Neumann Trace Research Articles

Dimensional reduction formulae for spectral traces and Casimir energies

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Refined von Neumann-type trace inequality and its application

The purpose of this note is to establish logarithmic submajorization inequalities that are associated with von Neumann's trace inequality for operators in finite von Neumann algebras. As an application, we extend some of of the results in Carlen and Lieb [Some matrix rearrangement inequalities. Ann Mat Pura Appl. 2006;185:S315–S324. doi: 10.1007/s10231-004-0147-z] and Chayes [Matrix rearrangement inequalities revisited; 2021. arXiv:2009.04032; Reverse Hölder, Minkowski, and Hanner inequalities for matrices; 2021. arXiv:2103.09915] to non-commutative L p spaces associated with finite von Neumann algebras. Finally, we provide an estimation of the geodesic distance on P .

Finite-Dimensional Observer-Based PI Regulation Control of a Reaction–Diffusion Equation

This article investigates the output feedback setpoint regulation control of a reaction–diffusion equation by means of boundary control. The considered reaction–diffusion plant may be open-loop unstable. The proposed control strategy consists of the coupling of a finite-dimensional observer and a PI controller in order to achieve the boundary setpoint regulation control of various system outputs such as the Dirichlet and Neumann traces. In this context, it is shown that the order of the finite-dimensional observer can always be selected large enough, with an explicit criterion, to achieve both the stabilization of the plant and the setpoint regulation of the system output.

Quantitative bounds on impedance-to-impedance operators with applications to fast direct solvers for PDEs

We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the Dirichlet-to-Neumann operator involves a decomposition of the domain into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain. Our estimates in particular ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann operator in terms of these impedance-to-impedance operators of the sub-domains. A key step in our proof is to obtain Neumann and Dirichlet boundary trace estimates on solutions of the impedance problem, which are of independent interest. In addition to the variable media setting, we also construct bounds for similar merge operators in the obstacle scattering problem.

A Borg–Levinson theorem for magnetic Schrödinger operators on a Riemannian manifold

We establish uniqueness and stability results for the inverse spectral problem of recovering the magnetic field and the electric potential in a Riemannian manifold from the knowledge of boundary spectral data of the corresponding magnetic Schrödinger operator with Dirichlet boundary condition. The spectral data consist in the knowledge of asymptotic properties, that we specify hereafter, of the sequence of eigenvalues and Neumann traces of the corresponding sequence of eigenfunctions. We also prove similar results for Schrödinger operators with Neumann boundary conditions. To our knowledge our results are the first ones involving such weak boundary spectral data.

Boundary output feedback stabilisation of a class of reaction–diffusion PDEs with delayed boundary measurement

This paper addresses the boundary output feedback stabilisation of a general class of 1-D reaction–diffusion PDEs with delayed boundary measurement. The output takes the form of either a Dirichlet or Neumann trace. The output delay can be arbitrarily large. The control strategy is composed of a finite-dimensional observer that is used to observe a delayed version of the first modes of the PDE and a predictor component that is employed to obtain the control input to be applied at the current time. For any given value of the output delay, we assess the stability of the resulting closed-loop system provided the order of the observer is selected large enough. Taking advantage of this result, we discuss the extension of the control strategy to the case of simultaneous input and output delays.

Output feedback stabilization of reaction–diffusion PDEs with a non-collocated boundary condition

This paper addresses the control design problem of output feedback stabilization of a reaction–diffusion PDE with a non-collocated boundary condition. More precisely, we consider a reaction–diffusion equation with a boundary condition describing a proportional relationship between the left and right Dirichlet traces. Such a boundary condition naturally emerges, e.g., in the context of reaction–diffusion partial differential equations presenting a transport term and with a periodic Dirichlet boundary condition. The control input takes the form of the left Neumann trace. Finally, the measurement is selected as a pointwise Dirichlet measurement located either in the domain or at the boundary. The adopted control strategy takes the form of a finite-dimensional controller coupling a state feedback and a finite-dimensional observer. The stability of the closed-loop system is obtained provided the order of the observer is selected to be large enough. Finally, we extend this result to the establishment of an input-to-state stability estimate with respect to an additive perturbation in the application of the boundary control.

