The study of diffeomorphism groups and their applications to problems in analysis and geometry has a long history. In geometric hydrodynamics, pioneered by V.~Arnold in the 1960s, one considers an ideal fluid flow as the geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms of the fluid domain with respect to the metric defined by the kinetic energy. Similar considerations on the space of densities lead to a geometric description of optimal mass transport and the Kantorovich-Wasserstein metric. Likewise, information geometry associated with the Fisher-Rao metric and the Hellinger distance has an equally beautiful infinite-dimensional geometric description and can be regarded as a higher-order Sobolev analogue of optimal transportation. In this work we review various metrics on diffeomorphism groups relevant to this approach and introduce appropriate topology, smooth structures and dynamics on the corresponding infinite-dimensional manifolds. Our main goal is to demonstrate how, alongside topological hydrodynamics, Hamiltonian dynamics and optimal mass transport, information geometry with its elaborate toolbox has become yet another exciting field for applications of geometric analysis on diffeomorphism groups.
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Gradient Descent (GD) and Conjugate Gradient (CG) methods are among the most effective iterative algorithms for solving unconstrained optimization problems, particularly in machine learning and statistical modeling, where they are employed to minimize cost functions. In these algorithms, tunable parameters, such as step sizes or conjugate parameters, play a crucial role in determining key performance metrics, like runtime and solution quality. In this work, we introduce a framework that models algorithm selection as a statistical learning problem, and thus learning complexity can be estimated by the pseudo-dimension of the algorithm group. We first propose a new cost measure for unconstrained optimization algorithms, inspired by the concept of primal-dual integral in mixed-integer linear programming. Based on the new cost measure, we derive an improved upper bound for the pseudo-dimension of gradient descent algorithm group by discretizing the set of step size configurations. Moreover, we generalize our findings from gradient descent algorithm to the conjugate gradient algorithm group for the first time, and prove the existence a learning algorithm capable of probabilistically identifying the optimal algorithm with a sufficiently large sample size.
We prove, for stably computably enumerable formal systems, direct analogues of the first and second incompleteness theorems of G\"odel. A typical stably computably enumerable set is the set of Diophantine equations with no integer solutions, and in particular such sets are generally not computably enumerable. And so this gives the first extension of the second incompleteness theorem to non classically computable formal systems. Let's motivate this with a somewhat physical application. Let $\mathcal{H} $ be the suitable infinite time limit (stabilization in the sense of the paper) of the mathematical output of humanity, specializing to first order sentences in the language of arithmetic (for simplicity), and understood as a formal system. Suppose that all the relevant physical processes in the formation of $\mathcal{H} $ are Turing computable. Then as defined $\mathcal{H} $ may \emph{not} be computably enumerable, but it is stably computably enumerable. Thus, the classical G\"odel disjunction applied to $\mathcal{H} $ is meaningless, but applying our incompleteness theorems to $\mathcal{H} $ we then get a sharper version of G\"odel's disjunction: assume $\mathcal{H} \vdash PA$ then either $\mathcal{H} $ is not stably computably enumerable or $\mathcal{H} $ is not 1-consistent (in particular is not sound) or $\mathcal{H} $ cannot prove a certain true statement of arithmetic (and cannot disprove it if in addition $\mathcal{H} $ is 2-consistent).
Computing the crossing number of a graph is one of the most classical problems in computational geometry. Both it and numerous variations of the problem have been studied, and overcoming their frequent computational difficulty is an active area of research. Particularly recently, there has been increased effort to show and understand the parameterized tractability of various crossing number variants. While many results in this direction use a similar approach, a general framework remains elusive. We suggest such a framework that generalizes important previous results, and can even be used to show the tractability of deciding crossing number variants for which this was stated as an open problem in previous literature. Our framework targets variants that prescribe a partial predrawing and some kind of topological restrictions on crossings. Additionally, to provide evidence for the non-generalizability of previous approaches for the partially crossing number problem to allow for geometric restrictions, we show a new more constrained hardness result for partially predrawn rectilinear crossing number. In particular, we show W-hardness of deciding Straight-Line Planarity Extension parameterized by the number of missing edges.
The proximal Galerkin finite element method is a high-order, low-iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of point-wise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. This work is accompanied by open-source implementations of our methods to facilitate reproduction and broader adoption.
