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We describe a `discretize-then-relax' strategy to globally minimize integral functionals over functions $u$ in a Sobolev space satisfying prescribed Dirichlet boundary conditions. The strategy applies whenever the integral functional depends polynomially on $u$ and its derivatives, even if it is nonconvex. The `discretize' step uses a bounded finite-element scheme to approximate the integral minimization problem with a convergent hierarchy of polynomial optimization problems over a compact feasible set, indexed by the decreasing size $h$ of the finite-element mesh. The `relax' step employs sparse moment-SOS relaxations to approximate each polynomial optimization problem with a hierarchy of convex semidefinite programs, indexed by an increasing relaxation order $\omega$. We prove that, as $\omega\to\infty$ and $h\to 0$, solutions of such semidefinite programs provide approximate minimizers that converge in $L^p$ to the global minimizer of the original integral functional if this is unique. We also report computational experiments that show our numerical strategy works well even when technical conditions required by our theoretical analysis are not satisfied.

We propose a matrix-free parallel two-level-deflation preconditioner combined with the Complex Shifted Laplacian preconditioner(CSLP) for the two-dimensional Helmholtz problems. The Helmholtz equation is widely studied in seismic exploration, antennas, and medical imaging. It is one of the hardest problems to solve both in terms of accuracy and convergence, due to scalability issues of the numerical solvers. Motivated by the observation that for large wavenumbers, the eigenvalues of the CSLP-preconditioned system shift towards zero, deflation with multigrid vectors, and further high-order vectors were incorporated to obtain wave-number-independent convergence. For large-scale applications, high-performance parallel scalable methods are also indispensable. In our method, we consider the preconditioned Krylov subspace methods for solving the linear system obtained from finite-difference discretization. The CSLP preconditioner is approximated by one parallel geometric multigrid V-cycle. For the two-level deflation, the matrix-free Galerkin coarsening as well as high-order re-discretization approaches on the coarse grid are studied. The results of matrix-vector multiplications in Krylov subspace methods and the interpolation/restriction operators are implemented based on the finite-difference grids without constructing any coefficient matrix. These adjustments lead to direct improvements in terms of memory consumption. Numerical experiments of model problems show that wavenumber independence has been obtained for medium wavenumbers. The matrix-free parallel framework shows satisfactory weak and strong parallel scalability.

In classical logic, "P implies Q" is equivalent to "not-P or Q". It is well known that the equivalence is problematic. Actually, from "P implies Q", "not-P or Q" can be inferred ("Implication-to-disjunction" is valid), while from "not-P or Q", "P implies Q" cannot be inferred in general ("Disjunction-to-implication" is not valid), so the equivalence between them is invalid. This work aims to remove exactly the incorrect Disjunction-to-implication from classical logic (CL). The paper proposes a logical system (IRL), which has the properties (1) adding Disjunction-to-implication to IRL is simply CL, and (2) Disjunction-to-implication is independent of IRL, i.e. either Disjunction-to-implication or its negation cannot be derived in IRL. In other words, IRL is just the sub-system of CL with Disjunction-to-implication being exactly removed.

Blumer et al. (1987, 1989) showed that any concept class that is learnable by Occam algorithms is PAC learnable. Board and Pitt (1990) showed a partial converse of this theorem: for concept classes that are closed under exception lists, any class that is PAC learnable is learnable by an Occam algorithm. However, their Occam algorithm outputs a hypothesis whose complexity is $\delta$-dependent, which is an important limitation. In this paper, we show that their partial converse applies to Occam algorithms with $\delta$-independent complexities as well. Thus, we provide a posteriori justification of various theoretical results and algorithm design methods which use the partial converse as a basis for their work.

Operator splitting is a popular divide-and-conquer strategy for solving differential equations. Typically, the right-hand side of the differential equation is split into a number of parts that are then integrated separately. Many methods are known that split the right-hand side into two parts. This approach is limiting, however, and there are situations when 3-splitting is more natural and ultimately more advantageous. The second-order Strang operator-splitting method readily generalizes to a right-hand side splitting into any number of operators. It is arguably the most popular method for 3-splitting because of its efficiency, ease of implementation, and intuitive nature. Other 3-splitting methods exist, but they are less well-known, and \rev{analysis and} evaluation of their performance in practice are scarce. We demonstrate the effectiveness of some alternative 3-split, second-order methods to Strang splitting on two problems: the reaction-diffusion Brusselator, which can be split into three parts that each have closed-form solutions, and the kinetic Vlasov--Poisson equations that is used in semi-Lagrangian plasma simulations. We find alternative second-order 3-operator-splitting methods that realize efficiency gains of 10\%--20\% over traditional Strang splitting. Our analysis for the practical assessment of efficiency of operator-splitting methods includes the computational cost of the integrators and can be used in method design.

This study performs an ablation analysis of Vector Quantized Generative Adversarial Networks (VQGANs), concentrating on image-to-image synthesis utilizing a single NVIDIA A100 GPU. The current work explores the nuanced effects of varying critical parameters including the number of epochs, image count, and attributes of codebook vectors and latent dimensions, specifically within the constraint of limited resources. Notably, our focus is pinpointed on the vector quantization loss, keeping other hyperparameters and loss components (GAN loss) fixed. This was done to delve into a deeper understanding of the discrete latent space, and to explore how varying its size affects the reconstruction. Though, our results do not surpass the existing benchmarks, however, our findings shed significant light on VQGAN's behaviour for a smaller dataset, particularly concerning artifacts, codebook size optimization, and comparative analysis with Principal Component Analysis (PCA). The study also uncovers the promising direction by introducing 2D positional encodings, revealing a marked reduction in artifacts and insights into balancing clarity and overfitting.

The discovery of equations with knowledge of the process origin is a tempting prospect. However, most equation discovery tools rely on gradient methods, which offer limited control over parameters. An alternative approach is the evolutionary equation discovery, which allows modification of almost every optimization stage. In this paper, we examine the modifications that can be introduced into the evolutionary operators of the equation discovery algorithm, taking inspiration from directed evolution techniques employed in fields such as chemistry and biology. The resulting approach, dubbed directed equation discovery, demonstrates a greater ability to converge towards accurate solutions than the conventional method. To support our findings, we present experiments based on Burgers', wave, and Korteweg--de Vries equations.

This article re-examines Lawvere's abstract, category-theoretic proof of the fixed-point theorem whose contrapositive is a `universal' diagonal argument. The main result is that the necessary axioms for both the fixed-point theorem and the diagonal argument can be stripped back further, to a semantic analogue of a weak substructural logic lacking weakening or exchange.

We revisit the relation between the gradient-flow equations and Hamilton's equations in information geometry. By regarding the gradient-flow equations as Huygens' equations in geometric optics, we have related the gradient flows to the geodesic flows induced by the geodesic Hamiltonian in an appropriate Riemannian geometry. The original evolution parameter $\textit{t}$ in the gradient-flow equations is related to the arc-length parameter in the associated Riemannian manifold by Jacobi-Maupertuis transformation. As a by-product, it is found the relation between the gradient-flow equation and replicator equations.

The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.