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The North Dakota Department of Public Instruction (NDDPI) uses a three-pillar approach to school renewal efforts: build relationships, cultivate opportunity, and inspire growth. A key product of this approach is the North Dakota School Renewal Guide, which includes practitioner-friendly school improvement strategies grounded in evidence-based improvement practices for schools identified as Targeted Support and Improvement (TSI) or Comprehensive Support and Improvement (CSI). The guide, now in its third edition, is updated annually based on feedback from the NDDPI, the North Dakota Regional Education Association (REA), and the TSI and CSI schools and districts using the guide. This report tells the story of how the NDDPI project team created and tested a calibration activity to increase inter-rater reliability ratings among REA staff, NDDPI representatives, and schools. By using procedures for consistent and accurate ratings, REA liaisons can provide schools with tailored coaching to support their improvement journeys, help schools track progress on goals and initiatives, and help schools collect the evidence required for them to exit TSI and CSI status. The report is intended as a resource for school leaders working to implement short-cycle school improvement goals.

The Monotonocity Principle states a monotonic relationship between a possibly non-linear material property and a proper corresponding boundary operator. The Monotonicity Principle (MP) has attracted great interest in the field of inverse problems, because of its fundamental role in developing real time imaging methods. Recently, with quite general assumptions, a MP in the presence of non linear materials has been established for elliptic PDE, such as those governing Electrical Resistance Tomography. Together with recently introduced imaging methods and algorithms based on MP, arises a fundamental question related to the Converse (of the MP). Indeed, the Converse of the MP is fundamental to define the theoretical limits of applicability of imaging methods and algorithms. Specifically, the Converse of the MP guarantees that the outer boundary of a nonlinear anomaly can be reconstructed by means of MP based imaging methods. In this paper, the Converse of the Monotonicity Principle for nonlinear anomaly embedded in a linear material is proved. The results is provided in a quite general setting for Electrical Resistance Tomography. Moreover, the nonlinear electrical conductivity of the anomaly, as function of the electric field, can be either bounded or not bounded from infinity and/or zero.

Metasurfaces, consisting of large arrays of interacting subwavelength scatterers, pose significant challenges for general-purpose computational methods due to their large electric dimensions and multiscale nature. This paper introduces an efficient boundary element method specifically tailored for metasurfaces, leveraging the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation. Our method combines the Multilevel Fast Multipole Algorithm (MLFMA) with a representation of the unknown equivalent surface current density by means of static modes, a set of entire domain basis functions dependent only on object shape but independent of the material and frequency. The compression of the number of unknowns enabled by the Static Mode Representation (SMR), combined with the \(\mathcal{O}(N \log N)\) complexity of MLFMA matrix-vector products, significantly reduces CPU time and memory requirements compared to classical MLFMA with RWG basis functions. We demonstrate the accuracy, time, and memory requirements of this method through several test cases including the full-wave simulation of a $100 \lambda \times 100 \lambda$ canonical metalens. The MLFMA-SMR method offers substantial benefits for the analysis and optimization of metasurfaces and metalenses.

A methodology to reduce the computational cost of time domain computations of eddy currents problems is proposed and implemented in a parallel computing environment. It is based on the modal decomposition of the current density and it is applicable even in presence of injected currents into the electrodes of a conducting domain. Using a theta-method integration algorithm, the performances of the the proposed approach are compared against those of a classical method based on the Cholesky factorization, for a case of interest from eddy current nondestructive testing. For this large eddy current problem (number of unknowns greater than 100k, number of time steps of interest equal to 100k) the proposed solution method is shown to be much faster than those based on standard time integration schemes., Comment: 18 pages, 14 Figures in 6 captions (1a, 1b, 2a, 2b, 2c, 2d, 3, 4a, 4b, 5a, 5b, 5c, 5d, 6)