A non-overlapping domain decomposition method with perfectly matched layer transmission conditions for the Helmholtz equation

It is well-known that the convergence rate of non-overlapping domain decomposition methods (DDMs) applied to the parallel finite-element solution of large-scale time-harmonic wave problems strongly depends on the transmission condition enforced at the interfaces between the subdomains. Transmission operators based on perfectly matched layers (PMLs) have proved to be well-suited for configurations with layered domain partitions. They are shown to be a good compromise between basic impedance conditions, which can lead to slow convergence, and computational expensive conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain. Unfortunately, the extension of the PML-based DDM for more general partitions with cross-points (where more than two subdomains meet) is rather tricky and requires some care.In this work, we present a non-overlapping substructured DDM with PML transmission conditions for checkerboard (Cartesian) decompositions that takes cross-points into account. In such decompositions, each subdomain is surrounded by PMLs associated to edges and corners. The continuity of Dirichlet traces at the interfaces between a subdomain and PMLs is enforced with Lagrange multipliers. This coupling strategy offers the benefit of naturally computing Neumann traces, which allows to use the PMLs as discrete operators approximating the exact Dirichlet-to-Neumann maps. Two possible Lagrange multiplier finite element spaces are presented, and the behavior of the corresponding DDM is analyzed on several numerical examples.

Boundary Integral Operators for the Heat Equation in Time-Dependent Domains

This article provides a functional analytical framework for boundary integral equations of the heat equation in time-dependent domains. More specifically, we consider a non-cylindrical domain in space-time that is the C^2-diffeomorphic image of a cylinder, i.e., the tensor product of a time interval and a fixed domain in space. On the non-cylindrical domain, we introduce Sobolev spaces, trace lemmata and provide the mapping properties of the layer operators. Here it is critical that the Neumann trace requires a correction term for the normal velocity of the moving boundary. Therefore, one has to analyze the situation carefully.

Local Transparent Boundary Conditions for Wave Propagation in Fractal Trees (II). Error and Complexity Analysis

This work is dedicated to a refined error analysis of the high-order transparent boundary conditions introduced in the companion work [8] for the weighted wave equation on a fractal tree. The construction of such boundary conditions relies on truncating the meromorphic series that represents the symbol of the Dirichlet-to-Neumann operator. The error induced by the truncation depends on the behaviour of the eigenvalues and the eigenfunctions of the weighted Laplacian on a self-similar metric tree. In this work we quantify this error by computing asymptotics of the eigenvalues and bounds for Neumann traces of the eigenfunctions. We prove the sharpness of the obtained bounds for a class of self-similar trees.

Predictor-based output feedback stabilization of an input delayed parabolic PDE with boundary measurement

This paper is concerned with the output feedback boundary stabilization of general 1-D reaction diffusion PDEs in the presence of an arbitrarily large input delay. We consider the cases of Dirichlet/Neumann/Robin boundary conditions for the both boundary control and boundary condition. The boundary measurement takes the form of an either Dirichlet or Neumann trace. The adopted control strategy is composed of a finite-dimensional observer estimating the first modes of the PDE coupled with a predictor to compensate the input delay. In this context, we show for any arbitrary value of the input delay that the control strategy achieves the exponential stabilization of the closed-loop system, for system trajectories evaluated in H1 norm (also in L2 norm in the case of a Dirichlet boundary measurement), provided the dimension of the observer is selected large enough. The reported proof of this result requires to perform both control design and stability analysis using simultaneously the (non-homogeneous) original version of the PDE and one of its equivalent homogeneous representations.

Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part I: the free space case

PurposeIn this paper, the authors revisit the computation of closed-form expressions of the topological indicator function for a one step imaging algorithm of two- and three-dimensional sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions) in the free space.Design/methodology/approachFrom the addition theorem for translated harmonics, explicit expressions of the scattered waves by infinitesimal circular (and spherical) holes subject to an incident plane wave or a compactly supported distribution of point sources are available. Then the authors derive the first-order term in the asymptotic expansion of the Dirichlet and Neumann traces and their surface derivatives on the boundary of the singular medium perturbation.FindingsAs the shape gradient of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.Originality/valueThe authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function that generates initial guesses in the iterated numerical solution of any shape optimization problem or imaging problems relying on time-harmonic acoustic wave propagation.

