Spectrally Accurate Fully Discrete Schemes for Some Nonlocal and Nonlinear Integrable PDEs via Explicit Formulas

These lectures are devoted to two integrable PDE on the line enjoying similar structures: the Benjamin–Ono equation and the Calogero–Moser derivative nonlinear Schrödinger equation. For both equations, a Lax pair of operators is introduced on the Hardy space of the upper-half plane, and is used to prove conservation laws and explicit formulae, and to study soliton and multisoliton solutions. In the special case of the Benjamin–Ono equation, the small dispersion limit with general initial data is proved to exist and is identified. These lectures were presented at the 2024 PDE Days, Centre Paul Langevin, Aussois, France.

The inverse problem which arises in the study of the integrable PDE proposed by V Novikov is solved for a class of discrete densities. The method of solution relies on the use of Cauchy biorthogonal polynomials. The explicit formulas are obtained directly from the analysis on the real axis without any additional transformations to the ‘string’-type boundary value problems known from prior works.

We present a perturbation method and a matrix method for formulating the explicit Fréchet derivatives for seismic body-wave waveform inversion in 3D general anisotropic, heterogeneous media. Theoretically, the two methods yield the same explicit formula valid for any class of anisotropy and are completely equivalent if the model parameterization in the inversion is the same as that used in the discretization scheme (unstructured or structured mesh) for forward modeling. Explicit formulas allow various model parameterization schemes that try to match the resolution capability of the data and possibly reduce the dimensions of the Jacobian matrix. Based on the general expressions, relevant formulas for isotropic and 2.5D and 3D tilted transversely isotropic (TTI) media are derived. Two computational schemes, constant-point and constant-block parameterization, offer effective and efficient means of forming the Jacobian matrix from the explicit Fréchet derivatives. The sensitivity patterns of the displacement vector to the independent model parameters in a weakly anisotropic medium clearly convey the imaging capability possible with seismic waveform inversion in such an anisotropic medium.

Nonlocal integrable PDEs from hierarchies of symmetry laws: The example of Pohlmeyer–Lund–Regge equation and its reflectionless potential solutions

Einstein–Weyl geometry is a triple ({mathbb {D}},g,omega ) where {mathbb {D}} is a symmetric connection, [g] is a conformal structure and omega is a covector such that bullet connection {mathbb {D}} preserves the conformal class [g], that is, {mathbb {D}}g=omega g; bullet trace-free part of the symmetrised Ricci tensor of {mathbb {D}} vanishes. Three-dimensional Einstein–Weyl structures naturally arise on solutions of second-order dispersionless integrable PDEs in 3D. In this context, [g] coincides with the characteristic conformal structure and is therefore uniquely determined by the equation. On the contrary, covector omega is a somewhat more mysterious object, recovered from the Einstein–Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge–Ampère type), the covector omega is also expressible in terms of the equation, thus providing an efficient ‘dispersionless integrability test’. The knowledge of g and omega provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein–Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein–Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation.

The Cox-Russ jump process pricing model is analysed using methods of nonstandard (i.e. infinitesimal) analysis, following the methodology developed by the authors in earlier papers. Explicit formulae are found for replicating strategies for European options. The existence of liftings for claims and strategies, and the concepts of D2-convergence and adapted discretisation schemes, introduced earlier by the authors to discuss the convergence of random walk pricing models to continuous-time models driven by Brownian motion, are analysed for Poisson jump price models

We present a tutorial introduction to methods for stabilization of systems with long input delays. The methods are based on techniques originally developed for boundary control of partial differential equations. We start with a consideration of linear systems, first with a known delay and then subject to a small uncertainty in the delay. Then we study linear systems with constant delays that are completely unknown, which requires an adaptive control approach. For linear systems, we also present a method for compensating arbitrarily large but known time-varying delays. Finally, we consider nonlinear control problems in the presence of arbitrarily long input delays.An enormous wealth of knowledge and research results exists for control of systems with state delays and input delays. Problems with long input delays, for unstable plants, represent a particular challenge. In fact, they were the first challenge to be tackled, in Otto J. M. Smith’s article [1], where the compensator known as the Smith predictor was introduced five decades ago. The Smith predictor’s value is in its ability to compensate for a long input or output delay in set point regulation or constant disturbance rejection problems. However, its major limitation is that, when the plant is unstable, it fails to recover the stabilizing property of a nominal controller when delay is introduced.A substantial modification to the Smith predictor, which removes its limitation to stable plants was developed three decades ago in the form of finite spectrum assignment (FSA) controllers [2, 3, 4]. More recent treatment of this subject can also be found in the books [5, 6]. In the FSA approach, the system $$ \dot {X}(t) = AX(t) + BU(t-D)\,, $$ (1) where X is the state vector, U is the control input (scalar in our consideration here), D is an arbitrarily long delay, and (A,B) is a controllable pair, is stabilized with the infinite-dimensional predictor feedback $$ U(t) = K\left[{\rm e}^{AD} X(t)+ \int_{t-D}^t {\rm e}^{A(t-\theta)} B U(\theta)d\theta\right]\,, $$ (2) where the gain K is chosen so that the matrix A + BK is Hurwitz. The word ‘predictor’ comes from the fact that the bracketed quantity is actually the future state X(t + D), expressed using the current state X(t) as the initial condition and using the controls U(θ) from the past time window [t − D,t]. Concerns are raised in [7] regarding the robustness of the feedback law (2) to digital implementation of the distributed delay (integral) term but are resolved with appropriate discretization schemes [8,9].One can view the feedback law (2) as being given implicitly, since U appears both on the left and on the right, however, one should observe that the input memory U(θ), θ ∈ [t − D,t] is actually a part of the state of the overall infinite-dimensional system, so the control law is actually given by an explicit full-state feedback formula. The predictor feedback (2) actually represents a particular form of boundary control, commonly encountered in the context of control of partial differential equations.Motivated by our recent efforts in solving boundary control problems for various classes of partial differential equations (PDEs) using the continuum version of the backstepping method [10,11], we review in this article various extensions to the predictor feedback design that we have recently developed, particularly for nonlinear and PDE systems. These extensions are the subject of our new book [12]. They include the extension of predictor feedback to nonlinear systems and PDEs with input delays, various robustness and inverse optimality results, a delay-adaptive design, an extension to time-varying delays, and observer design in the presence of sensor delays and PDE dynamics. This article is a tutorial introduction to these design tools and concludes with a brief review of some open problems and research opportunities.KeywordsBoundary ControlInput DelaySmith PredictorHyperbolic PDEsActuator DelayThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of $N$-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.

