Partial Differential Operators Research Articles

Killing Operator for the Kerr Metric

When D: E → F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: Jq(E) → F=F0 that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1: F0 → F1. When D is involutive, that is when the corresponding system Rq = ker (Φ) is involutive, this procedure provides successive first order involutive operators D1, ..., Dn. Though D1 ο D = 0 implies ad (D) ο ad(D1) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D1) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When Rq is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.

Duality of generalized Hardy and BMO spaces associated with singular partial differential operator

Duality of generalized Hardy and BMO spaces associated with singular partial differential operator

A sharp-interface mesoscopic model for dendritic growth

The grain envelope model (GEM) describes the growth of envelopes of dendritic crystal grains during solidification. Numerically the growing envelopes are usually tracked using an interface capturing method employing a phase field equation on a fixed grid. Such an approach describes the envelope as a diffuse interface, which can lead to numerical artefacts that are possibly detrimental. In this work, we present a sharp-interface formulation of the GEM that eliminates such artefacts and can thus track the envelope with high accuracy. The new formulation uses an adaptive meshless discretization method to solve the diffusion in the liquid around the grains. We use the ability of the meshless method to operate on scattered nodes to accurately describe the interface, i.e., the envelope. The proposed algorithm combines parametric surface reconstruction, meshless discretization of parametric surfaces, global solution construction procedure and partial differential operator approximation using monomials as basis functions. The approach is demonstrated on a two-dimensional h-adaptive solution of diffusive growth of dendrites and assessed by comparison to a conventional diffuse-interface fixed-grid GEM.

Propagation of anisotropic Gelfand–Shilov wave front sets

We show a result on propagation of the anisotropic Gelfand–Shilov wave front set for linear operators with Schwartz kernel which is a Gelfand–Shilov ultradistribution of Beurling type. This anisotropic wave front set is parametrized by two positive parameters relating the space and frequency variables. The anisotropic Gelfand–Shilov wave front set of the Schwartz kernel of the operator is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by a partial differential operator defined by a symbol which is a polynomial with real coefficients and order at least two.

Deformed Calogero–Moser Operators and Ideals of Rational Cherednik Algebras

We introduce a class of hyperplane arrangements $$\mathcal {A}$$ in $${\mathbb {C}}^n$$ that generalise the locus configurations of Chalykh, Feigin and Veselov. To such an arrangement we associate a second order partial differential operator of Calogero–Moser type and prove that this operator is completely integrable (in the sense that its centraliser in $$\mathcal {D}({\mathbb {C}}^n\!\setminus \!\mathcal {A})$$ contains a maximal commutative subalgebra of Krull dimension n). Our approach is based on the study of shift operators and associated ideals in spherical Cherednik algebras that may be of independent interest. Examples include all known completely integrable deformations of Calogero–Moser operators with rational potentials. In addition, we construct new families of examples, including a BC-type generalisation of the deformed Calogero-Moser operators recently found by Gaiotto and Rapčák. We describe these examples in a unified representation-theoretic framework of rational Cherednik algebras.

Homogenization of a parabolic equation in a perforated domain with a unilateral dynamic boundary condition: critical case

In this paper, we study the homogenization of a parabolic equation given in a domain perforated by ''tiny'' balls. On the boundary of these perforations, a unilateral dynamic boundary constraints are specified. We address the so-called ''critical'' case that is characterized by a relation between the coefficient in the boundary condition, the period of the structure and the size of the holes. In this case, the homogenized equation contains a nonlocal ''strange'' term. This term is obtained as a solution of the variational problem involving ordinary differential operator.

On a Class of Stochastic Hyperbolic Equations with Double Characteristics

We study the effect of Gaussian perturbations on a hyperbolic partial differential equation with double characteristics in two spatial dimensions. The coefficients of our partial differential operator depend polynomially on the space variables, while the noise is additive, white in time and coloured in space. We provide a sufficient condition on the spectral measure of the covariance functional describing the noise that allows for the existence of a random field solution for the resulting stochastic partial differential equation. Our approach is based on explicit computations for the fundamental solution of the partial differential operator and its Fourier transform.

Linear differential operators with distribution coefficients of various singularity orders

In this paper, the linear differential expression of order with distribution coefficients of various singularity orders is considered. We obtain the associated matrix for the regularization of this expression. Furthermore, we present new statements of inverse spectral problems that consist in the recovery of differential operators from the Weyl matrix on the half‐line and on a finite interval. The uniqueness theorems for these two inverse problems are proved by developing the method of spectral mappings. To the best of the author's knowledge, inverse problems for higher order differential operators with distribution coefficients on the half‐line have not been studied before. For the finite interval case, we for the first time consider the issue of recovering coefficients of the boundary conditions.

