In the literature about interpolation and approximation, the classical approach consists in using linear methods. It is known that these methods are convergent and stable for smooth functions but that they introduce Gibbs effect and diffusion when dealing with piecewise continuous functions. These numerical effects make it difficult to use linear methods for solving many problems where solutions present discontinuities or where we need to deal with discontinuous data. Examples are the solutions of PDEs that arise from conservation laws and that contain shocks in the flux or the approximation of piecewise continuous functions in CAGD (computer aided geometric design). In the past years, many methods that were originally designed for solving problems in the context of conservation laws where lately applied in other fields such as interpolation or CAGD. This is the case of ENO (essentially non oscilatory), WENO (weighted essentially non oscillatory) methods and, more recently, the IIM (inmerssed interface method). The PI of this proyect is a world leading expert in the IIM method. During the past 30 years his research has been focused on the design and the application of this method in the solution of a myriad of problems that arise in the fluid dynamics field. More specifically, the IIM has been applyed to the solution of elliptic and parabolic problems with moving and steady interfaces. Even though, this method has not been yet applied to the solution of conservation laws. In Sergio Amat, Z. Li, Juan Ruiz. On a New Algorithm for Function Approximation with Full Accuracy in the Presence of Discontinuities Based on the Immersed Interface Method. J. Sci. Comput., 75 (3), 1500-1534 (2018), the authors use the IIM to design an adapted Newton interpolation that is capable of rising the accuracy close to discontinuities of the function or its derivatives. This method is capable of compete with Harten's ENO-subcell resolution algorithm. ENO-SR was originally designed to deal with the linearly degenerate characteristic field in the Euler equations in order to improve the resolution of contact discontinuities. Our first objective is to check if the IIM applied to this kind of problems can improve the results attained by ENO-SR. Our second objective stands in the line opened in the article by Amat, Li and Ruiz in the sense that we plan to apply the IIM to other classical methods that are often used in CAGD. We are already working in a Hermite spline nonlinearly modified using the IIM in order to adapt this kind of interpolation to the presence of discontinuities. Summarizing, our second objective is to explore the possibilities of the IIM in the CAGD field.

Overview: This proposal concerns the development and analysis of some new ideas for interface, and fluid structure interactions problems and applications based structured meshes. In the past, the PI and PI������������������s collaborators have developed the immersed interface method (IIM), non-conforming and conforming immersed finite methods (IFE), and augmented IIM and IFEM for interface problems. Since then IIM and IFE methods have attracted a lot of interests in the literature. In this proposal, we propose several novel ideas that would make significant steps forward in these research directions. Our proposed new methods will be backed up by rigorous mathematical analysis and numerical experiments. Applications include elasticity problems, anisotropic elliptic interface problems, optimal control of interface problems, fluid structure interactions in modeling tissue mechanics coupled with cell biology, and tip propagation of a crack using the the Mumford-Shah energy. Intellectual Merit: Many important applications in computational fluid mechanics, mathematical biology, material sciences, porous media flow \emph{etc.}, involve interface and fluid structure interaction problems. Solving such problems is always challenging because the problems involve discontinuities in physical parameters, free boundaries, moving interfaces, or irregular domains, and different types of governing equations. In this proposal, we propose some new innovative finite difference and finite element methods based on structured meshes. Physical interface or boundary conditions are enforced so that high order accuracy can be achieved using structured meshes. We will also apply the new methods to several important applications that either have not been solved or not to the level of satisfaction. For some applications, accurate sharp interface solutions are necessary to get meaningful results. Our proposed methods will be built on our previous research on interface problems and are based on structured meshes such as Cartesian meshes that are not necessarily aligned with the interface in two and three dimensions. Our proposed research projects include: (1), new augmented methods for a fluid structure interaction between a fluid flow modeled by Stokes or Navier-Stokes equations and a porous media modeled by the Darcy������������������s law; (2), a new computational framework for accurate gradient computation on a boundary or interface with accurate solution globally; (3), a new second order symmetric, consistent, and parameter free immersed finite element method; (4), a new SVD free augmented IFE method for elliptic interface problems with non-homogenous jump conditions; (5), applications of the new methods for cell deformation in different layers (porous media, Stokes flow, Navier-Stokes flow) and scales; (6), numerical simulations of a crack prorogation based on the Mumford-Shah minimizer. Broader Impacts: The impact of the proposal will be significant if the projects described in this proposal succeed, which we are confident from our preliminary results and our experiences in this area. (1) We will have an efficient framework for multi-phase and multi-physics problems; (2) We will have a new computational framework for computing a high order gradient on a boundary or interface; (3) We will have new efficient immersed finite element methods (in both 2D and 3D) based on a new formulation and on structured meshes; The proposed new methods are based on sharp interface models (no smearing) in which the jump conditions are enforced. (4) We will be able to solve some important application problems in elasticity and fluid structure interaction between a porous media and a fluid flow. (5) This proposal will have positive effect on education by attracting graduate students and possible postdoctoral researchers to conduct research in this area. Some components of the projects will be designed as undergraduate projects. Our research results can lead to a software package useful to computational science involving discontinuities and singularities, free boundary/moving interface, multi-phase and multi-physics, and irregular domain problems.

