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In this paper, first a great number of inverse problems which arise in instrumentation, in computer imaging systems and in computer vision are presented. Then a common general forward modeling for them is given and the corresponding…

Simulation models, in particular agent-based models, are gaining popularity in economics. The considerable flexibility they offer, as well as their capacity to reproduce a variety of empirically observed behaviours of complex systems, give…

A method for adaptive model order reduction for nonsmooth discrete element simulation is developed and analysed in numerical experiments. Regions of the granular media that collectively move as rigid bodies are substituted with rigid bodies…

A common goal throughout science and engineering is to solve optimization problems constrained by computational models. However, in many cases a high-fidelity numerical emulation of systems cannot be optimized due to code complexity and…

Computer simulation models are widely used to study complex physical systems. A related fundamental topic is the inverse problem, also called calibration, which aims at learning about the values of parameters in the model based on…

When data contains measurement errors, it is necessary to make assumptions relating the observed, erroneous data to the unobserved true phenomena of interest. These assumptions should be justifiable on substantive grounds, but are often…

Considering the increasing size of available data, the need for statistical methods that control the finite sample bias is growing. This is mainly due to the frequent settings where the number of variables is large and allowed to increase…

These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental…

Model selection in machine learning (ML) is a crucial part of the Bayesian learning procedure. Model choice may impose strong biases on the resulting predictions, which can hinder the performance of methods such as Bayesian neural networks…

In the framework of BEPU (Best Estimate plus Uncertainty) methodology, the uncertainties involved in the simulations must be quantified to prove that the investigated design is acceptable. The output uncertainties are usually calculated by…

Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to…

The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…

This work proposes a machine-learning framework for constructing statistical models of errors incurred by approximate solutions to parameterized systems of nonlinear equations. These approximate solutions may arise from early termination of…

Recent advances in reconstruction methods for inverse problems leverage powerful data-driven models, e.g., deep neural networks. These techniques have demonstrated state-of-the-art performances for several imaging tasks, but they often do…

The development of algorithms for automation of subtasks during robotic surgery can be accelerated by the availability of realistic simulation environments. In this work, we focus on one aspect of the realism of a surgical simulator, which…

Neural networks have been identified as powerful tools for the study of complex systems. A noteworthy example is the neural network differential equation (NN DE) solver, which can provide functional approximations to the solutions of a wide…

Model correction is essential for reliable PDE learning when the governing physics is misspecified due to simplified assumptions or limited observations. In the machine learning literature, existing correction methods typically operate in…

Consider scene understanding problems such as predicting where a person is probably reaching, or inferring the pose of 3D objects from depth images, or inferring the probable street crossings of pedestrians at a busy intersection. This…

Neural networks are among the most accurate supervised learning methods in use today. However, their opacity makes them difficult to trust in critical applications, especially when conditions in training may differ from those in practice.…

In this article, we consider computing expectations w.r.t. probability measures which are subject to discretization error. Examples include partially observed diffusion processes or inverse problems, where one may have to discretize time…