In this paper we present a first non-iterative imaging method for nonlinear materials, based on Monotonicity Principle. Specifically, we deal with the inverse obstacle problem, where the aim is to retrieve a nonlinear anomaly embedded in linear known background. The Monotonicity Principle (MP) is a general property for various class of PDEs, that has recently generalized to nonlinear elliptic PDEs. Basically, it states a monotone relation between the point-wise value of the unknown material property and the boundary measurements. It is at the foundation of a class of non-iterative imaging methods, characterized by a very low execution time that makes them ideal candidates for real-time applications. In this work, we develop an inversion method that overcomes some of the peculiar difficulties in practical application of MP to imaging of nonlinear materials, preserving the feasibility for real-time applications. For the sake of clarity, we focus on a specific application, i.e. the Magnetostatic Permeability Tomography where the goal is retrieving the unknown (nonlinear) permeability by boundary measurements in DC operations. This choice is motivated by applications in the inspection of boxes and containers for security. Reconstructions from simulated data prove the effectiveness of the presented method.

In this paper we consider the inverse problem of electrical conductivity retrieval starting from boundary measurements, in the framework of Electrical Resistance Tomography (ERT). In particular, the focus is on non-iterative reconstruction algorithms, compatible with real-time applications. In this work a new non-iterative reconstruction method for Electrical Resistance Tomography, termed Kernel Method, is presented. The imaging algorithm deals with the problem of retrieving the shape of one or more anomalies embedded in a known background. The foundation of the proposed method is given by the idea that if there exists a current flux at the boundary (Neumann data) able to produce the same voltage measurements on two different configurations, with and without the anomaly, respectively, then the corresponding electric current density for the problem involving only the background material vanishes in the region occupied by the anomaly. Coherently with this observation, the Kernel Method consists in (i) evaluating a proper current flux at the boundary $g$, (ii) solving one direct problem on a configuration without anomaly and driven by $g$, (iii) reconstructing the anomaly from the spatial plot of the power density as the region in which the power density vanishes. This new tomographic method has a very simple numerical implementation at a very low computational cost. Beside theoretical results and justifications of our method, we present a large number of numerical examples to show the potential of this new algorithm.

Inverse problems, which are related to Maxwell's equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behaviour of some unknown physical property, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.

This paper is focused on the Monotonicity Principle (MP) for nonlinear materials with piecewise growth exponent. This results are relevant because enables the use of a fast imaging method based on MP, to the wide class of problems with two or more materials, where at least one is nonlinear. The treatment is very general and allows to model a wide variety of practical configurations such as, for instance, Superconducting (SC) or Perfect Electrical Conducting (PEC) or Perfect Electrical Insulating (PEI) materials. A key role is played by the average Dirichlet-to-Neumann operator, introduced in [Corbo Esposito et. al, Inverse Problems 2021], where the MP for a single type of nonlinearity was treated. Realistic numerical examples confirm the theoretical findings.

This paper refers to an imaging problem in the presence of nonlinear materials. Specifically, the problem we address falls within the framework of Electrical Resistance Tomography and involves two different materials, one or both of which are nonlinear. Tomography with nonlinear materials in the early stages of developments, although breakthroughs are expected in the not-too-distant future. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted $p_0$-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the $p_0$-Laplacian in inverse problems with nonlinear materials. Moreover, when $p_0=2$, this result allows all the imaging methods and algorithms developed for linear materials to be brought into the arena of problems with nonlinear materials. The main result of this work is that for "small" Dirichlet data, (i) one material can be replaced by a perfect electric conductor and (ii) the other material can be replaced by a material giving rise to a weighted $p_0$-Laplace problem., Comment: arXiv admin note: substantial text overlap with arXiv:2309.15206

This paper is inspired by an imaging problem encountered in the framework of Electrical Resistance Tomography involving two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of developments, although breakthroughs are expected in the not-too-distant future. We consider nonlinear constitutive relationships which, at a given point in the space, present a behaviour for large arguments that is described by monomials of order p and q. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted p-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the $p-$Laplacian in inverse problems with nonlinear materials. Moreover, when p=2, this provides a powerful bridge to bring all the imaging methods and algorithms developed for linear materials into the arena of problems with nonlinear materials. The main result of this work is that for "large" Dirichlet data in the presence of two materials of different order (i) one material can be replaced by either a perfect electric conductor or a perfect electric insulator and (ii) the other material can be replaced by a material giving rise to a weighted p-Laplace problem.