On a generalized Lamé-Navier system in ℝ3

Abstract This paper is devoted to a fundamental system of equations in Linear Elasticity Theory: the famous Lamé-Navier system. The Clifford algebra language allows us to rewrite this system in terms of the Euclidean Dirac operator, which at the same time suggests a very natural generalization involving the so-called structural sets. We are interested in finding some structures in the solutions of these generalized Lamé-Navier systems. Using MATLAB we also implement algorithms to compute with such partial differential operators as well as to verify some theoretical results obtained in the paper.

Scientific Machine Learning Enables Multiphysics Digital Twins of Large-Scale Electronic Chips

We propose a scientific machine learning (SciML) algorithm toward 3-D dynamic digital twins, which represent multiphysics coupling effects in large-scale electronic chips. The SciML is a burgeoning topic, and here we refer to an organic fusion of scientific computing, model order reduction, and machine learning (ML) method. The proposed model order reduction compresses multiphysics information by a data-driven non-intrusive technique, thus getting rid of the access to backend source code; the proposed ML intrinsically infers the partial differential equation operators encoding the physical process. Numerical experiments showcase that the proposed digital twins have superior properties in real-time intelligent computing and generalization capability in predictive modeling. Nevertheless, it should be mentioned that the presented work is still in an early stage of intelligent digital twins.

Financial Interpretation of Feller’s Factorization

The infinitesimal generator of a time-homogeneous univariate diffusion process is a second-order linear ordinary differential operator. Feller (1952) famously factorized this generator into successive differentiations with respect to scale and speed measure. Later, Feller (1957) also factored an extended generator that loads also on the identity operator in a particular way. We provide a novel financial interpretation of these factorization results and show that they produce an operator representation of a conditionally linear risk-return tradeoff when the conditioning variable evolves like a one-dimensional diffusion process.

Classification of 𝔞𝔣𝔣(3|1)-invariant binary differential operators on weighted densities and cohomology

In this paper, we consider the [Formula: see text]-module structure on the spaces of differential bilinear operators acting on the superspaces of weighted densities. We classify [Formula: see text]-invariant binary differential operators acting on the spaces of weighted densities. This result allows us to compute the first [Formula: see text]-relative differential cohomology of [Formula: see text] with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. Finally, we deduce the classification of the [Formula: see text]-relative extension of [Formula: see text] modules [Formula: see text]

The Functional Dissipativity of Certain Systems of Partial Differential Equations

We study the functional dissipativity of the Dirichlet problem for systems of partial differential operators of the form ∂h( hk(x)∂k), where hk are m × m matrices with complex-valued $$ {L}_{loc}^1 $$ entries. In the particular case of operator ∂h( h(x)∂h), where h are m × m matrices, we obtain algebraic necessary and sufficient conditions. We give three different notions of functional ellipticity and investigate relations between them and the functional dissipativity for the operators in question.

The four-state problem and convex integration for linear differential operators

We show that the four-state problem for general linear differential operators is flexible. The only flexibility result available in this context is the one for the five-state problem for the curl operator due to B. Kirchheim and D. Preiss, [27, Section 4.3], and its generalization [21]. To build our counterexample, we extend the convex integration method introduced by S. Müller and V. Šverák in [33] to linear operators that admit a potential, and we exploit the notion of largeTNconfiguration introduced by C. Förster and L. Székelyhidi in [21].

Super-localization of elliptic multiscale problems

Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a d d -dimensional domain. The application of the inverse operator to some standard finite element space defines an approximation space with uniform algebraic approximation rates with respect to the mesh size parameter H H . This holds even for under-resolved rough coefficients. However, the true challenge of numerical homogenization is the localized computation of a localized basis for such an operator-dependent approximation space. This paper presents a novel localization technique that leads to a super-exponential decay of its basis relative to H H . This suggests that basis functions with supports of width O ( H | log ⁡ H | ( d − 1 ) / d ) \mathcal O(H|\log H|^{(d-1)/d}) are sufficient to preserve the optimal algebraic rates of convergence in H H without pre-asymptotic effects. A sequence of numerical experiments illustrates the significance of the new localization technique when compared to the so far best localization to supports of width O ( H | log ⁡ H | ) \mathcal O(H|\log H|) .

