Multidimensional Setting Research Articles

A generalization of Kantorovich operators for convex compact subsets

In this article, we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number, and a sequence of Borel probability measures. By considering special cases of these parameters for particular convex compact subsets, we obtain the classical Kantorovich operators defined in the 1-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, we discuss the preservation of Lipschitz-continuity and of convexity.

Maximum Entropy Estimation via Gauss-LP Quadratures

We present an approximation method to a class of parametric integration problems that naturally appear when solving the dual of the maximum entropy estimation problem. Our method builds up on a recent generalization of Gauss quadratures via an infinite-dimensional linear program, and utilizes a convex clustering algorithm to compute an approximate solution which requires reduced computational effort. It shows to be particularly appealing when looking at problems with unusual domains and in a multi-dimensional setting. As a proof of concept we apply our method to an example problem on the unit disc.

The role of multidimensional instabilities in direct initiation of gaseous detonations in free space

We numerically investigate the direct initiation of detonations driven by the propagation of a blast wave into a unconfined gaseous combustible mixture to study the role played by multidimensional instabilities in direct initiation of stable and unstable detonations. To this end, we first model the dynamics of unsteady propagation of detonation using the one-dimensional compressible Euler equations with a one-step chemical reaction model and cylindrical geometrical source terms. Subsequently, we use two-dimensional compressible Euler equations with just the chemical reaction source term to directly model cylindrical detonations. The one-dimensional results suggest that there are three regimes in the direct initiation for stable detonations, that the critical energy for mildly unstable detonations is not unique, and that highly unstable detonations are not self-sustainable. These phenomena agree well with one-dimensional theories and computations available in the literature. However, our two-dimensional results indicate that one-dimensional approaches are valid only for stable detonations. In mildly and highly unstable detonations, one-dimensional approaches break down because they cannot take the effects and interactions of multidimensional instabilities into account. In fact, instabilities generated in multidimensional settings yield the formation of strong transverse waves that, on one hand, increase the risk of failure of the detonation and, on the other hand, lead to the initiation of local over-driven detonations that enhance the overall self-sustainability of the global process. The competition between these two possible outcomes plays an important role in the direct initiation of detonations.

Adaptive wavelet multivariate regression with errors in variables

In the multidimensional setting, we consider the errors-in- variables model. We aim at estimating the unknown nonparametric multivariate regression function with errors in the covariates. We devise an adaptive estimators based on projection kernels on wavelets and a deconvolution operator. We propose an automatic and fully data driven procedure to select the wavelet level resolution. We obtain an oracle inequality and optimal rates of convergence over anisotropic Hölder classes. Our theoretical results are illustrated by some simulations.

Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case

A multi-dimensional junction is obtained by identifying the boundaries of a finite number of copies of an Euclidian half-space. The main contribution of this article is the construction of a multidimensional vertex test function G(x, y). First, such a function has to be sufficiently regular to be used as a test function in the viscosity solution theory for quasi-convex Hamilton-Jacobi equations posed on a multi-dimensional junction. Second, its gradients have to satisfy appropriate compatibility conditions in order to replace the usual quadratic pe-nalization function |x -- y| 2 in the proof of strong uniqueness (comparison principle) by the celebrated doubling variable technique. This result extends a construction the authors previously achieved in the network setting. In the multi-dimensional setting, the construction is less explicit and more delicate. Mathematical Subject Classification: 35F21, 49L25, 35B51.

Pricing multidimensional financial derivatives with stochastic volatilities using the dimensional-adaptive combination technique

In this paper, we present a new and general approach to price derivatives based on the Black–Scholes partial differential equation (BS-PDE) in a multidimensional setting. The first ingredient in our approach is the dimensional-adaptive sparse grid combination technique, which, in the case of underlying models with stochastic volatilities, allows for inhomogeneous discretization levels of the dimensional axes. Thus, by applying the dimensional-adaptive combination technique to such problems, one may achieve higher numerical efficiency. We combine this approach with a stretched grid discretization that is derived from the underlying’s stochastic differential equation (SDE) in a general manner. This stretching enables us to employ efficient geometrical multigrid solvers, even for the strong anisotropic convection and diffusion coefficients that frequently occur in application. Our combination of the dimensional-adaptive sparse grid combination technique with SDE-based grid stretching and an efficient multigrid solver represents a new approach designed to enable derivative pricing by directly solving PDEs in higher dimensions than were possible before. The numerical results outlined in the paper demonstrate the efficacy of this new approach and of our implementation method, which entails pricing various derivatives with up to twelve dimensions in a general and simple manner.

