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Regularization is a long-standing challenge for ill-posed linear inverse problems, and a prototype is the Fredholm integral equation of the first kind with additive Gaussian measurement noise. We introduce a new RKHS regularization adaptive to measurement data and the underlying linear operator. This RKHS arises naturally in a variational approach, and its closure is the function space in which we can identify the true solution. Also, we introduce a small noise analysis to compare regularization norms by sharp convergence rates in the small noise limit. Our analysis shows that the RKHS- and $L^2$-regularizers yield the same convergence rate when their optimal hyper-parameters are selected using the true solution, and the RKHS-regularizer has a smaller multiplicative constant. However, in computational practice, the RKHS regularizer significantly outperforms the $L^2$-and $l^2$-regularizers in producing consistently converging estimators when the noise level decays or the observation mesh refines., Comment: 25 pages

In this work, we investigate the propagation of electromagnetic waves in the Cole-Cole dispersive medium by using the discontinuous Galerkin (DG) method to solve the coupled time-domain Maxwell's equations and polarization equation. We define a new and sharpened total energy function for the Cole-Cole model, which better describes the behaviors of the energy than what is available in the current literature. A major theme in the time-domain numerical modeling of this problem has been tackling the difficulty of handling the nonlocal term involved in the time-domain polarization equation. Based on the diffusive representation and the quadrature formula, we derive an approximate system, where the convolution kernel is replaced by a finite number of auxiliary variables that satisfy local-in-time ordinary differential equations. To ensure the resulted approximate system is stable, a nonlinear constrained optimization numerical scheme is established to determine the quadrature coefficients. By a special choice of the numerical fluxes and projections, we obtain {for the constant coefficient case } an optimal-order convergence result for the semi-discrete DG scheme. The temporal discretization is achieved by the standard two-step backward difference formula and a fast algorithm with linear complexity is constructed. Numerical examples are provided for demonstrating the efficiency of the proposed algorithm, validating the theoretical results and illustrating the behaviors of the energy., Comment: 24 pages, 6 figures, 3 tables

This is the first part of the review article which focuses on theory and applications of Herglotz-Nevanlinna functions in material sciences. It starts with the definition of scalar valued Herglotz-Nevanlinna functions and explains in detail the theorems that are pertinent to applications, followed by a short overview of the matrix-valued and operator-valued versions of these functions and the properties that carry over from scalar cases. The theory is complemented by some applications from electromagnetics that are related to the sum rules. More applications of Herglotz Nevanlinnna functions in material sciences can be found in Part II., Comment: 26 pages, 1 figure

Part II of the review article focuses on the applications of Herglotz-Nevanlinna functions in material sciences. It presents a diverse set of applications with details and the role of Herglotz-Nevanlinna functions clearly pointed out. This paper is concluded by a collection of existent generalizations of the class of Herglotz-Nevanlinna functions that are motivated by potential applications., Comment: 37 pages

This paper concerns the Time-Domain Full Waveform Inversion (FWI) for dispersive and dissipative poroelastic materials. The forward problem is an initial boundary value problem (IBVP) of the poroelastic equations with a memory term; the FWI is formulated as a minimization problem of a least-square misfit function with the (IBVP) as the constraint. In this paper, we derive the adjoint problem of this minimization problem, whose solution can be applied to computed the direction of steepest descent in the iterative process for minimization. The adjoint problem has a similar numerical structure as the forward problem and hence can be solved by the same numerical solver. Because the tracking of the energy evolution plays an important role in the FWI for dissipative and dispersive equations, the energy analysis of the forward system is also carried out in this paper., Comment: 14 pages

Inspired by applications, we study the effect of interface slip on the effective wave propagation in poroelastic composites. The current literature on the homogenization for the poroelastic wave equations are all based on the no-slip interface condition posed on the micro-scale. However, for certain pore fluids, the no-slip conditions are known to be physically invalid. Even though there are results in a few papers regarding porous media with slip condition on the interface, they are for porous media with rigid solid matrix rather than an elastic one. For the former case, the equations for the micro-scale are posed only in the pore space and the slip on the interface involves only the fluid velocity and the fluid stress. For the latter case, both the fluid equations and the elastic equations are posed in the respective phases and the slip conditions involve the velocities on both sides of the interface, rather than just the fluid side. With this slip condition, a variational boundary value problem governing the small vibrations of a periodic mixture of an elastic solid and a slightly viscous fluid is studied in the paper. The method of two-scale convergence is used to obtain the macroscopic behavior of the solution and to identify the role played by the slip interface condition.

We present a fundamentally new regularization method for the solution of the Fredholm integral equation of the first kind, in which we incorporate solutions corresponding to a range of Tikhonov regularizers into the end result. This method identifies solutions within a much larger function space, spanned by this set of regularized solutions, than is available to conventional regularizaton methods. Each of these solutions is regularized to a different extent. In effect, we combine the stability of solutions with greater degrees of regularization with the resolution of those that are less regularized. In contrast, current methods involve selection of a single, or in some cases several, regularization parameters that define an optimal degree of regularization. Because the identified solution is within the span of a set of differently-regularized solutions, we call this method \textit{span of regularizations}, or SpanReg. We demonstrate the performance of SpanReg through a non-negative least squares analysis employing a Gaussian basis, and demonstrate the improved recovery of bimodal Gaussian distribution functions as compared to conventional methods. We also demonstrate that this method exhibits decreased dependence of the end result on the optimality of regularization parameter selection. We further illustrate the method with an application to myelin water fraction mapping in the human brain from experimental magnetic resonance imaging relaxometry data. We expect SpanReg to be widely applicable as an effective new method for regularization of inverse problems., Comment: 22 pages, 7 figures