Resonant modes determine the response of electromagnetic devices, including dielectric and plasmonic resonators. Relying on the degrees of freedom that metamaterials provide, this contribution shows how to design, at will, the resonant modes of a dielectric object placed in an unbounded space. Specifically, the proposed method returns in analytical form the spatial distribution of the dielectric susceptibility tensor for which the object exhibits resonances at prescribed frequencies and spatial distribution of the polarization. Together with the synthesis of the material, two key concepts are introduced: the controlled tunability of the resonant modes and the number of essential modes, i.e. the number of modes that uniquely characterize the spatial distribution of the dielectric susceptibility. Moreover, this approach can be applied to design the resonant modes of any system where the constitutive relationship is linear and local.

This paper introduces dimensional analysis in Non-Destructive Testing & Evaluation (NDT&E) problems. This is the first time that this approach is adopted in the framework of NDT\&E, and the paper opens to the development of probes and methods to simultaneously estimate several parameters with a simple approach. The most important theorem of dimensional analysis is the Buckingham's \textpi \ theorem, based on the concept that the laws of the physics do not depend on the particular set of units chosen. The core of this theorem is a systematic reduction in the number of variables describing a physical problem. This reduction is equal to $k$, the number of fundamental dimensions required to describe the variables of the physical problem in its original setting. This makes the approach ideal when the number of variables of the physical problem is not much greater than $k$. In this work, we demonstrate the effectiveness of the approach for the simple problem of the simultaneous estimation of the thickness and electrical conductivity of a conducting plate, via Eddy Current Testing. The approach is original, effective, efficient, and currently patent-pending. All the aspects, from theory to experimental validation, are provided, and it is proved that the proposed method achieves a very good accuracy over a wide range of thicknesses and electrical conductivities. Moreover, the proposed method is compatible with the in-line and real-time estimation of thickness and electrical conductivity in an industrial environment.

We introduce the longitudinal and transverse static surface modes and use them to solve the full-wave electromagnetic scattering problem from penetrable objects. The longitudinal static modes are the eigenmodes with zero surface curl of the electrostatic integral operator that gives the tangential component of the electric field, as a function of the surface charge density. The transverse static modes are the eigenmodes with zero surface divergence of the magnetostatic integral operator that returns the tangential component of the vector potential, as a function of the surface current distribution. The static modes only depend on the shape of the object, thus, the same static basis can be used regardless of the frequency of operation and of the material constituting the object. We expand the unknown surface currents of the Poggio-Miller-Chang-Harrington-Wu-Tsai surface integral equations in terms of the static surface modes and solve them using the Galerkin-projection scheme. The static modes expansion allows the regularization of the singular integral operators and yields a drastic reduction of the number of unknowns compared to a discretization based on sub-domain basis functions. The introduced expansion significantly reduces the cpu-time required for the numerical solution of the scattering problem from particle arrays.

The inverse problem treated in this article consists in reconstructing the electrical conductivity from the free response of the system in the magneto-quasi-stationary (MQS) limit. The MQS limit corresponds to a diffusion PDE. In this framework, a key role is played by the Monotonicity Principle, that is a monotone relation connecting the unknown material property to the (measured) free-response. MP is relevant as basis of noniterative and real-time imaging methods. Monotonicity Principles have been found in many different physical problems governed by PDEs of different nature. Despite its rather general nature, each different physical/mathematical context requires to discover the proper operator showing MP. For doing this, it is necessary to develop ad-hoc mathematical approaches tailored on the specific framework. In this article, we prove a monotonic relationship between the electrical resistivity and the time constants characterizing the free-response for MQS systems. The key result is the representation of the induced current density through a modal representation. The main result is based on the analysis of an elliptic eigenvalue problem, obtained from separation of variables., Comment: 25 pages, 4 figures