Professor Kazi Ito was selected by the Institute for Mathematics and its Applications (IMA) directors and annual program organizers to participate as a General Member for 4.0 months during the period September 1, 2015-December 15, 2015. The IMA 2015-16 annual program is Control Theory and its Applications. The selection was made on the basis of Professor Ito's knowledge of and expertise in the area of research. While a participant, Professor Ito will participate in research based activities including collaborating with IMA postdoctoral fellows and other visitors and participating in IMA workshops, tutorials, seminars and other activities.

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 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 have developed a numerical method for accurately evaluating scattering waves from elastic targets buried in sediment. It is capable of modeling the scattering of sonar signals by undersea mines located in or near the seabed in littoral environments with smooth or rippled water/sediment interfaces. It is based on a preconditioned iterative method that couples the near-field finite element and the fast far-field Helmholtz solver to evaluate the scattered field with hundreds of wavelength. It enables us to evaluate the scattered field for higher frequencies 15-25 kHz. Our objective is to validate and verify the accuracy of the method by error analysis and compare it against experiments at NSWCPC.

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

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. The contractor shall 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. Both theoretical and computational issues for the Tikhonov regularization shall be investigated. Proposed algorithms shall be analyzed and the verification and validation of algorithms shall be performed based on numerical tests. Also, the proposed method shall be applied to the feature selection and classification method for sonar imaging analysis.

We propose research with three objectives: (i) Development of a conceptual, theoretical, and computational framework for dynamic pursuit/evasion games with uncertainty/stochasticity in the context of network security games or information warfare; (ii) Development of dynamic object identification and stochastic resonance techniques for detecting subliminal objects based on the Fokker-Planck equation; (iii) Development of innovative computational approaches for a general classes of Fokker-Planck systems. In one case we will develop operator splitting ideas; in the other we propose to develop a mapping framework to convert Fokker-Planck models to an equivalent probabilistic formulation permitting fast efficient computational solutions.

The proposed activity develops and analyzes efficient numerical methods for time-harmonic acoustic wave propagation problems in fluids and solids. Particularly, exterior two-dimensional and three-dimensional problems for acoustic scattering in heterogenous media and fluid-structure scattering are considered. Acoustical geological surveys and scattering by solids in layered media are used as model problems. The discretization of the governing PDE models is performed using finite elements on orthogonal rectangular meshes with local adaptation to boundaries and interfaces. Special phase error reducing low-order finite elements are studied and employed. Such discretizations are well-suited for scattering problems in complicated, large domains with varying material parameters and interface conditions. The aim is to solve three-dimensional time-harmonic wave propagation problems in which the diameter of the computational domain is from tens to hundreds wavelengths. These problems leads to very large systems of linear equations which can have from millions to several billions unknowns. Numerical methods for this frequency range are under active research, and no efficient method still seem to exist. The proposed solution methods for the large scale linear systems are based on the GMRES method with a nonoverlapping domain decomposition preconditioner. Very efficient subdomain preconditioners are constructed using fast direct solvers by means of domain embedding. This novelty makes the proposed approach very fast.

We propose research with three objectives: (i) Development of a conceptual, theoretical, and computational framework for dynamic electromagnetic pursuit/evasion games with uncertainty/stochasticity; (ii) Development of inverse problem methodologies (generalized sensitivity functions; asymptotic standard errors) for estimation of infinite dimensional functional parameters including probability measures and temporal/spatial dependent functions in complex nonlinear dynamical systems; (iii) Development of a theoretical and computational framework for a general class of nonlinear complex nodal network models with inherent uncertainties.

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.