Nonsmooth data error estimates for fully discrete finite element approximations of semilinear parabolic equations in Banach space

The primary objective of this work is to study the inverse problem of identifying a stochastic parameter in partial differential equations with random data. In the framework of stochastic Sobolev spaces, we prove the Lipschitz continuity and the differentiability of the parameter-to-solution map and provide a new derivative characterization. We introduce a new energy-norm based modified output least-squares (OLS) objective functional and prove its smoothness and convexity. For stable inversion, we develop a regularization framework and prove an existence result for the regularized stochastic optimization problem. We also consider the OLS based stochastic optimization problem and provide an adjoint approach to compute the derivative of the OLS-functional. In the finite-dimensional noise setting, we give a parameterization of the inverse problem. We develop a computational framework by using the stochastic Galerkin discretization scheme and derive explicit discrete formulas for the considered objective functionals and their gradient. We provide detailed computational results to illustrate the feasibility and efficacy of the developed inversion framework. Encouraging numerical results demonstrate some of the advantages of the new framework over the existing approaches.

The convergence and stability of full discretization scheme for stochastic age-structured population models

In this paper some results are reviewed concerning the characterization of inverses of symmetric tridiagonal and block tridiagonal matrices as well as results concerning the decay of the elements of the inverses. These results are obtained by relating the elements of inverses to elements of the Cholesky decompositions of these matrices. This gives explicit formulas for the elements of the inverse and gives rise to stable algorithms to compute them. These expressions also lead to bounds for the decay of the elements of the inverse for problems arising from discretization schemes.

Finding matrix representations is an important part of operator theory. Calculating such a discretization scheme is equally important for the numerical solution of operator equations. Traditionally in both fields, this was done using bases. Recently, frames have been used here. In this paper, we apply fusion frames for this task, a generalization motivated by a block representation, respectively, a domain decomposition. We interpret the operator representation using fusion frames as a generalization of fusion Gram matrices. We present the basic definition of U-fusion cross Gram matrices of operators for a bounded operator U. We give necessary and sufficient conditions for their (pseudo-)invertibility and present explicit formulas for the (pseudo-)inverse. More precisely, our attention is on how to represent the inverse and pseudo-inverse of such matrices as U-fusion cross Gram matrices. In particular, we characterize fusion Riesz bases and fusion orthonormal bases by such matrices. Finally, we look at which perturbations of fusion Bessel sequences preserve the invertibility of the fusion Gram matrix of operators.

PurposeMagnetic polarizability tensors (MPTs) provide an economical characterisation of conducting magnetic metallic objects and their spectral signature can aid in the solution of metal detection inverse problems, such as scrap metal sorting, searching for unexploded ordnance in areas of former conflict and security screening at event venues and transport hubs. In this work, the authors aim to discuss methods for efficiently building large dictionaries for classification approaches.Design/methodology/approachPrevious work has established explicit formulae for MPT coefficients, underpinned by a rigorous mathematical theory. To assist with the efficient computation of MPTs at differing parameters and objects of interest, this work applies new observations about the way the MPT coefficients can be computed. Furthermore, the authors discuss discretisation strategies for hp-finite elements on meshes of unstructured tetrahedra combined with prismatic boundary layer elements for resolving thin skin depths and using an adaptive proper orthogonal decomposition (POD) reduced-order modelling methodology to accelerate computations for varying parameters.FindingsThe success of the proposed methodologies is demonstrated using a series of examples. A significant reduction in computational effort is observed across all examples. The authors identify and recommend a simple discretisation strategy and improved accuracy is obtained using adaptive POD.Originality/valueThe authors present novel computations, timings and error certificates of MPT characterisations of realistic objects made of magnetic materials. A novel postprocessing implementation is introduced and an adaptive POD algorithm is demonstrated.

Symmetrization for fractional Neumann problems