The foundations of spectral computations via the Solvability Complexity Index hierarchy

The problem of computing spectra of operators is arguably one of the most investigated areas of computational mathematics. However, the problem of computing spectra of general bounded infinite matrices has only recently been solved. We establish some of the foundations of computational spectral theory through the Solvability Complexity Index (SCI) hierarchy, an approach closely related to Smale’s program on the foundations of computational mathematics and McMullen’s results on polynomial root finding with rational maps. Infinite-dimensional problems yield an intricate infinite classification theory, determining which spectral problems can be solved and with what types of algorithms. We provide answers to many longstanding open questions on the existence of algorithms. For example, we show that spectra can be computed, with error control, from point sampling operator coefficients for large classes of partial differential operators on unbounded domains. Further results include: computing spectra of (possibly unbounded) operators on graphs and separable Hilbert spaces with error control; determining if the spectrum intersects a compact set; the computational spectral gap problem and computing spectral classifications at the bottom of the spectrum; and computing discrete spectra, multiplicities, eigenspaces and determining if the discrete spectrum is non-empty. Moreover, the positive results with error control can be used in computer-assisted proofs. In contrast, the negative results preclude computer-assisted proofs for classes of operators as a whole. Our proofs are constructive, yielding a library of new algorithms and techniques that handle problems that before were out of reach. We demonstrate these algorithms on challenging problems, giving concrete examples of the failure of traditional approaches (e.g., “spectral pollution”) compared to the introduced techniques.

Oscillation and Concentration in Sequences of PDE Constrained Measures

We show that for constant rank partial differential operators mathscr {A} whose wave cones are spanning, generalized Young measures generated by bounded sequences of mathscr {A}-free measures can be characterized by duality with mathscr {A}-quasiconvex integrands of linear growth. This includes a characterization of the concentration effects in such sequences that allows us to conclude that, in sharp contrast to the oscillation effects, the concentration always has mathscr {A}-free structure.

Existence Results for an m-Point Mixed Fractional-Order Problem at Resonance on the Half-Line

This work considers the existence of solutions for a mixed fractional-order boundary value problem at resonance on the half-line. The Mawhin’s coincidence degree theory will be used to prove existence results when the dimension of the kernel of the linear fractional differential operator is equal to two. An example is given to demonstrate the main result obtained.

Recursive moment computation in Filon methods and application to high-frequency wave scattering in two dimensions

Abstract We study the efficient approximation of highly oscillatory integrals using Filon methods. A crucial step in the implementation of these methods is the accurate and fast computation of the Filon quadrature moments. In this work we demonstrate how recurrences can be constructed for a wide class of oscillatory kernel functions, based on the observation that many physically relevant kernel functions are in the null space of a linear differential operator whose action on the Filon interpolation basis is represented by a banded (infinite) matrix. We discuss in further detail the application to two classes of particular interest: integrals with algebraic singularities and stationary points and integrals involving a Hankel function. We provide rigorous stability results for the moment computation for the first of these classes and demonstrate how the corresponding Filon method results in an accurate approximation at truly frequency-independent cost. For the Hankel kernel, we derive error estimates that describe the convergence behaviour of the method in terms of frequency and number of Filon quadrature points. Finally, we show how Filon methods with recursive moment computation can be applied to efficiently compute integrals arising in hybrid numerical-asymptotic collocation methods for high-frequency wave scattering on a screen.

Local limit theorems for complex functions on [formula omitted

The local (central) limit theorem precisely describes the behavior of iterated convolution powers of a probability distribution on the d-dimensional integer lattice, Zd. Under certain mild assumptions on the distribution, the theorem says that the convolution powers are well-approximated by a single scaled Gaussian density which we call an attractor. When such distributions are allowed to take on complex values, their convolution powers exhibit new and striking behaviors not seen in the probabilistic setting. Following works of I. J. Schoenberg, T. N. E. Greville, P. Diaconis, and L. Saloff-Coste, the author and L. Saloff-Coste provided a complete description of local limit theorems for the class of finitely supported complex-valued functions on Z. For convolution powers of complex-valued functions on Zd, much less is known. In a previous work by the author and L. Saloff-Coste, local limit theorems were established for complex-valued functions whose Fourier transform is maximized in absolute value at so-called points of positive homogeneous type and, in that case, the resultant attractors are generalized heat kernels corresponding to a class of higher order partial differential operators. By considering the possibility that the Fourier transform can be maximized in absolute value at points of imaginary homogeneous type, a notion motivated by V. Thomée, this article extends previous work of the author and L. Saloff-Coste to broaden the class of complex-valued functions for which it is possible to obtain local limit theorems. These local limit theorems contain attractors given by certain oscillatory integrals and their convergence is established using a generalized polar-coordinate integration formula, due to H. Bui and the author, and the Van der Corput lemma. The article also extends recent results on sup-norm type estimates of H. Bui and the author.