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Isogeometric analysis has brought a paradigm shift in integrating computational simulations with geometric designs across engineering disciplines. This technique necessitates analysis-suitable parameterization of physical domains to fully harness the synergy between Computer-Aided Design and Computer-Aided Engineering analyses. The existing methods often fix boundary parameters, leading to challenges in elongated geometries such as fluid channels and tubular reactors. This paper presents an innovative solution for the boundary parameter matching problem, specifically designed for analysis-suitable parameterizations. We employ a sophisticated Schwarz-Christoffel mapping technique, which is instrumental in computing boundary correspondences. A refined boundary curve reparameterization process complements this. Our dual-strategy approach maintains the geometric exactness and continuity of input physical domains, overcoming limitations often encountered with the existing reparameterization techniques. By employing our proposed boundary parameter method, we show that even a simple linear interpolation approach can effectively construct a satisfactory analysis-suitable parameterization. Our methodology offers significant improvements over traditional practices, enabling the generation of analysis-suitable and geometrically precise models, which is crucial for ensuring accurate simulation results. Numerical experiments show the capacity of the proposed method to enhance the quality and reliability of isogeometric analysis workflows.

Generative diffusion models have achieved spectacular performance in many areas of generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, variational inference and stochastic calculus, in this paper we show that many aspects of these models can be understood using the tools of equilibrium statistical mechanics. Using this reformulation, we show that generative diffusion models undergo second-order phase transitions corresponding to symmetry breaking phenomena. We show that these phase-transitions are always in a mean-field universality class, as they are the result of a self-consistency condition in the generative dynamics. We argue that the critical instability that arises from the phase transitions lies at the heart of their generative capabilities, which are characterized by a set of mean field critical exponents. Furthermore, using the statistical physics of disordered systems, we show that memorization can be understood as a form of critical condensation corresponding to a disordered phase transition. Finally, we show that the dynamic equation of the generative process can be interpreted as a stochastic adiabatic transformation that minimizes the free energy while keeping the system in thermal equilibrium.

The ParaDiag family of algorithms solves differential equations by using preconditioners that can be inverted in parallel through diagonalization. In the context of optimal control of linear parabolic PDEs, the state-of-the-art ParaDiag method is limited to solving self-adjoint problems with a tracking objective. We propose three improvements to the ParaDiag method: the use of alpha-circulant matrices to construct an alternative preconditioner, a generalization of the algorithm for solving non-self-adjoint equations, and the formulation of an algorithm for terminal-cost objectives. We present novel analytic results about the eigenvalues of the preconditioned systems for all discussed ParaDiag algorithms in the case of self-adjoint equations, which proves the favorable properties the alpha-circulant preconditioner. We use these results to perform a theoretical parallel-scaling analysis of ParaDiag for self-adjoint problems. Numerical tests confirm our findings and suggest that the self-adjoint behavior, which is backed by theory, generalizes to the non-self-adjoint case. We provide a sequential, open-source reference solver in Matlab for all discussed algorithms.

Regression models that incorporate smooth functions of predictor variables to explain the relationships with a response variable have gained widespread usage and proved successful in various applications. By incorporating smooth functions of predictor variables, these models can capture complex relationships between the response and predictors while still allowing for interpretation of the results. In situations where the relationships between a response variable and predictors are explored, it is not uncommon to assume that these relationships adhere to certain shape constraints. Examples of such constraints include monotonicity and convexity. The scam package for R has become a popular package to carry out the full fitting of exponential family generalized additive modelling with shape restrictions on smooths. The paper aims to extend the existing framework of shape-constrained generalized additive models (SCAM) to accommodate smooth interactions of covariates, linear functionals of shape-constrained smooths and incorporation of residual autocorrelation. The methods described in this paper are implemented in the recent version of the package scam, available on the Comprehensive R Archive Network (CRAN).