Integration by parts for nonsymmetric fractional-order operators on a halfspace

For a strongly elliptic pseudodifferential operator L of order 2a (0<a<1) with real kernel, we show an integration-by-parts formula for solutions of the homogeneous Dirichlet problem, in the model case where the operator is x-independent with homogeneous symbol, considered on the halfspace R+n. The new aspect compared to (−Δ)a is that L is nonsymmetric, having both an even and an odd part. Hence it satisfies a μ-transmission condition where generally μ≠a.We present a complex method, relying on a factorization in factors holomorphic in ξn in the lower or upper complex halfplane, using order-reducing operators combined with a decomposition principle originating from Wiener and Hopf. This is in contrast to a real, computational method presented very recently by Dipierro, Ros-Oton, Serra and Valdinoci. Our method allows μ in a larger range than they consider.Another new contribution is the (model) study of “large” solutions of nonhomogeneous Dirichlet problems when μ>0. Here we deduce a “halfways Green's formula” for L:∫R+nLuv¯dx−∫R+nuL⁎v‾dx=c∫Rn−1γ0(u/xnμ−1)γ0(v¯/xnμ⁎)dx′, when u solves a nonhomogeneous Dirichlet problem for L, and v solves a homogeneous Dirichlet problem for L⁎; μ⁎=2a−μ. Finally, we show a full Green's formula, when both u and v solve nonhomogeneous Dirichlet problems; here both Dirichlet and Neumann traces of u and v enter, as well as a first-order pseudodifferential operator over the boundary.

Recovering the Initial Data of the Wave Equation from Neumann Traces

We study the problem of recovering the initial data ($f, 0$) of the standard wave equation from the Neumann trace (the normal derivative) of the solution on the boundary of convex domains in arbitr...

Well-posedness and qualitative behaviour of the Mullins-Sekerka problem with ninety-degree angle boundary contact

We show local well-posedness for a Mullins-Sekerka system with ninety degree angle boundary contact. We will describe the motion of the moving interface by a height function over a fixed reference surface. Using the theory of maximal regularity together with a linearization of the equations and a localization argument we will prove well-posedness of the full nonlinear problem via the contraction mapping principle. Here one difficulty lies in choosing the right space for the Neumann trace of the height function and showing maximal L_p{-}L_q-regularity for the linear problem. In the second part we show that solutions starting close to certain equilibria exist globally in time, are stable, and converge to an equilibrium solution at an exponential rate.

Neumann trace tracking of a constant reference input for 1-D boundary controlled heat-like equations with delay

This paper discusses the Proportional Integral (PI) regulation control of the left Neumann trace of a one-dimensional reaction-diffusion equation with a delayed right Dirichlet boundary control. Specifically, a PI controller is designed based on a finite-dimensional truncated model that captures the unstable dynamics of the original infinite-dimensional system. In this setting, the control input delay is handled by resorting to the Artstein transformation. The stability of the resulting infinite-dimensional system, as well as the tracking of a constant reference signal in the presence of a constant distributed perturbation, is assessed based on the introduction of an adequate Lyapunov function. The theoretical results are illustrated with numerical simulations.