Detection of abrupt spatial changes in physical properties representing unique geometric features such as buried objects, cavities, and fractures is an important problem in geophysics and many engineering disciplines. In this context, simultaneous spatial field and geometry estimation methods that explicitly parameterize the background spatial field and the geometry of the embedded anomalies are of great interest. This paper introduces an advanced inversion procedure for simultaneous estimation using the domain independence property of the Karhunen-Lo\`eve (K-L) expansion. Previous methods pursuing this strategy face significant computational challenges. The associated integral eigenvalue problem (IEVP) needs to be solved repeatedly on evolving domains, and the shape derivatives in gradient-based algorithms require costly computations of the Moore-Penrose inverse. Leveraging the domain independence property of the K-L expansion, the proposed method avoids both of these bottlenecks, and the IEVP is solved only once on a fixed bounding domain. Comparative studies demonstrate that our approach yields two orders of magnitude improvement in K-L expansion gradient computation time. Inversion studies on one-dimensional and two-dimensional seepage flow problems highlight the benefits of incorporating geometry parameters along with spatial field parameters. The proposed method captures abrupt changes in hydraulic conductivity with a lower number of parameters and provides accurate estimates of boundary and spatial-field uncertainties, outperforming spatial-field-only estimation methods.
A new variant of the GMRES method is presented for solving linear systems with the same matrix and subsequently obtained multiple right-hand sides. The new method keeps such properties of the classical GMRES algorithm as follows. Both bases of the search space and its image are maintained orthonormal that increases the robustness of the method. Moreover there is no need to store both bases since they are effectively represented within a common basis. Along with it our method is theoretically equivalent to the GCR method extended for a case of multiple right-hand sides but is more numerically robust and requires less memory. The main result of the paper is a mechanism of adding an arbitrary direction vector to the search space that can be easily adopted for flexible GMRES or GMRES with deflated restarting.
This study presents a novel representation learning model tailored for dynamic networks, which describes the continuously evolving relationships among individuals within a population. The problem is encapsulated in the dimension reduction topic of functional data analysis. With dynamic networks represented as matrix-valued functions, our objective is to map this functional data into a set of vector-valued functions in a lower-dimensional learning space. This space, defined as a metric functional space, allows for the calculation of norms and inner products. By constructing this learning space, we address (i) attribute learning, (ii) community detection, and (iii) link prediction and recovery of individual nodes in the dynamic network. Our model also accommodates asymmetric low-dimensional representations, enabling the separate study of nodes' regulatory and receiving roles. Crucially, the learning method accounts for the time-dependency of networks, ensuring that representations are continuous over time. The functional learning space we define naturally spans the time frame of the dynamic networks, facilitating both the inference of network links at specific time points and the reconstruction of the entire network structure without direct observation. We validated our approach through simulation studies and real-world applications. In simulations, we compared our methods link prediction performance to existing approaches under various data corruption scenarios. For real-world applications, we examined a dynamic social network replicated across six ant populations, demonstrating that our low-dimensional learning space effectively captures interactions, roles of individual ants, and the social evolution of the network. Our findings align with existing knowledge of ant colony behavior.
For several types of information relations, the induced rough sets system RS does not form a lattice but only a partially ordered set. However, by studying its Dedekind-MacNeille completion DM(RS), one may reveal new important properties of rough set structures. Building upon D. Umadevi's work on describing joins and meets in DM(RS), we previously investigated pseudo-Kleene algebras defined on DM(RS) for reflexive relations. This paper delves deeper into the order-theoretic properties of DM(RS) in the context of reflexive relations. We describe the completely join-irreducible elements of DM(RS) and characterize when DM(RS) is a spatial completely distributive lattice. We show that even in the case of a non-transitive reflexive relation, DM(RS) can form a Nelson algebra, a property generally associated with quasiorders. We introduce a novel concept, the core of a relational neighborhood, and use it to provide a necessary and sufficient condition for DM(RS) to determine a Nelson algebra.
Boundary-preserving weak approximations of some semilinear stochastic partial differential equations
We propose and analyse a boundary-preserving numerical scheme for the weak approximations of some stochastic partial differential equations (SPDEs) with bounded state-space. We impose regularity assumptions on the drift and diffusion coefficients only locally on the state-space. In particular, the drift and diffusion coefficients may be non-globally Lipschitz continuous and superlinearly growing. The scheme consists of a finite difference discretisation in space and a Lie--Trotter splitting followed by exact simulation and exact integration in time. We prove weak convergence of optimal order 1/4 for globally Lipschitz continuous test functions of the scheme by proving strong convergence towards a strong solution driven by a different noise process. Boundary-preservation is ensured by the use of Lie--Trotter time splitting followed by exact simulation and exact integration. Numerical experiments confirm the theoretical results and demonstrate the effectiveness of the proposed Lie--Trotter-Exact (LTE) scheme compared to existing methods for SPDEs.
Parameter inference is essential when interpreting observational data using mathematical models. Standard inference methods for differential equation models typically rely on obtaining repeated numerical solutions of the differential equation(s). Recent results have explored how numerical truncation error can have major, detrimental, and sometimes hidden impacts on likelihood-based inference by introducing false local maxima into the log-likelihood function. We present a straightforward approach for inference that eliminates the need for solving the underlying differential equations, thereby completely avoiding the impact of truncation error. Open-access Jupyter notebooks, available on GitHub, allow others to implement this method for a broad class of widely-used models to interpret biological data.
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