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This paper presents a novel stochastic optimisation methodology to perform empirical Bayesian inference in semi-blind image deconvolution problems. Given a blurred image and a parametric class of possible operators, the proposed optimisation approach automatically calibrates the parameters of the blur model by maximum marginal likelihood estimation, followed by (non-blind) image deconvolution by maximum-a-posteriori estimation conditionally to the estimated model parameters. In addition to the blur model, the proposed approach also automatically calibrates the noise variance as well as any regularisation parameters. The marginal likelihood of the blur, noise variance, and regularisation parameters is generally computationally intractable, as it requires calculating several integrals over the entire solution space. Our approach addresses this difficulty by using a stochastic approximation proximal gradient optimisation scheme, which iteratively solves such integrals by using a Moreau-Yosida regularised unadjusted Langevin Markov chain Monte Carlo algorithm. This optimisation strategy can be easily and efficiently applied to any model that is log-concave, and by using the same gradient and proximal operators that are required to compute the maximum-a-posteriori solution by convex optimisation. We provide convergence guarantees for the proposed optimisation scheme under realistic and easily verifiable conditions and subsequently demonstrate the effectiveness of the approach with a series of deconvolution experiments and comparisons with alternative strategies from the state of the art.

A topical challenge for algorithms in general and for automatic image categorization and generation in particular is presented in the form of a drawing for AI to understand. In a second vein, AI is challenged to produce something similar from verbal description. The aim of the paper is to highlight strengths and deficiencies of current Artificial Intelligence approaches while coarsely sketching a way forward. A general lack of encompassing symbol-embedding and (not only) -grounding in some bodily basis is made responsible for current deficiencies. A concomitant dearth of hierarchical organization of concepts follows suite. As a remedy for these shortcomings, it is proposed to take a wide step back and to newly incorporate aspects of cybernetics and analog control processes. It is claimed that a promising overarching perspective is provided by the Ouroboros Model with a valid and versatile algorithmic backbone for general cognition at all accessible levels of abstraction and capabilities. Reality, rules, truth, and Free Will are all useful abstractions according to the Ouroboros Model. Logic deduction as well as intuitive guesses are claimed as produced on the basis of one compartmentalized memory for schemata and a pattern-matching, i.e., monitoring process termed consumption analysis. The latter directs attention on short (attention proper) and also on long times scales (emotional biases). In this cybernetic approach, discrepancies between expectations and actual activations (e.g., sensory precepts) drive the general process of cognition and at the same time steer the storage of new and adapted memory entries. Dedicated structures in the human brain work in concert according to this scheme.

With the increasing availability of large scale datasets, computational power and tools like automatic differentiation and expressive neural network architectures, sequential data are now often treated in a data-driven way, with a dynamical model trained from the observation data. While neural networks are often seen as uninterpretable black-box architectures, they can still benefit from physical priors on the data and from mathematical knowledge. In this paper, we use a neural network architecture which leverages the long-known Koopman operator theory to embed dynamical systems in latent spaces where their dynamics can be described linearly, enabling a number of appealing features. We introduce methods that enable to train such a model for long-term continuous reconstruction, even in difficult contexts where the data comes in irregularly-sampled time series. The potential for self-supervised learning is also demonstrated, as we show the promising use of trained dynamical models as priors for variational data assimilation techniques, with applications to e.g. time series interpolation and forecasting.

This work implements and numerically tests the direct reconstruction algorithm introduced in [Garde & Hyv\"onen, SIAM J. Math. Anal., 2024] for two-dimensional linearized electrical impedance tomography. Although the algorithm was originally designed for a linearized setting, we numerically demonstrate its functionality when the input data is the corresponding change in the current-to-voltage boundary operator. Both idealized continuum model and practical complete electrode model measurements are considered in the numerical studies, with the examined domain being either the unit disk or a convex polygon. Special attention is paid to regularizing the algorithm and its connections to the singular value decomposition of a truncated linearized forward map, as well as to the explicit triangular structures originating from the properties of the employed Zernike polynomial basis for the conductivity.

Of all the possible projection methods for solving large-scale Lyapunov matrix equations, Galerkin approaches remain much more popular than minimal-residual ones. This is mainly due to the different nature of the projected problems stemming from these two families of methods. While a Galerkin approach leads to the solution of a low-dimensional matrix equation per iteration, a matrix least-squares problem needs to be solved per iteration in a minimal-residual setting. The significant computational cost of these least-squares problems has steered researchers towards Galerkin methods in spite of the appealing properties of minimal-residual schemes. In this paper we introduce a framework that allows for modifying the Galerkin approach by low-rank, additive corrections to the projected matrix equation problem with the two-fold goal of attaining monotonic convergence rates similar to those of minimal-residual schemes while maintaining essentially the same computational cost of the original Galerkin method. We analyze the well-posedness of our framework and determine possible scenarios where we expect the residual norm attained by two low-rank-modified variants to behave similarly to the one computed by a minimal-residual technique. A panel of diverse numerical examples shows the behavior and potential of our new approach.

