Numerical Homogenization of Heterogeneous Fractional Laplacians

In this paper, we develop a numerical multiscale method to solve the fractional Laplacian with a heterogeneous diffusion coefficient. When the coefficient is heterogeneous, this adds to the computational costs. Moreover, the fractional Laplacian is a nonlocal operator in its standard form, however the Caffarelli-Silvestre extension allows for a localization of the equations. This adds a complexity of an extra spacial dimension and a singular/degenerate coefficient depending on the fractional order. Using a sub-grid correction method, we correct the basis functions in a natural weighted Sobolev space and show that these corrections are able to be truncated to design a computationally efficient scheme with optimal convergence rates. A key ingredient of this method is the use of quasi-interpolation operators to construct the fine scale spaces. Since the solution of the extended problem on the critical boundary is of main interest, we construct a projective quasi-interpolation that has both $d$ and $d+1$ dimensional averages over subsets in the spirit of the Scott-Zhang operator. We show that this operator satisfies local stability and local approximation properties in weighted Sobolev spaces. We further show that we can obtain a greater rate of convergence for sufficient smooth forces, and utilizing a global $L^2$ projection on the critical boundary. We present some numerical examples, utilizing our projective quasi-interpolation in dimension $2+1$ for analytic and heterogeneous cases to demonstrate the rates and effectiveness of the method

Estimation of Nonparametric Functions in Simultaneous Equations Models, with an Application to Consumer Demand

We present a method for consistently estimating nonparametric functions and distributions in simultaneous equations models. This method is used to identify and estimate a random utility model of consumer demand. Our identification conditions for this particular model extend the results of Houthakker (1950), Uzawa (1971) and Mas-Colell (1977), where a deterministic utility function is uniquely recovered from its deterministic demand function.

A hierarchical finite element Monte Carlo method for stochastic two-scale elliptic equations

We consider two-scale elliptic equations whose coefficients are random. In particular, we study two cases: in the first case, the coefficients are obtained from an ergodic dynamical system acting on a probability space, and in the second the case, the coefficients are periodic in the microscale but are random. We suppose that the coefficients also depend on the macroscopic slow variables. While the effective coefficient of the ergodic homogenization problem is deterministic, to approximate it, it is necessary to solve cell equations in a large but finite size “truncated" cube and compute an approximated effective coefficient from the solution of this equation. This approximated effective coefficient is, however, realization dependent; and the deterministic effective coefficient of the homogenization problem can be approximated by taking its expectation. In the periodic random setting, the effective coefficient for each realization are obtained from the solutions of cell equations which are posed in the unit cube, but to compute its average by the Monte Carlo method, we need to consider many uncorrelated realizations to accurately approximate the average. Straightforward employment of finite element approximation and the Monte Carlo method to compute this expectation with the same level of finite element resolution and the same number of Monte Carlo samples at every macroscopic point is prohibitively expensive. We develop a hierarchical finite element Monte Carlo algorithm to approximate the effective coefficients at a dense hierarchical network of macroscopic points. The method requires an optimal level of complexity that is essentially equal to that for computing the effective coefficient at one macroscopic point, and achieves essentially the same accuracy. The levels of accuracy for solving cell problems and for the Monte Carlo sampling are chosen according to the level in the hierarchy that the macroscopic points belong to. Solutions and the effective coefficients at the points where the cell problems are solved with higher accuracy and the effective coefficients are approximated with a larger number of Monte Carlo samples are employed as correctors for the effective coefficient at those points at which the cell problems are solved with lower accuracy and fewer Monte Carlo samples. The method combines the hierarchical finite element method for solving cell problems at a dense network of macroscopic points with the optimal complexity developed in D. L. Brown, Y. Efendiev and V. H. Hoang, Multiscale Model. Simul. 11 (2013), with a hierarchical Monte Carlo sampling algorithm that uses different number of samples at different macroscopic points depending on the level in the hierarchy that the macroscopic points belong to. Proof of concept numerical examples confirm the theoretical results

Uncertainty quantification for random fields estimated from effective moduli of elasticity

The stochastic finite element method is a useful tool to calculate the response of systems subject to uncertain parameters and has been applied extensively to analyse structures composed of randomly heterogeneous materials. The methodology to estimate the parameters of the random field underlying a stochastic finite element model often utilises the midpoint approximation wherein material properties that are measured over a sample volume are treated as point observations of the random field at the centroid of the sample volume. This paper investigates the error induced by this approximation for the case of effective moduli of elasticity resulting from tensile loading as well as 3 and 4-point bending. A computer experiment has been performed consisting of the generation of synthetic stiffness profiles from a lognormal stochastic process, the calculation of effective properties as weighted harmonic averages and the estimation of random field parameters through the method of moments. The uncertainty in the parameter estimates is quantified and a recommendation is made as to which bending test is superior for obtaining random field parameter estimates with reference to the statistics of the base process and the tensile loading condition

Testable Restrictions on the Equilibrium Manifold

We present a finite system of polynomial inequalities in unobservable variables and market data that observations on market prices, individual incomes and aggregate endowments must satisfy to be consistent with the equilibrium behavior of some pure trade economy. Quantifier elimination is used to derive testable propositions on finite data sets for the pure trade model.General equilibrium, nonparametric restrictions, quantifier elimination

A bioprinted cardiac patch composed of cardiac-specific extracellular matrix and progenitor cells for heart repair

Congenital heart defects are present in 8 of 1000 newborns and palliative surgical therapy has increased survival. Despite improved outcomes, many children develop reduced cardiac function and heart failure requiring transplantation. Human cardiac progenitor cell (hCPC) therapy has potential to repair the pediatric myocardium through release of reparative factors, but therapy suffers from limited hCPC retention and functionality. Decellularized cardiac extracellular matrix hydrogel (cECM) improves heart function in animals, and human trials are ongoing. In the present study, a 3D-bioprinted patch containing cECM for delivery of pediatric hCPCs is developed. Cardiac patches are printed with bioinks composed of cECM, hCPCs, and gelatin methacrylate (GelMA). GelMA-cECM bioinks print uniformly with a homogeneous distribution of cECM and hCPCs. hCPCs maintain >75% viability and incorporation of cECM within patches results in a 30-fold increase in cardiogenic gene expression of hCPCs compared to hCPCs grown in pure GelMA patches. Conditioned media from GelMA-cECM patches show increased angiogenic potential (>2-fold) over GelMA alone, as seen by improved endothelial cell tube formation. Finally, patches are retained on rat hearts and show vascularization over 14 d in vivo. This work shows the successful bioprinting and implementation of cECM-hCPC patches for potential use in repairing damaged myocardium