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A major challenge in computed tomography is reconstructing objects from incomplete data. An increasingly popular solution for these problems is to incorporate deep learning models into reconstruction algorithms. This study introduces a novel approach by integrating a Fourier neural operator (FNO) into the Filtered Backprojection (FBP) reconstruction method, yielding the FNO back projection (FNO-BP) network. We employ moment conditions for sinogram extrapolation to assist the model in mitigating artefacts from limited data. Notably, our deep learning architecture maintains a runtime comparable to classical filtered back projection (FBP) reconstructions, ensuring swift performance during both inference and training. We assess our reconstruction method in the context of the Helsinki Tomography Challenge 2022 and also compare it against regular FBP methods.

Motivated by a recent work on a preconditioned MINRES for flipped linear systems in imaging, in this note we extend the scope of that research for including more precise boundary conditions such as reflective and anti-reflective ones. We prove spectral results for the matrix-sequences associated to the original problem, which justify the use of the MINRES in the current setting. The theoretical spectral analysis is supported by a wide variety of numerical experiments, concerning the visualization of the spectra of the original matrices in various ways. We also report numerical tests regarding the convergence speed and regularization features of the associated GMRES and MINRES methods. Conclusions and open problems end the present study.

The main reason for query model's prominence in complexity theory and quantum computing is the presence of concrete lower bounding techniques: polynomial and adversary method. There have been considerable efforts to give lower bounds using these methods, and to compare/relate them with other measures based on the decision tree. We explore the value of these lower bounds on quantum query complexity and their relation with other decision tree based complexity measures for the class of symmetric functions, arguably one of the most natural and basic sets of Boolean functions. We show an explicit construction for the dual of the positive adversary method and also of the square root of private coin certificate game complexity for any total symmetric function. This shows that the two values can't be distinguished for any symmetric function. Additionally, we show that the recently introduced measure of spectral sensitivity gives the same value as both positive adversary and approximate degree for every total symmetric Boolean function. Further, we look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. Finally, we study how large certificate complexity and block sensitivity can be as compared to sensitivity for symmetric functions (even up to constant factors). We show tight separations, i.e., give upper bounds on possible separations and construct functions achieving the same.

Fourth-order variational inequalities are encountered in various scientific and engineering disciplines, including elliptic optimal control problems and plate obstacle problems. In this paper, we consider additive Schwarz methods for solving fourth-order variational inequalities. Based on a unified framework of various finite element methods for fourth-order variational inequalities, we develop one- and two-level additive Schwarz methods. We prove that the two-level method is scalable in the sense that the convergence rate of the method depends on $H/h$ and $H/\delta$ only, where $h$ and $H$ are the typical diameters of an element and a subdomain, respectively, and $\delta$ measures the overlap among the subdomains. This proof relies on a new nonlinear positivity-preserving coarse interpolation operator, the construction of which was previously unknown. To the best of our knowledge, this analysis represents the first investigation into the scalability of the two-level additive Schwarz method for fourth-order variational inequalities. Our theoretical results are verified by numerical experiments.

Boundary value problems involving elliptic PDEs such as the Laplace and the Helmholtz equations are ubiquitous in mathematical physics and engineering. Many such problems can be alternatively formulated as integral equations that are mathematically more tractable. However, an integral-equation formulation poses a significant computational challenge: solving large dense linear systems that arise upon discretization. In cases where iterative methods converge rapidly, existing methods that draw on fast summation schemes such as the Fast Multipole Method are highly efficient and well-established. More recently, linear complexity direct solvers that sidestep convergence issues by directly computing an invertible factorization have been developed. However, storage and computation costs are high, which limits their ability to solve large-scale problems in practice. In this work, we introduce a distributed-memory parallel algorithm based on an existing direct solver named ``strong recursive skeletonization factorization.'' Specifically, we apply low-rank compression to certain off-diagonal matrix blocks in a way that minimizes computation and data movement. Compared to iterative algorithms, our method is particularly suitable for problems involving ill-conditioned matrices or multiple right-hand sides. Large-scale numerical experiments are presented to show the performance of our Julia implementation.

