adherence to physical conservation laws. However, when approximating eigenvalues, this significant benefit for the solution's accuracy implies that the solution operator cannot be compact. To address this, the solution operator must be confined to a subset of the solution that excludes the stresses. Yet, due to compatibility conditions, the trial space for the other solution components typically does not yield the desired accuracy. The second part of this talk will therefore explore the Least-Squares method as a remedy to these challenges [3].

To conclude this talk, we will emphasize the integration of those methods within global solution strategies, with a particular focus on the challenges regarding model order reduction methods [4].

[1] W. Prager, J. Synge. Approximations in elasticity based on the concept of function space.

Quarterly of Applied Mathematics 5(3), 1947.

[2] FB, K. Bernhard, M. Moldenhauer, G. Starke. Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity, Numerical Methods for Partial Differential Equations 37(4), 2021.

[3] FB, D. Boffi. First order least-squares formulations for eigenvalue problems, IMA Journal of Numerical Analysis 42(2), 2023.

[4] FB, D. Boffi, A. Halim. A reduced order model for the finite element approximation of eigenvalue problems,Computer Methods in Applied Mechanics and Engineering 404, 2023.

In recent years, we have witnessed the emergence of scientific machine learning as a data-driven tool for the analysis, by means of deep-learning techniques, of data produced by computational science and engineering applications.  At the core of these methods is the supervised training algorithm to learn the neural network realization, a highly non-convex optimization problem that is usually solved using stochastic gradient methods.

However, distinct from deep-learning practice, scientific machine-learning training problems feature a much larger volume of smooth data and better characterizations of the empirical risk functions, which make them suited for conventional solvers for unconstrained optimization.

In this talk, we empirically demonstrate the superior efficacy of a trust region method based on the Gauss-Newton approximation of the Hessian in improving the generalization errors arising from regression tasks when learning surrogate models for a wide range of scientific machine-learning techniques and test cases. All the conventional solvers tested, including L-BFGS and inexact Newton with line-search, compare favorably, either in terms of cost or accuracy, with the adaptive first-order methods used to validate the surrogate models.

A novel framework for hierarchical low-rank matrices is proposed that combines an adaptive hierarchical partitioning of the matrix with low-rank approximation. One typical application is the approximation of discretized functions on rectangular domains; the flexibility of the format makes it possible to deal with functions that feature singularities in small, localized regions. To deal with time evolution and relocation of singularities, the partitioning can be dynamically adjusted based on features of the underlying data. Our format can be leveraged to efficiently solve linear systems with Kronecker product structure, as they arise from discretized partial differential equations (PDEs). For this purpose, these linear systems are rephrased as linear matrix equations and a recursive solver is derived from low-rank updates of such equations. 
We demonstrate the effectiveness of our framework for stationary and time-dependent, linear and nonlinear PDEs, including the Burgers' and Allen–Cahn equations.

This is a joint work with Daniel Kressner and Stefano Massei.