Objectives The living world is embedded in the physical world. Small organisms live in a world of diffusion. The rest of us require a system of ducts for transport of all the materials we need to live. Systems of branched ducts are found in lung, kidney, mammary gland, and many other organs. This project?s primary objective is to understand the mechanisms of tissue dynamics that create branched syst-ems. It also aims to clarify the best ways to work with continuum mechanical models of morphogenesis to realistically describe mechanics and transport in developing tissues. Additionally, the project will develop new numerical methods for mixture models with interfaces. We will then test these models in a real time living branching system, the early embryonic lung. Methods This project coordinates a lung biologist, a tissue modeler, and an expert in numerical methods to study the mechanisms of branching morphogenesis. They will work together to o translate verbal hypotheses into a set of mathematical models o design and perform numerical experiments to test mechanism feasibility and to generate predictions o develop high-accuracy stable numerical methods to solve the PDEs of mixtures by front tracking o create simulations of the model partial differential equations o design and perform experiments to test the model predictions and calibrate the model o refine models and hypotheses.

We propose research with three objectives: (i) Material interrogation with uncertainty; (ii) Theoretical/computational efforts on random/stochastic dynamical systems: Estimation and control; and (iii) Just-in-Time networks with stochasticity/uncertainty.

This proposal concerns the development and analysis of a systematical new numerical methods for Navier Stokes equations involving free boundary and moving interface. Applications include one-phase and two-phase (with discontinuous viscosity and density) moving contact line problems; interfaces with open-ended interface; incompressible (nonextensible) interfaces in incompressible flows, interfaces with masses and other applications in computational fluid dynamics (CFD)

The objective of the effort discussed in this project is to develop an innovative interdisciplinary program on high-fidelity simulation of energy recovery from oil shale and other unconventional energy resources using advanced computational fluid dynamics methodology. A primary focus of this project is to establish a research center that will provide our expertise and capability in numerical simulation and modeling to address some challenging problems in different INL (Idaho national laboratory)research project and eventually be funded by INL.

This proposal is to renew our ARO research grant ``Sharp Interface Methods for Moving Interface/Free Boundary Problems and Applications'' with new focus and projects. One of objectives of the proposal is to develop optimal control and inverse methods for flow and scattering problems using the accurate forward solver that we plan to develop; and to carry out the stability and sensitivity analysis with respect to physical parameters and functions including shape of domains for the problems of interest. Another objective of this proposal is to develop accurate sharp interface methods based on Cartesian grids for incompressible flows with free boundary and moving interface, and EM and elastic wave propagations (solver for forward problems). Our proposed work will be based on the new augmented immersed interface method in which some augmented variables are introduced to simplify problems, get accurate discritizations, and utilize existing fast solvers. In order to cope with multi-physics, multi-scales, and local nature of the problems, local refinement and adaptivity, and domain decomposition technique will be utilized..The application areas include simulations and control of moving contact lines of droplet in one or two-phase flows, moving objects with fixed or deformable boundaries with low or high Reynolds number (for example, turbulent flows); incompressible (non-extensible) interfaces in an incompressible flow, forward and inverse scattering with applications to the object detection and classification.