Optimal Stopping at Random Intervention Times

In decision problems, frictions as well as constraints play an increasingly important role. Especially, optimal timing problems can be affected by potentially “non-rational” behavior of the decision maker which is not incorporated in the standard theory. A relevant problem of this kind is the real option to abandon a project. Limited cognitive resources and external restrictions to option exercise may result in a suboptimal outcome. The term inattention can summarize such frictions and constraints. In this paper, we address this issue by proposing a Markovian model to value American-style contracts of agents who are temporarily inattentive. Exercise decisions maximizing the contract’s payoff are not admissible continuously but at random intervention times arriving with possibly state and time dependent intensities. An optimal stopping problem provides the contract value. It is converted to optimal control which, given sufficient regularity, induces a characterisation in terms of a partial integro differential equation. We consider three numerical approaches, forward improvement iteration, least squares Monte-Carlo and finite differences, each corresponding to one particular characterization of the contract value. Our adapted least squares Monte-Carlo method can treat complex and possibly multi-dimensional settings.

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calder\'on-Zygmund theory we prove that these operators are bounded on weighted $L^p$ spaces, $1 < p < \infty$, and from weighted $L^1$ to weighted weak $L^1$. We also obtain similar results for analogous set of operators in the closely related multi-dimensional Laguerre-symmetrized framework. The latter emerges from a symmetrization procedure proposed recently by the first two authors. As a by-product of the main developments we get some new results in the multi-dimensional Laguerre function setting of convolution type.

On the Kesten–Goldie constant

We consider the stochastic difference equation on where is an i.i.d. sequence of random variables and is an initial distribution. Under mild contractivity hypotheses the sequence converges in law to a random variable S, which is the unique solution of the random difference equation . We prove that under the Kesten–Goldie conditionswhere is the Kesten–Goldie constant is the Cramér coefficient of and . Thus, on one side we describe the behaviour of the th moments of the process , and on the other we obtain an alternative formula for . The results are further extended to a class of Lipschitz iterated systems and to a multidimensional setting.

Existence and stability of planar shocks of viscous scalar conservation laws with space-periodic flux

The goal of this paper is to prove the existence and stability of shocks for viscous scalar conservation laws with space periodic flux, in the multi-dimensional case. Such a result had been proved by the first author in one space dimension, but the extension to a multi-dimensional setting makes the existence proof non-trivial. We construct approximate solutions by restricting the size of the domain and then passing to the limit as the size of the domain goes to infinity. One of the key steps is a “normalization” procedure, which ensures that the limit objects obtained by the approximation scheme are indeed shocks. The proofs rely on elliptic PDE theory rather than ODE arguments as in the 1d case. Once the existence of shocks is proved, their stability follows from classical arguments based on the theory of dynamical systems.

The Mega Distributed Lag Model

This paper attempts to describe the graphical behavior of the distributed lag model in an infinite coordinate space. The “mega distributed lag model” (MDL) is a mathematical framework that can examine the simultaneous interrelationships between all involved variables. The multidimensional graphical setting simultaneously reveals all non-linear exposure — response dependencies and delayed effects between lagged and dependent variables — which two-dimensional figures overwhelmingly fail to capture. Under the Omnia Mobilis assumption, each distribution lag function is indexed with respect to time and space. The Mega distributed lag model observes multiple trends in full motion, the final output (determinant) of which is called “the JIM-coefficient”. Hence, this paper tries to analyze different approaches of lag distribution models that can help in the construction of our new model. The mega distributed lag model (MDL) is moving from the uses of the classic 2-dimensional and 3-dimensional graphical modeling to a multidimensional graphical modeling in Econometrics. Finally, this model is an extension of those explored earlier in the field of Econographicology.

Reducing complexes in multidimensional persistent homology theory

Forman's discrete Morse theory appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for one-dimensional filtrations. This paper is perhaps the first attempt in the direction of extending such algorithms to multidimensional filtrations. An initial framework related to Morse matchings for the multidimensional setting is proposed, and a matching algorithm given by King, Knudson, and Mramor is extended in this direction. The correctness of the algorithm is proved, and its complexity analyzed. The algorithm is used for establishing a reduction of a simplicial complex to a smaller but not necessarily optimal cellular complex. First experiments with filtrations of triangular meshes are presented.

A concept of absolute continuity and its characterization in terms of convergence in variation

We study the notion of φ‐absolute continuity, providing several equivalent definitions, and we prove a characterization of the space of φ‐absolutely continuous functions in terms of convergence in variation for a family of Mellin integral operators in the multidimensional setting.

A Review on Approximation Results for Integral Operators in the Space of Functions of Bounded Variation

We present a review on recent approximation results in the space of functions of bounded variation for some classes of integral operators in the multidimensional setting. In particular, we present estimates and convergence in variation results for both convolution and Mellin integral operators with respect to the Tonelli variation. Results with respect to a multidimensional concept ofφ-variation in the sense of Tonelli are also presented.