In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg--Landau equations. To this aim, we employ high-order exponential methods of splitting and Lawson type for the time integration. These schemes enjoy favorable stability properties and, in particular, do not show restrictions on the time step size due to the underlying stiffness of the models. The needed actions of matrix exponentials are efficiently realized with pointwise operations in Fourier space (when the model is considered with periodic boundary conditions) or by using a tensor-oriented approach that suitably employs the so-called $\mu$-mode products (when the semidiscretization in space is performed with finite differences). The overall effectiveness of the approach is demonstrated by running simulations on a variety of two- and three-dimensional (systems of) complex Ginzburg--Landau equations with cubic and cubic-quintic nonlinearities, which are widely considered in literature to model relevant physical phenomena. In fact, in all instances high-order exponential-type schemes can outperform standard techniques to integrate in time the models under consideration, i.e., the well-known split-step method and the explicit fourth-order Runge--Kutta integrator.

Deep learning methods have access to be employed for solving physical systems governed by parametric partial differential equations (PDEs) due to massive scientific data. It has been refined to operator learning that focuses on learning non-linear mapping between infinite-dimensional function spaces, offering interface from observations to solutions. However, state-of-the-art neural operators are limited to constant and uniform discretization, thereby leading to deficiency in generalization on arbitrary discretization schemes for computational domain. In this work, we propose a novel operator learning algorithm, referred to as Dynamic Gaussian Graph Operator (DGGO) that expands neural operators to learning parametric PDEs in arbitrary discrete mechanics problems. The Dynamic Gaussian Graph (DGG) kernel learns to map the observation vectors defined in general Euclidean space to metric vectors defined in high-dimensional uniform metric space. The DGG integral kernel is parameterized by Gaussian kernel weighted Riemann sum approximating and using dynamic message passing graph to depict the interrelation within the integral term. Fourier Neural Operator is selected to localize the metric vectors on spatial and frequency domains. Metric vectors are regarded as located on latent uniform domain, wherein spatial and spectral transformation offer highly regular constraints on solution space. The efficiency and robustness of DGGO are validated by applying it to solve numerical arbitrary discrete mechanics problems in comparison with mainstream neural operators. Ablation experiments are implemented to demonstrate the effectiveness of spatial transformation in the DGG kernel. The proposed method is utilized to forecast stress field of hyper-elastic material with geometrically variable void as engineering application.

We introduce a novel quantum programming language featuring higher-order programs and quantum controlflow which ensures that all qubit transformations are unitary. Our language boasts a type system guaranteeingboth unitarity and polynomial-time normalization. Unitarity is achieved by using a special modality forsuperpositions while requiring orthogonality among superposed terms. Polynomial-time normalization isachieved using a linear-logic-based type discipline employing Barber and Plotkin duality along with a specificmodality to account for potential duplications. This type discipline also guarantees that derived values havepolynomial size. Our language seamlessly combines the two modalities: quantum circuit programs upholdunitarity, and all programs are evaluated in polynomial time, ensuring their feasibility.

Iterated conditional expectation (ICE) g-computation is an estimation approach for addressing time-varying confounding for both longitudinal and time-to-event data. Unlike other g-computation implementations, ICE avoids the need to specify models for each time-varying covariate. For variance estimation, previous work has suggested the bootstrap. However, bootstrapping can be computationally intense and sensitive to the number of resamples used. Here, we present ICE g-computation as a set of stacked estimating equations. Therefore, the variance for the ICE g-computation estimator can be consistently estimated using the empirical sandwich variance estimator. Performance of the variance estimator was evaluated empirically with a simulation study. The proposed approach is also demonstrated with an illustrative example on the effect of cigarette smoking on the prevalence of hypertension. In the simulation study, the empirical sandwich variance estimator appropriately estimated the variance. When comparing runtimes between the sandwich variance estimator and the bootstrap for the applied example, the sandwich estimator was substantially faster, even when bootstraps were run in parallel. The empirical sandwich variance estimator is a viable option for variance estimation with ICE g-computation.

Computing the regularized solution of Bayesian linear inverse problems as well as the corresponding regularization parameter is highly desirable in many applications. This paper proposes a novel iterative method, termed the Projected Newton method (PNT), that can simultaneously update the regularization parameter and solution step by step without requiring any high-cost matrix inversions or decompositions. By reformulating the Tikhonov regularization as a constrained minimization problem and writing its Lagrangian function, a Newton-type method coupled with a Krylov subspace method, called the generalized Golub-Kahan bidiagonalization, is employed for the unconstrained Lagrangian function. The resulting PNT algorithm only needs solving a small-scale linear system to get a descent direction of a merit function at each iteration, thus significantly reducing computational overhead. Rigorous convergence results are proved, showing that PNT always converges to the unique regularized solution and the corresponding Lagrangian multiplier. Experimental results on both small and large-scale Bayesian inverse problems demonstrate its excellent convergence property, robustness and efficiency. Given that the most demanding computational tasks in PNT are primarily matrix-vector products, it is particularly well-suited for large-scale problems.