In this paper, we show that the permeability of a porous {material} and that of a bubbly fluid are limiting cases of the complexified version of the two-fluid models posed in {Lipton_Avellaneda_1990}. We assume the viscosity of the inclusion fluid is $z\mu_1$ and the viscosity of the hosting fluid is $\mu_1\in \mathbb{R}^+$, $z\in\mathbb{C}$. The proof is carried out by the construction of solutions for large $|z|$ and small $|z|$ with an iteration process and the analytic continuation. Moreover, we also show that for a fixed microstructure, the permeabilities of these three cases share the same integral representation formula (IRF) in Equation (99) with different values of contrast parameter $s:=1/(z-1)$, as long as $s$ is outside the interval $[-\frac{2E_2^2}{1+2E_2^2},-\frac{1}{1+2E_1^2}]$, where the positive constants $E_1$ and $E_2$ are the extension constants that depend only on the geometry of the periodic pore space of the material., Comment: 25 pages

In this paper, we investigate wave propagation in orthotropic poroelastic media by studying the time-domain poroelastic equations. Both the low frequency Biot's (LF-Biot) equations and the Biot-Johnson-Koplik-Dashen (Biot-JKD) models are considered. In LF-Biot equations, the dissipation terms are proportional to the relative velocity between the fluid and the solid by a constant. Contrast to this, the dissipation terms in the Biot-JKD model are in the form of time convolution (memory) as a result of the frequency-dependence of fluid-solid interaction at the underlying microscopic scale in the frequency domain. The dynamic tortuosity and permeability described by Darcy's law are two crucial factors in this problem, and highly linked to the viscous force. In the Biot model, the key difficulty is to handle the viscous term when the pore fluid is viscous flow. In the Biot-JKD dynamic permeability model, the convolution operator involves order $1/2$ shifted fractional derivatives in the time domain, which is challenging to discretize. In this work, we utilize the multipoint Pad$\acute{e}$ (or Rational) approximation for Stieltjes function to approximate the dynamic tortuosity and then obtain an augmented system of equations which avoids storing the solutions of the past time. The Runge-Kutta discontinuous Galerkin (RKDG) method is used to compute the numerical solution, and numerical examples are presented to demonstrate the high order accuracy and stability of the method.

In this paper, we derive the augmented Biot-JKD equations, where the memory terms in the original Biot-JKD equations are dealt with by introducing auxiliary dependent variables. The evolution in time of these new variables are governed by ordinary differential equations whose coefficients can be rigorously computed from the JKD dynamic tortuosity function $T^D(\omega)$ by utilizing its Stieltjes function representation derived in \cite{ou2014on-reconstructi}, where an algorithm for computing the pole-residue representation of the JKD tortuosity is also proposed. The two numerical schemes presented in the current work for computing the poles and residues representation of $T^D(\omega)$ improve the previous scheme in the sense that they interpolate the function at infinite frequency and have much higher accuracy than the one proposed in \cite{ou2014on-reconstructi}.

This article focuses on the mathematical problem of reconstructing dynamic permeability $K(\omega)$ of two-phase composites from data at different frequencies, utilizing the analytic structure of the Stieltjes function representation of $K(\omega)$ derived by Avellaneda and Tortquato in \cite{avellaneda1991rigorous-link-b}, which is valid for all pore space geometry. The integral representation formula (IRF) for dynamic tortuosity is derived and its analytic structure exploited for reconstructing the function from a finite data set. All information of pore-space microstructure is contained in the measure of the IRF. The theory of multipoint Pad\'{e} approximates for Stieltjes functions guarantees the existence of relaxation kernels that can approximate the dynamic permeability function and the dynamic tortuosity function with high accuracy. In this paper, a numerical algorithm is proposed for computing the relaxation time and the corresponding strength for each element in the relaxation kernels; in the frequency domain, the approximation can be regarded as approximating the Stieltjes function by rational functions with simple poles and positive residues. With the idea from dehomogenization, we also established the exact relations between the moments of the positive measures in the IRFs of permeability and tortuosity with two important parameters in the theory of poroelasticity: the infinite-frequency tortuosity $\alpha_\infty$ for general case and the weighted volume-to-surface ratio $\Lambda$ for the JKD model. From these relations, we suggest a new way for evaluating these two microstructure-dependent parameters from a finite data set of permeability and given values of porosity $\phi$ and pore fluid kinetic viscosity $\nu$. Numerical results for JKD permeability and tortuosity are presented., Comment: 20 pages, 3 figures

This paper investigates existence of the nonstandard Pade approximants introduced by Cherkaev and Zhang in J. Comp. Phys. 2009 for approximating the spectral function of composites from effective properties at different frequencies. The spectral functions contain microstructure information. Since this reconstruction problem is ill-posed [9], the well-performed Pade approach is noteworthy and requires further investigations. In this paper, we validate the assumption that the effective dielectric component of interest can be approximated by Pade approximants whose denominator has nonzero power one term. We refer to this as the nonstandard Pade approximant, in contrast to the standard approximants with nonzero constant terms. For composites whose spectral function assumes infinitely many different values, the proof is carried by using classic results for Stieltjes functions. For those with spectral functions having only finitely many different values, we prove the results by utilizing a special product decomposition of the coefficient matrix of the Pade system. The results in this paper can be considered as an extension of the Pade theory for Stieltjes functions whose spectral function take infinitely many different values to those taking only finitely many values. In the literature, the latter is usually excluded from the definition of Stieltjes functions because they correspond to rational functions, hence convergence of their Pade approximants is trivial. However, from an inverse problem point of view, our main concern is the existence of the nonstandard Pade approximants, rather than their convergence. The results in this paper provide a mathematical foundation for applying the Pade approach for reconstructing the spectral functions of composites whose microstructure is not a priori known., Comment: 16 pages