Explicit time integration schemes coupled with Galerkin discretizations of time-dependent partial differential equations require solving a linear system with the mass matrix at each time step. For applications in structural dynamics, the solution of the linear system is frequently approximated through so-called mass lumping, which consists in replacing the mass matrix by some diagonal approximation. Mass lumping has been widely used in engineering practice for decades already and has a sound mathematical theory supporting it for finite element methods using the classical Lagrange basis. However, the theory for more general basis functions is still missing. Our paper partly addresses this shortcoming. Some special and practically relevant properties of lumped mass matrices are proved and we discuss how these properties naturally extend to banded and Kronecker product matrices whose structure allows to solve linear systems very efficiently. Our theoretical results are applied to isogeometric discretizations but are not restricted to them.

In this work, we derive a $\gamma$-robust a posteriori error estimator for finite element approximations of the Allen-Cahn equation with variable non-degenerate mobility. The estimator utilizes spectral estimates for the linearized steady part of the differential operator as well as a conditional stability estimate based on a weighted sum of Bregman distances, based on the energy and a functional related to the mobility. A suitable reconstruction of the numerical solution in the stability estimate leads to a fully computable estimator.

Reconstructing a dynamic object with affine motion in computerized tomography (CT) leads to motion artifacts if the motion is not taken into account. In most cases, the actual motion is neither known nor can be determined easily. As a consequence, the respective model that describes CT is incomplete. The iterative RESESOP-Kaczmarz method can - under certain conditions and by exploiting the modeling error - reconstruct dynamic objects at different time points even if the exact motion is unknown. However, the method is very time-consuming. To speed the reconstruction process up and obtain better results, we combine the following three steps: 1. RESESOP-Kacmarz with only a few iterations is implemented to reconstruct the object at different time points. 2. The motion is estimated via landmark detection, e.g. using deep learning. 3. The estimated motion is integrated into the reconstruction process, allowing the use of dynamic filtered backprojection. We give a short review of all methods involved and present numerical results as a proof of principle.

Machine learning (ML) methods, which fit to data the parameters of a given parameterized model class, have garnered significant interest as potential methods for learning surrogate models for complex engineering systems for which traditional simulation is expensive. However, in many scientific and engineering settings, generating high-fidelity data on which to train ML models is expensive, and the available budget for generating training data is limited. ML models trained on the resulting scarce high-fidelity data have high variance and are sensitive to vagaries of the training data set. We propose a new multifidelity training approach for scientific machine learning that exploits the scientific context where data of varying fidelities and costs are available; for example high-fidelity data may be generated by an expensive fully resolved physics simulation whereas lower-fidelity data may arise from a cheaper model based on simplifying assumptions. We use the multifidelity data to define new multifidelity Monte Carlo estimators for the unknown parameters of linear regression models, and provide theoretical analyses that guarantee the approach's accuracy and improved robustness to small training budgets. Numerical results verify the theoretical analysis and demonstrate that multifidelity learned models trained on scarce high-fidelity data and additional low-fidelity data achieve order-of-magnitude lower model variance than standard models trained on only high-fidelity data of comparable cost. This illustrates that in the scarce data regime, our multifidelity training strategy yields models with lower expected error than standard training approaches.

We consider fully discrete finite element approximations for a semilinear optimal control system of partial differential equations in two cases: for distributed and Robin boundary control. The ecological predator-prey optimal control model is approximated by conforming finite element methods mimicking the spatial part, while a discontinuous Galerkin method is used for the time discretization. We investigate the sensitivity of the solution distance from the target function, in cases with smooth and rough initial data. We employ low, and higher-order polynomials in time and space whenever proper regularity is present. The approximation schemes considered are with and without control constraints, driving efficiently the system to desired states realized using non-linear gradient methods.

Deep learning has achieved widespread success in medical image analysis, leading to an increasing demand for large-scale expert-annotated medical image datasets. Yet, the high cost of annotating medical images severely hampers the development of deep learning in this field. To reduce annotation costs, active learning aims to select the most informative samples for annotation and train high-performance models with as few labeled samples as possible. In this survey, we review the core methods of active learning, including the evaluation of informativeness and sampling strategy. For the first time, we provide a detailed summary of the integration of active learning with other label-efficient techniques, such as semi-supervised, self-supervised learning, and so on. We also summarize active learning works that are specifically tailored to medical image analysis. Additionally, we conduct a thorough comparative analysis of the performance of different AL methods in medical image analysis with experiments. In the end, we offer our perspectives on the future trends and challenges of active learning and its applications in medical image analysis.