We are organizing a workshop on "Fluid-Structure Interaction Problems" at the National Center for Theoretical Sciences (NCTS) at Taiwan, May 27-29, 2011. As one of international organizer, the PI is writing this grant for fund to support US participants to this international event.

The main objective of the proposal concerns the development, analysis, implementation, and application of efficient and accurate numerical methods for interface problems, and problems on irregular domains, with applications in computational fluid dynamics, scattering problems, control of incompressible flows modeled by the Navier-Stokes equations, imaging processing, and data mining. Our approach is based on the immersed interface method and provides very sharp interface treatment for discontinuous media, free boundary and moving interface problems. Under the support of the current AFOSR grant, we have made significant progress for interface problems for the time-dependent incompressible flows, the two-phase flows, and compact fourth order MAC schemes for the Stokes equations, and algebraic domain decomposition/embedding method. It is our objective to improve the performance of our methods in terms of efficient and advanced implementation and widening applications of our interface technique, including interfacial flows with surfactant and contact angle problems, capillary flows with insoluble surfactant, open ended interface problems, and scattering from a thin shell. A critical step for advancement includes incorporating adaptivity into our methods in order to cope multi-physics, multi-scaled and local nature of computations. Our effort will focus on the improvement and advancement of our methods and develop new advanced methods based on local refinement near the interface, domain decomposition method and higher order discretizations. Another type of problems of interest are those defined on heterogeneous and random media, and/or infinite domains. These problems have many applications in electro-magnetics, elastic wave propagation in random media, and structural design problems. The accurate discretization along the material interface or boundary may be crucial to develop reliable numerical methods. It is also essential to develop a fast linear solver for the resulting systems. The discretization of such problems leads to very large systems of linear equations which can have from millions ($10^6$) to billions ($10^9$) unknowns. Our solution method utilizes the sparse subspace reduction and will be further improved by the overlapping domain decomposition technique. Our research will also focus on the following new directions that address AFOSR interests in the information science, data mining, uncertainty analysis and statistical optimization. Effective methods for data mining that include the representation, compression and classification of large and complex data sets become increasingly important. For example, our objective includes the reduced-order representation of feedback gain distributions, adaptive mesh generation for finite-element analysis, clustering and segmentation of spatially distributed data for data mining, and information retrieval. Our approach uses the equal volume CVT (Centroidal Voronoi Tessellation). CVT has proven its applicability and feasibility for performing such scientific tasks. The equal volume property of the Voronoi Tessellation is our novelty and should improve the effectiveness and applicability of CVT methodologies significantly. An efficient algorithm based on sequential Gauss-Newton method is developed and its convergence analysis will be carried. Tikhonov regularization techniques play an important role in ill-posed inverse problems and imaging analysis. The regularization parameter must be carefully selected to obtain accurate solutions and the quality of solutions heavily depends on the fine tuning of the regularization parameter. We develop and analyze a novel method of determining the regularization parameter in Tikhonov regularization for a general class of inverse problems and Bayesian imaging formulation. The method detects the noise level of the data and selects the regularization parameter automatically. Our approach is very flexible and can be readily applied to the relevant problems to AFOSR's in

This proposal concerns the development of efficient numerical methods and associated theory for interface problems in time-harmonic acoustic, elastic, and electro-magnetic wave propagation problems in three-dimensional domains. Particularly, we consider exterior problems for acoustic scattering in heterogeneous media, and electro-magnetic scattering by conducting bodies. Some applications for these problems are geological surveys, the detection of solids in sediment, and a radar detection. The developed methods are based on a finite element and/or a finite difference discretization of the corresponding partial differential equation models. Especially, our primary focus is to apply and adapt the immersed interface method to treat the inhomogeneities and the interface conditions. So, our proposed approach is well suited for scattering problems in complicated domains with varying material parameters and interface conditions.