Impossibility of Dimension Reduction in the Nuclear Norm

Let \({\mathsf {S}}_1\) (the Schatten–von Neumann trace class) denote the Banach space of all compact linear operators \(T:\ell _2\rightarrow \ell _2\) whose nuclear norm \(\Vert T\Vert _{{\mathsf {S}}_1}=\sum _{j=1}^\infty \upsigma _j(T)\) is finite, where \(\{\upsigma _j(T)\}_{j=1}^\infty \) are the singular values of T. We prove that for arbitrarily large \(n\in {\mathbb {N}}\) there exists a subset \({\mathscr {C}}\subseteq {\mathsf {S}}_1\) with \(|{\mathscr {C}}|=n\) that cannot be embedded with bi-Lipschitz distortion O(1) into any \(n^{o(1)}\)-dimensional linear subspace of \({\mathsf {S}}_1\). \({\mathscr {C}}\) is not even a O(1)-Lipschitz quotient of any subset of any \(n^{o(1)}\)-dimensional linear subspace of \({\mathsf {S}}_1\). Thus, \({\mathsf {S}}_1\) does not admit a dimension reduction result á la Johnson and Lindenstrauss (Conference in modern analysis and probability, American Mathematical Society, Providence, 1984), which complements the work of Harrow et al. (Automata, languages and programming (ICALP 2011), Springer, Heidelberg, 2011) on the limitations of quantum dimension reduction under the assumption that the embedding into low dimensions is a quantum channel. Such a statement was previously known with \({\mathsf {S}}_1\) replaced by the Banach space \(\ell _1\) of absolutely summable sequences via the work of Brinkman and Charikar (J ACM 52(5):766–788, 2005). In fact, the above set \({\mathscr {C}}\) can be taken to be the same set as the one that Brinkman and Charikar considered, viewed as a collection of diagonal matrices in \({\mathsf {S}}_1\). The challenge is to demonstrate that \({\mathscr {C}}\) cannot be faithfully realized in an arbitrary low-dimensional subspace of \({\mathsf {S}}_1\), while Brinkman and Charikar obtained such an assertion only for subspaces of \({\mathsf {S}}_1\) that consist of diagonal operators (i.e., subspaces of \(\ell _1\)). We establish this by proving that the Markov 2-convexity constant of any finite dimensional linear subspace X of \({\mathsf {S}}_1\) is at most a universal constant multiple of \(\sqrt{\log \dim (X)}\).

PI Regulation of a Reaction–Diffusion Equation With Delayed Boundary Control

The general context of this work is the feedback control of an infinite-dimensional system so that the closed-loop system satisfies a fading-memory property and achieves the setpoint tracking of a given reference signal. More specifically, this paper is concerned with the Proportional Integral (PI) regulation control of the left Neumann trace of a one-dimensional reaction-diffusion equation with a delayed right Dirichlet boundary control. In this setting, the studied reaction-diffusion equation might be either open-loop stable or unstable. The proposed control strategy goes as follows. First, a finite-dimensional truncated model that captures the unstable dynamics of the original infinite-dimensional system is obtained via spectral decomposition. The truncated model is then augmented by an integral component on the tracking error of the left Neumann trace. After resorting to the Artstein transformation to handle the control input delay, the PI controller is designed by pole shifting. Stability of the resulting closed-loop infinite-dimensional system, consisting of the original reaction-diffusion equation with the PI controller, is then established thanks to an adequate Lyapunov function. In the case of a time-varying reference input and a time-varying distributed disturbance, our stability result takes the form of an exponential Input-to-State Stability (ISS) estimate with fading memory. Finally, another exponential ISS estimate with fading memory is established for the tracking performance of the reference signal by the system output. In particular, these results assess the setpoint regulation of the left Neumann trace in the presence of distributed perturbations that converge to a steady-state value and with a time-derivative that converges to zero. Numerical simulations are carried out to illustrate the efficiency of our control strategy.

Hp-version time domain boundary elements for the wave equation on quasi-uniform meshes

Solutions to the wave equation in the exterior of a polyhedral domain or a screen in R3 exhibit singular behavior from the edges and corners. We present quasi-optimal hp-explicit estimates for the approximation of the Dirichlet and Neumann traces of these solutions for uniform time steps and (globally) quasi-uniform meshes on the boundary. The results are applied to an hp-version of the time domain boundary element method. Numerical examples confirm the theoretical results for the Dirichlet problem both for screens and polyhedral domains.

The INTERNODES Method for Non-conforming Discretizations of PDEs

INTERNODES is a general purpose method to deal with non-conforming discretizations of partial differential equations on 2D and 3D regions partitioned into two or several disjoint subdomains. It exploits two intergrid interpolation operators, one for transfering the Dirichlet trace across the interfaces, and the other for the Neumann trace. In this paper, in every subdomain the original problem is discretized by either the finite element method (FEM) or the spectral element method (SEM or hp-FEM), using a priori non-matching grids and piecewise polynomials of different degrees. Other discretization methods, however, can be used. INTERNODES can also be applied to heterogeneous or multiphysics problems, that is, problems that feature different differential operators inside adjacent subdomains. For instance, in this paper we apply the INTERNODES method to a Stokes–Darcy coupled problem that models the filtration of fluids in porous media. Our results highlight the flexibility of the method as well as its optimal rate of convergence with respect to the grid size and the polynomial degree.