Research Statement Catalin S. Trenchea

Transcription

1 Research Statement Catalin S. Trenchea Department of Mathematics Phone: (412) University of Pittsburgh Fax: (412) Pittsburgh PA Research accomplishments. [16] A generalized stochastic collocation approach to constrained optimization for random data identification problems. We introduced a novel stochastic parameter identification algorithm that integrates an adjoint-based deterministic algorithm with the sparse grid stochastic collocation FEM approach. This allows for decoupled, moderately high dimensional, parameterized computations of the stochastic optimality system, where at each collocation point, deterministic analysis and techniques can be utilized. The advantage of our approach is that it allows for the optimal identification of statistical moments (mean value, variance, covariance, etc.) or even the whole probability distribution of the input random fields, given the probability distribution of some responses of the system (quantities of physical interest). [27] Stability of two IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations. We proved the stability of two partitioned methods for decoupling a multi-physics problem. This answered the 1963 unsolved problem of stability of Crank-Nicolson leap-frog method. [35] Stability of partitioned IMEX methods for systems of evolution equations with skew-symmetric coupling. Here I generalize the work [27], providing a novel proof for a whole family of second order IMEX methods that contains CNLF and BDF2-AB2. [36] Unconditional stability of a partitioned IMEX methods for magnetohydrodynamic flows. Using the Elsässer variables I decoupled the full MHD equations at high magnetic Reynolds numbers into two subproblems of half the size, using the same solver for each variable. [25] Analysis of stability and errors of IMEX methods for magnetohydrodynamics flows at small magnetic Reynolds number. We introduce a implicit-explicit method where the MHD equations can be evolved in time by calls to the NSE and Maxwell codes, each optimized for the subproblem s respective physics. [20] Numerical analysis of modular regularization methods for the BDF2 time discretization of the NSE. We give a complete stability analysis and error estimates of an uncoupled, modular regularization algorithm for approximation of the Navier- Stokes equations. [24] Analysis of Long Time Stability and Errors of Two Partitioned Methods for Uncoupling Evolutionary Groundwater - Surface Water Flows. We analyzed and tested two such partitioned (non-iterative, domain decomposition) methods for the fully evolutionary Stokes-Darcy problem, that model the multi-physics coupling of groundwater to surface water, allowing the use of the best groundwater and surface water codes. [34] Second order implicit for local effects and explicit for nonlocal effects is unconditionally stable. The second order method considered is implicit in local and stabilizing terms in the underlying PDE and explicit in nonlocal and unstabilizing terms. Unconditional stability and convergence of the numerical IMEX scheme are proved by the energy method and by algebraic techniques. This stability result is surprising because usually when different methods are combined, the stability properties of the least stable method plays a determining role in the combination. [5],[10]. An efficient and robust numerical algorithm for estimating parameters in Turing systems. We introduce a general methodology for parameter identification in reaction-diffusion systems that display pattern formation via the mechanism of diffusion-driven instability. [21] Modular Nonlinear Filter Stabilization of Methods for Higher Reynolds Numbers Flow. This is a breakthrough work: we develop nonlinear filter stabilizations which tune the amount and location of eddy viscosity to the local flow structures and whether the nonlinearity tends to break marginally resolved scales down to smaller scales or let the local structure persist. [4] Improved accuracy in regularization models of incompressible flow via adaptive nonlinear filtering. We study adaptive nonlinear filtering in the Leray regularization model for incompressible, viscous Newtonian flow. The filtering radius is locally adjusted so that resolved flow regions and coherent flow structures are 1

