Dimensional Function Space Research Articles

Deep operator networks for bioheat transfer problems with parameterized laser source functions

To numerically characterize optothermal interactions at the laser-tissue interface during tissue imaging or surgery, the bioheat transfer partial differential equation (PDE) needs to be solved with a laser excitation source function. Considering various possible experimental scenarios, the PDEs become parametric and need to be solved for several optical and laser parameters. Classical numerical solvers that are used for performing such simulations are computationally expensive and time consuming. In this work, we use the recent deep operator network (DeepONet) framework for learning the mapping between infinite dimensional function spaces of the parameterized laser source functions and the solutions of the parameterized bioheat transfer PDE. We propose a new neural network architecture, Res-DeepONet, for training a purely physics-informed DeepONet with laser heating functions comprising of Gaussian beam profiles. We also showcase the versatility and robustness of our network architecture within purely data-driven and hybrid training frameworks for applications where labeled ground truth data may be accessible. The proposed neural network architecture exhibits excellent generalization performance on unseen Gaussian source functions with 10× higher accuracy when compared to traditional feedforward network architectures. Furthermore, zero-shot performance studies highlight the exceptional extrapolation capabilities of our network with 50× higher accuracy compared to traditional DeepONet architectures at inference times that are two orders of magnitude faster than classical solvers.

Infinitesimal gradient boosting

We define infinitesimal gradient boosting as a limit of the popular tree-based gradient boosting algorithm from machine learning. The limit is considered in the vanishing-learning-rate asymptotic, that is when the learning rate tends to zero and the number of gradient trees is rescaled accordingly. For this purpose, we introduce a new class of randomized regression trees bridging totally randomized trees and Extra Trees and using a softmax distribution for binary splitting. Our main result is the convergence of the associated stochastic algorithm and the characterization of the limiting procedure as the unique solution of a nonlinear ordinary differential equation in a infinite dimensional function space. Infinitesimal gradient boosting defines a smooth path in the space of continuous functions along which the training error decreases, the residuals remain centered and the total variation is well controlled.

Staircasing effect for minimizers of the one-dimensional discrete Perona–Malik functional

We consider the one-dimensional Perona–Malik functional, that is the energy associated to the celebrated forward-backward equation introduced by P. Perona and J. Malik in the context of image processing, with the addition of a forcing term. We discretize the functional by restricting its domain to a finite dimensional space of piecewise constant functions, and by replacing the derivative with a difference quotient. We investigate the asymptotic behavior of minima and minimizers as the discretization scale vanishes. In particular, if the forcing term has bounded variation, we show that any sequence of minimizers converges in the sense of varifolds to the graph of the forcing term, but with tangent component which is a combination of the horizontal and vertical directions. If the forcing term is more regular, we also prove that minimizers actually develop a microstructure that looks like a piecewise constant function at a suitable scale, which is intermediate between the macroscopic scale and the scale of the discretization.

Generalized bent functions with large dimension

By a fundamental result by Mesnager et al. in 2018, a generalized bent function (originally defined as a class of functions from an $ n $-dimensional vector space $ \mathbb{V}_n^{(p)} $ into a cyclic group), is a bent function $ g: \mathbb{V}_n^{(p)}\rightarrow \mathbb{F}_p $ with a partition $ \mathcal{P} $ of $ \mathbb{V}_n^{(p)} $, such that for every function $ C $ which is constant on the sets of $ \mathcal{P} $, the function $ g + C $ is bent. The set of these bent functions forms then an affine space of dimension $ |\mathcal{P}| \le p^{n/2} $. This characterization of generalized bent functions is much more comprehensive than any earlier description.In this article, we analyse some classes of bent functions under this perspective. As shown earlier, Maiorana-McFarland bent functions permit the largest possible partitions, giving rise to $ p^{n/2} $-dimensional affine bent function spaces. The reason behind is their characterization as the bent functions, which are affine restricted to the $ n/2 $-dimensional affine subspaces of a trivial cover of $ \mathbb{V}_n^{(p)} $. We will show that maximal possible partitions can also be obtained for other classes of (regular) bent functions. Most notably, these classes are described as bent functions which are affine restricted to non-trivial covers of $ \mathbb{V}_n^{(p)} $. We investigate (largest) partitions for other types of bent functions, including weakly regular and non-weakly respectively non-dual bent functions. We round off the article giving partitions for bent functions obtained from bent partitions, which includes the partial spread class, and partitions for Carlet's class $ \mathcal{C} $ and $ \mathcal{D} $.