Multidimensional Potential Burgers Turbulence

We consider the multidimensional generalised stochastic Burgers equation in the space-periodic setting: $$\frac{\partial \mathbf{u}}{\partial t}+(\nabla f(\mathbf{u}) \cdot \nabla) \mathbf{u}-\nu \Delta \mathbf{u}= \nabla \eta, \quad t \geq 0, \, \mathbf{x} \in\mathbb{T}^d=(\mathbb{R}/ \mathbb{Z})^d,$$ under the assumption that u is a gradient. Here f is strongly convex and satisfies a growth condition, ν is small and positive, while η is a random forcing term, smooth in space and white in time. For solutions u of this equation, we study Sobolev norms of u averaged in time and in ensemble: each of these norms behaves as a given negative power of ν. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. These quantities behave as a power of the norm of the relevant parameter, which is respectively the separation l in the physical space and the wavenumber k in the Fourier space. Our bounds do not depend on the initial condition and hold uniformly in \({\nu}\). We generalise the results obtained for the one-dimensional case in [10], confirming the physical predictions in [4, 30]. Note that the form of the estimates does not depend on the dimension: the powers of \({\nu, |{\mathbf{k}}|, \ell}\) are the same in the one- and the multi-dimensional setting.

Spectral analysis and structure preserving preconditioners for fractional diffusion equations

Fractional partial order diffusion equations are a generalization of classical partial differential equations, used to model anomalous diffusion phenomena. When using the implicit Euler formula and the shifted Grunwald formula, it has been shown that the related discretizations lead to a linear system whose coefficient matrix has a Toeplitz-like structure. In this paper we focus our attention on the case of variable diffusion coefficients. Under appropriate conditions, we show that the sequence of the coefficient matrices belongs to the Generalized Locally Toeplitz class and we compute the symbol describing its asymptotic eigenvalue/singular value distribution, as the matrix size diverges. We employ the spectral information for analyzing known methods of preconditioned Krylov and multigrid type, with both positive and negative results and with a look forward to the multidimensional setting. We also propose two new tridiagonal structure preserving preconditioners to solve the resulting linear system, with Krylov methods such as CGNR and GMRES. A number of numerical examples show that our proposal is more effective than recently used circulant preconditioners.

Variational formulation and stability analysis of a three dimensional superelastic model for shape memory alloys

This paper presents a variational framework for the three-dimensional macroscopic modelling of superelastic shape memory alloys in an isothermal setting. Phase transformation is accounted through a unique second order tensorial internal variable, acting as the transformation strain. Postulating the total strain energy density as the sum of a free energy and a dissipated energy, the model depends on two material scalar functions of the norm of the transformation strain and a material scalar constant. Appropriate calibration of these material functions allows to render a wide range of constitutive behaviours including stress-softening and stress-hardening. The quasi-static evolution problem of a domain is formulated in terms of two physical principles based on the total energy of the system: a stability criterion, which selects the local minima of the total energy, and an energy balance condition, which ensures the consistency of the evolution of the total energy with respect to the external loadings. The local phase transformation laws in terms of Kuhn–Tucker relations are deduced from the first-order stability condition and the energy balance condition.The response of the model is illustrated with a numerical traction–torsion test performed on a thin-walled cylinder. Evolutions of homogeneous states are given for proportional and non-proportional loadings. Influence of the stress-hardening/softening properties on the evolution of the transformation domain is emphasized. Finally, in view of an identification process, the issue of stability of homogeneous states in a multi-dimensional setting is answered based on the study of second-order derivative of the total energy. Explicit necessary and sufficient conditions of stability are provided.

Burgess bounds for multi-dimensional short mixed character sums

This paper proves Burgess bounds for short mixed character sums in multi-dimensional settings. The mixed character sums we consider involve both an exponential evaluated at a real-valued multivariate polynomial f, and a product of multiplicative Dirichlet characters. We combine a multi-dimensional Burgess method with recent results on multi-dimensional Vinogradov Mean Value Theorems for translation–dilation invariant systems in order to prove character sum bounds in k≥1 dimensions that recapture the Burgess bound in dimension 1. Moreover, we show that by embedding any given polynomial f into an advantageously chosen translation–dilation invariant system constructed in terms of f, we may in many cases significantly improve the bound for the associated character sum, due to a novel phenomenon that occurs only in dimensions k≥2.

One-sided band-limited approximations of some radial functions

We construct majorants and minorants of a Gaussian function in Euclidean space that have Fourier transforms supported in a box. The majorants that we construct are shown to be extremal and our minorants are shown to be asymptotically extremal as the sides of the box become uniformly large. We then adapt the Distribution and Gaussian Subordination methods of [12] to the multidimensional setting to obtain majorants and minorants for a class of radial functions. Periodic analogues of the main results are proven and applications to Hilbert-type inequalities are given.

A generalization of the quantum Bohm identity: Hyperbolic CFL condition for Euler–Korteweg equations

In this note, we propose a surprising and important generalization of the quantum Bohm potential identity. This formula allows us to design an original conservative extended formulation of Euler–Korteweg systems and the construction of a numerical scheme with entropy stability property under a hyperbolic CFL condition in the multi-dimensional setting. To the authors' knowledge, this generalization of the quantum Bohm identity strongly improves what is already known for simulation of such a dispersive system and is also important for theoretical studies on compressible Navier–Stokes equations with degenerate viscosities.