Complex interval arithmetic is a powerful tool for the analysis of computational errors. The naturally arising rectangular, polar, and circular (together called primitive) interval types are not closed under simple arithmetic operations, and their use yields overly relaxed bounds. The later introduced polygonal type, on the other hand, allows for arbitrarily precise representation of the above operations for a higher computational cost. We propose the polyarcular interval type as an effective extension of the previous types. The polyarcular interval can represent all primitive intervals and most of their arithmetic combinations precisely and has an approximation capability competing with that of the polygonal interval. In particular, in antenna tolerance analysis it can achieve perfect accuracy for lower computational cost then the polygonal type, which we show in a relevant case study. In this paper, we present a rigorous analysis of the arithmetic properties of all five interval types, involving a new algebro-geometric method of boundary analysis.

In this contribution we apply an adaptive model hierarchy, consisting of a full-order model, a reduced basis reduced order model, and a machine learning surrogate, to parametrized linear-quadratic optimal control problems. The involved reduced order models are constructed adaptively and are called in such a way that the model hierarchy returns an approximate solution of given accuracy for every parameter value. At the same time, the fastest model of the hierarchy is evaluated whenever possible and slower models are only queried if the faster ones are not sufficiently accurate. The performance of the model hierarchy is studied for a parametrized heat equation example with boundary value control.

We consider the numerical behavior of the fixed-stress splitting method for coupled poromechanics as undrained regimes are approached. We explain that pressure stability is related to the splitting error of the scheme, not the fact that the discrete saddle point matrix never appears in the fixed-stress approach. This observation reconciles previous results regarding the pressure stability of the splitting method. Using examples of compositional poromechanics with application to geological CO$_2$ sequestration, we see that solutions obtained using the fixed-stress scheme with a low order finite element-finite volume discretization which is not inherently inf-sup stable can exhibit the same pressure oscillations obtained with the corresponding fully implicit scheme. Moreover, pressure jump stabilization can effectively remove these spurious oscillations in the fixed-stress setting, while also improving the efficiency of the scheme in terms of the number of iterations required at every time step to reach convergence.

Diffusion models have demonstrated remarkable performance in generation tasks. Nevertheless, explaining the diffusion process remains challenging due to it being a sequence of denoising noisy images that are difficult for experts to interpret. To address this issue, we propose the three research questions to interpret the diffusion process from the perspective of the visual concepts generated by the model and the region where the model attends in each time step. We devise tools for visualizing the diffusion process and answering the aforementioned research questions to render the diffusion process human-understandable. We show how the output is progressively generated in the diffusion process by explaining the level of denoising and highlighting relationships to foundational visual concepts at each time step through the results of experiments with various visual analyses using the tools. Throughout the training of the diffusion model, the model learns diverse visual concepts corresponding to each time-step, enabling the model to predict varying levels of visual concepts at different stages. We substantiate our tools using Area Under Cover (AUC) score, correlation quantification, and cross-attention mapping. Our findings provide insights into the diffusion process and pave the way for further research into explainable diffusion mechanisms.

We consider the problem of sketching a set valuation function, which is defined as the expectation of a valuation function of independent random item values. We show that for monotone subadditive or submodular valuation functions satisfying a weak homogeneity condition, or certain other conditions, there exist discretized distributions of item values with $O(k\log(k))$ support sizes that yield a sketch valuation function which is a constant-factor approximation, for any value query for a set of items of cardinality less than or equal to $k$. The discretized distributions can be efficiently computed by an algorithm for each item's value distribution separately. Our results hold under conditions that accommodate a wide range of valuation functions arising in applications, such as the value of a team corresponding to the best performance of a team member, constant elasticity of substitution production functions exhibiting diminishing returns used in economics and consumer theory, and others. Sketch valuation functions are particularly valuable for finding approximate solutions to optimization problems such as best set selection and welfare maximization. They enable computationally efficient evaluation of approximate value oracle queries and provide an approximation guarantee for the underlying optimization problem.