The vanishing learning rate asymptotic for linearL2-boosting

We investigate the asymptotic behaviour of gradient boosting algorithms when the learning rate converges to zero and the number of iterations is rescaled accordingly. We mostly considerL2-boosting for regression with linear base learner as studied in P. Bühlmann and B. Yu,J. Am. Statist. Assoc.98(2003) 324–339 and analyze also a stochastic version of the model where subsampling is used at each step (J.H. Friedman,Computat. Statist. Data Anal.38(2002) 367–378). We prove a deterministic limit in the vanishing learning rate asymptotic and characterize the limit as the unique solution of a linear differential equation in an infinite dimensional function space. Besides, the training and test error of the limiting procedure are thoroughly analyzed. We finally illustrate and discuss our result on a simple numerical experiment where the linearL2-boosting operator is interpreted as a smoothed projection and time is related to its number of degrees of freedom.

Global Stabilization of Delayed Feedback Financial System Involved in Advertisement under Impulsive Disturbance

Diffusion is an inevitable important factor in advertising dynamic systems. However, previous literature did not involve this important diffusion factor, and only involved the local stability of the advertising model. This paper develops a global stability criterion for the impulsive advertising dynamic model with a feedback term under the influence of diffusion. Since global stability requires the unique existence of equilibrium points, variational methods are employed to solve it in the infinite dimensional function space, and then a global stability criterion of the system is derived by way of the impulse inequality lemma and orthogonal decomposition of a class of Sobolev spaces. Numerical simulations verify the effectiveness of the proposed method.

Functional diversity patterns reveal different elevations shaping Himalayan amphibian assemblages, highlighting the importance of morphologically extreme individuals

Biological diversity is a concept that contains multiple facets, of which functional diversity is considered to be more integrative to describe the relationship between biodiversity and ecosystem functioning. In recent decades, functional diversity patterns have been assessed along elevational gradients in plants and birds. However, few empirical studies have been conducted to reveal amphibian functional diversity patterns between elevational zonations on mountains. In the present study, we investigated amphibian ecomorphologically functional diversity patterns in different elevational zonations of Nepal Himalaya. Our results indicated that functional space occupied by amphibian assemblages differed in size between elevational zonations, with assemblages in zonation-2 (500–1300 m) occupying the highest proportion. In terms of the position in functional space, assemblages in low elevational zonations (zonation 1 & 2) differed significantly from those in high elevational zonations (zonation 3–5). Distribution of individuals in the five assemblages along the six axes of the functional space was significantly different between most pairs of comparisons, while the difference varied between assemblages along axes. Interestingly, despite morphologically extreme amphibians accounted for small proportions of the total number of amphibians, they filled a large proportion of the six dimensional functional space. More importantly, most of the extreme amphibians were also the most functional uniqueness individuals, and they were widely distributed in all the zonations. Therefore, functional vulnerability of amphibian assemblages exists throughout all the elevational zonations. Accordingly, extreme amphibians with most functional uniqueness should be in particular well protected to maintain amphibian functional diversity. Further research can explore the mechanism underlying the unique amphibian functional diversity patterns in Nepal Himalaya.

An Online Projection Estimator for Nonparametric Regression in Reproducing Kernel Hilbert Spaces.

The goal of nonparametric regression is to recover an underlying regression function from noisy observations, under the assumption that the regression function belongs to a prespecified infinite-dimensional function space. In the online setting, in which the observations come in a stream, it is generally computationally infeasible to refit the whole model repeatedly. As yet, there are no methods that are both computationally efficient and statistically rate optimal. In this paper, we propose an estimator for online nonparametric regression. Notably, our estimator is an empirical risk minimizer in a deterministic linear space, which is quite different from existing methods that use random features and a functional stochastic gradient. Our theoretical analysis shows that this estimator obtains a rate-optimal generalization error when the regression function is known to live in a reproducing kernel Hilbert space. We also show, theoretically and empirically, that the computational cost of our estimator is much lower than that of other rate-optimal estimators proposed for this online setting.

Pseudo-Fubini entire functions on the plane in the sense of Riemann

We prove the existence of a maximal dimensional vector space of real functions on the real plane all of whose nonzero members are bounded, entire, non-Lebesgue-integrable, and satisfy the equality of the two iterated integrals given in the conclusion of the Fubini theorem, with the additional property that these iterated integrals exist in the Riemann sense, but not in the Lebesgue one. Moreover, this vector space is dense in the space of smooth functions on the plane under its natural topology.

Pseudo-Fubini Real-Entire Functions on the Plane

In this note, it is proved the existence of a mathfrak {c}-dimensional vector space of real-entire functions all of whose nonzero members are non-integrable in the sense of Lebesgue but yet their two iterated integrals exist as real numbers and coincide. Moreover, it is shown that this vector space can be chosen to be dense in the space of all real C^infty -functions on the plane endowed with the topology of uniform convergence on compacta for all derivatives of all orders. If the condition of being entire is dropped, then a closed infinite dimensional subspace satisfying the same properties can be obtained.

A mass-, kinetic energy- and helicity-conserving mimetic dual-field discretization for three-dimensional incompressible Navier-Stokes equations, part I: Periodic domains

We introduce a mimetic dual-field discretization which conserves mass, kinetic energy and helicity for three-dimensional incompressible Navier-Stokes equations. The discretization makes use of a conservative dual-field mixed weak formulation where two evolution equations of velocity are employed and dual representations of the solution are sought for each variable. A temporal discretization, which staggers the evolution equations and handles the nonlinearity such that the resulting discrete algebraic systems are linear and decoupled, is constructed. The spatial discretization is mimetic in the sense that the finite dimensional function spaces form a discrete de Rham complex. Conservation of mass, kinetic energy and helicity in the absence of dissipative terms is proven at the discrete level. Proper dissipation rates of kinetic energy and helicity in the viscous case are also proven. Numerical tests supporting the method are provided.

Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method

Inferring the latent structure of complex nonlinear dynamical systems in a data driven setting is a challenging mathematical problem with an ever increasing spectrum of applications in sciences and engineering. Koopman operator-based linearization provides a powerful framework that is suitable for identification of nonlinear systems in various scenarios. A recently proposed method by Mauroy and Goncalves is based on lifting the data snapshots into a suitable finite dimensional function space and identification of the infinitesimal generator of the Koopman semigroup. This elegant and mathematically appealing approach has good analytical (convergence) properties, but numerical experiments show that software implementation of the method has certain limitations. More precisely, with the increased dimension that guarantees theoretically better approximation and ultimate convergence, the numerical implementation may become unstable and it may even break down. The main sources of numerical difficulties are the computations of the matrix representation of the compressed Koopman operator and its logarithm. This paper addresses the subtle numerical details and proposes a new implementation algorithm that alleviates these problems.

Almost Phaseless Sampling for Spline Spaces with Arbitrary Knots*

This article deals with almost phaseless sampling for spaces generated by B-splines with arbitrary knots. We provide a necessary and sufficient condition, which is a characterization of an almost phaseless sampling set, to admit phase recovery for almost all B-spline functions on an interval. Our results show that only N + 1 intensity measurements are sufficient to recover phase information for almost all functions in N dimensional B-spline function spaces with arbitrary knots.

A conjugate gradient method for distributed optimal control problems with nonhomogeneous Helmholtz equation

A conjugate gradient algorithm with strong Wolfe-Powell line search for distributed optimal control problem is proposed. The optimal system has been discussed in [1]. The proposed algorithm is employed to solve the problem in infinite dimensional function space. With low complexity, it is suitable for large-scale problem. The sufficient descent condition of conjugate gradient and the existence of iterative step are proved. The algorithm also has global convergence property and linear convergence rate. At last, numerical experiments are presented to illustrate the efficiency and the convergence rate of the proposed algorithm.

Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system

<p style='text-indent:20px;'>At the very beginning of the theory of finite dynamical systems, it was discovered that some relatively simple systems, even of ordinary differential equations, can generate very complicated (chaotic) behaviors. Furthermore these systems are extremely sensitive to perturbations, in the sense that trajectories with close but different initial data may diverge exponentially. Very often, the trajectories of such chaotic systems are localized, up to some transient process, in some subset of the phase space, the so-called strange attractors. Such subset have a very complicated geometric structure. They accumulate the nontrivial dynamics of the system.</p><p style='text-indent:20px;'>For a distributed system, whose time evolution is usually governed by partial differential equations (PDEs), the phase space X is (a subset of) an infinite dimensional function space. We will thus speak of infinite dimensional dynamical systems. Since the global existence and uniqueness of solutions has been proven for a large class of PDEs arising from different domains as Mechanics and Physics, it is therefore natural to investigate whether the features, in particular the notion of attractor, obtained for dynamical systems generated by systems of ODEs generalizes to systems of PDEs.</p><p style='text-indent:20px;'>In this paper we give a positive aftermath by proving the existence of pullback <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{D} $\end{document}</tex-math></inline-formula>-attractor. The key point is to find a bounded family of pullback <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{D} $\end{document}</tex-math></inline-formula>-absorbing sets then we apply the decomposition techniques and a method used in previous works to verify the pullback <inline-formula><tex-math id="M4">\begin{document}$ w $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M5">\begin{document}$ \mathcal{D} $\end{document}</tex-math></inline-formula>-limit compactness. It is based on the concept of the Kuratowski measure of noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions.</p>

Study on Change of Topography in Water Area with Field Measurement

Disastrous flood events in recent years include 2018 Japan floods (July 2018), Typhoon 19 (Hagibis, October 2019) and the following heavy rain event, and July 2020 heavy rain disaster. Such disastrous flood events are expected to occur more frequently as the climate change progresses, and it is indispensable to update information regarding water areas such as rivers, reservoirs, and coastal waters. This study demonstrates practical techniques to analyze underwater topography. Longitude and latitude components of ellipsoidal data recorded by a GPS receiver are projected to a rectangular coordinate system, and results are combined with vertical components including data recorded by an echo sounder unit so that tracks on an underwater floor are obtained. A mapping on a finite dimensional space of continuous functions over a triangular mesh is formulated, and its fixed point gives rise to an underwater floor that fits the tracks. Our techniques are illustrated with data obtained in measurement conducted in a reservoir called Kojima Lake located in Okayama Prefecture, Japan.

Entropy Numbers of Finite Dimensional Mixed-Norm Balls and Function Space Embeddings with Small Mixed Smoothness

We study the embedding mathrm {id}: ell _p^b(ell _q^d) rightarrow ell _r^b(ell _u^d) and prove matching bounds for the entropy numbers e_k(mathrm {id}) provided that 0<p<rle infty and 0<qle ule infty . Based on this finding, we establish optimal dimension-free asymptotic rates for the entropy numbers of embeddings of Besov and Triebel–Lizorkin spaces of small dominating mixed smoothness, which gives a complete answer to an open problem mentioned in the recent monograph by Dũng, Temlyakov, and Ullrich. Both results rely on a novel covering construction recently found by Edmunds and Netrusov.

On Well-Posedness for General Hierarchy Equations of Gross–Pitaevskii and Hartree Type

Gross–Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs emerging naturally from the mean field theory of Bose gases. Their solutions are known to be related to initial value problems, in particular the Gross–Pitaevskii and Hartree equations. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques. In this article, we introduce a new approach for the study of such hierarchy equations by firstly establishing a duality between them and certain Liouville equations, and secondly, solving the uniqueness and existence questions for the latter. As an outcome, we formulate a hierarchy equation starting from any initial value problem which is U(1)-invariant and prove a general principle which can be stated formally as follows: In particular, several new well-posedness results, as well as a counterexample to uniqueness for the Gross–Pitaevskii hierarchy equation, are proved. The novelty in our work lies in the aforementioned duality and the use of Liouville equations with powerful transport techniques extended to infinite dimensional functional spaces.

The core variety and representing measures in the truncated moment problem

The truncated moment problem asks for conditions so that a linear functional L on the vector space of real n-variable polynomials of degree at most d can be written as integration with respect to a positive Borel measure μ on Rn. More generally, let L act on a finite dimensional space of Borel-measurable functions defined on a T1 topological space S. Using an iterative geometric construction, we associate to L a subset of S called the \textit{core variety} CV(L). Our main result is that L has a representing measure μ if and only if CV(L) is nonempty. In this case, L has a finitely atomic representing measure, and the union of the supports of such measures is precisely CV(L).

An adaptive nested source term iteration for radiative transfer equations

We propose a new approach to the numerical solution of radiative transfer equations with certified a posteriori error bounds. A key role is played by stable Petrov--Galerkin type variational formulations of parametric transport equations and corresponding radiative transfer equations. This allows us to formulate an iteration in a suitable, infinite dimensional function space that is guaranteed to converge with a fixed error reduction per step. The numerical scheme is then based on approximately realizing this iteration within dynamically updated accuracy tolerances that still ensure convergence to the exact solution. To advance this iteration two operations need to be performed within suitably tightened accuracy tolerances. First, the global scattering operator needs to be approximately applied to the current iterate within a tolerance comparable to the current accuracy level. Second, parameter dependent linear transport equations need to be solved, again at the required accuracy of the iteration. To ensure that the stage dependent error tolerances are met, one has to employ rigorous a posteriori error bounds which, in our case, rest on a Discontinuous Petrov--Galerkin (DPG) scheme. These a posteriori bounds are not only crucial for guaranteeing the convergence of the perturbed iteration but are also used to generate adapted parameter dependent spatial meshes. This turns out to significantly reduce overall computational complexity. Since the global operator is only applied, we avoid the need to solve linear systems with densely populated matrices. Moreover, the approximate application of the global scatterer accelerated through low-rank approximation and matrix compression techniques. The theoretical findings are illustrated and complemented by numerical experiments with non-trivial scattering kernels.