2 not filtered-out. The method decouples the problem so that the filtering becomes linear at each timestep and is decoupled from the system. [26] The Das-Moser commutator closure for filtering through a boundary is well posed. When filtering through a wall with constant averaging radius, in addition to the subfilter scale stresses, a non-closed commutator term arises. We consider a proposal of Das and Moser to close the commutator error term by embedding it in an optimization problem, and showed that this optimization based closure, with a small modification, leads to a well posed problem. [19][18] Large eddy simulation for turbulent magnetohydrodynamic flows. Approximate deconvolution models for magnetohydrodynamics. We consider the family of approximate deconvolution models (ADM) and the zeroth order model (LES) for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove existence and uniqueness of solutions, we prove that the solutions to the ADM-MHD equations converge to the solution of the MHD equations in a weak sense as the averaging radii converge to zero, and we derive a bound on the modeling error. We prove that the energy and helicity of the models are conserved, and the models preserve the Alfèn waves. [23] Bounds on Energy, Magnetic Helicity and Cross Helicity Dissipation Rates of Approximate Deconvolution Models of Turbulence for MHD Flows. For body force driven turbulence, we prove directly from the model s equations of motion bounds on the model s time averaged energy dissipation rate, time averaged cross helicity dissipation rate and magnetic helicity dissipation rate. [11] Analysis of Nonlinear Spectral Eddy-Viscosity Models of Turbulence. [17] Mathematical Architecture of Approximate Deconvolution Models of Turbulence. [22] Theory of the NS-ω model: a complement to the NS-α model. [12] The velocity tracking problem for MHD flows with distributed magnetic field controls. [7] Optimal control of a nutrient-phytoplankton-zooplankton-fish system. [6] Finite element approximations of spatially extended predator-prey interactions with the Holling type II functional response. [9] A second-order, three level finite element approximation of an experimental substrate inhibition model [8] Spatiotemporal dynamics of two generic predator-prey models. [15] Analysis of an optimal control problem for the three-dimensional coupled modified Navier-Stokes and Maxwell equation. [13] Analysis and Discretization of an Optimal Control Problem for the Time- Periodic MHD Equations. [31] Optimal control of an elliptic equation under periodic conditions. [33] Periodic optimal control of the Boussinesq equation. [3] Noncooperative optimization of controls for time periodic Navier-Stokes systems with multiple solutions. [32] Internal optimal control of the periodic Euler-Bernoulli equation. [30] Optimal control of the periodic string equation with internal control. [28] Identification for nonlinear periodic wave equation. [29] Hamilton-Jacobi equation and optimality conditions for control systems governed by semilinear parabolic equations with boundary control. 2. Research interests. My research is interdisciplinary in nature, combining mathematical analysis, numerical analysis and scientific computing. My main interest is in the analysis and the numerical computations of models that emerge from physical and life sciences, with specific attention to the direct and inverse problems (optimal control / parameter identification) in connection with uncertainty quantification, fluid dynamics, magneto-hydrodynamics, large eddy simulation of turbulent flows and mathematical biology. My expertise lies in the mathematical and numerical analysis of continuous, semi-discrete and fullydiscrete space-time discretizations of deterministic and stochastic partial differential equations, convergence, error estimates, and the development of stable and accurate numerical algorithms. Among the mathematical areas and tools I use are: calculus of variations, adjoint based methods for optimal control, finite element method, collocation and sparse grids for stochastic partial differential equations, semigroup theory, functional analysis, convex analysis, numerical analysis for stiff and nonstiff ordinary differential equations. Modular computations of multi-domain, multi-physics. Many important applications require the accurate solution of multi-domain [24, 27], multi-physics coupling [27, 35], e.g., the coupling of groundwater flows with surface flows [24], fluid-magnetic field interaction [25, 36], atmosphere-ocean coupling, and fluid-structure interaction. One example that occurs in my recent research is the groundwater and surface water interaction. The essential problems of estimation of the penetration of a plume of pollution into groundwater and remediation after such a penetration are that (i) the coupled problem in both sub-regions are inherently time dependent, (ii) the different physical processes suggest that codes optimized for each 2

3 sub-process need to be used for solution of the coupled problem and (iii) the large domains plus the need to compute for several turn-over times for reliable statistics require calculations over long time intervals. One approach, for which numerical analysis and test problems are relatively easy, is monolithic discretization by an implicit method followed by iterative solution of the system with subregion uncoupling in the preconditioner. In practical computing, partitioned schemes (and associated IMEX methods) have been used extensively in the computational practice of multi-domain, multiphysics applications. However, before my work, there was very little rigorous theory of stability, convergence etc. of partitioned methods. They are often motivated by available codes for subproblems and allow parallel, non-iterative uncoupling into independent sub-physics system per time step. This modularity capability is paramount since it enables, with the minimal extra cost of a script function, the exploitation of highly optimized (up-to-date, continuously upgraded as our understanding of physical processes improves, national laboratory supported) legacy codes. The partitioned methods, which require one (per sub domain) solve per time step, are also very attractive from the viewpoint of computational complexity compared to monolithic methods (requiring one coupled, non symmetric system of roughly double in size). Some of my recent publications have aimed at developing new methods and a theoretical foundation for partitioned methods. They address the stability of modular computational algorithms for groundwater and surface water coupling [24, 27], fluid and magnetic field interaction [25, 36], and the general case of ordinary differential equation with a skew symmetric coupling [27, 35]. Large eddy simulations in hydrodynamics and magnetohydrodynamics flows. Turbulence is an omnipresent phenomenon: when fluids are set into motion, turbulence tends to develop. When the fluid is electrically conducting, the turbulent motions are accompanied by magnetic fluctuations. Although terrestrial conducting fluids are infrequent (but include liquid metals and the flow of liquid sodium in the cooling ducts of a fast-breeder reactor), the extraterrestrial world abounds in plasma (99% of all material is plasma-ionized gas). For MagnetoHydroDynamics (MHD) turbulence, numerical simulations play a greater role than they play for hydrodynamic turbulence, since laboratory experiments are practically impossible and astrophysical systems (solar-wind turbulence, the most important system of high-reynolds-number MHD accessible to in situ measurements) are too complex to be comparable with theoretical results. MHD turbulence modulates hydrodynamic turbulence. The beginning point is to apply, find limitations and then generalize the formalism developed for the latter. There are, however, processes in MHD turbulence that have no hydrodynamic counterpart. These processes stem from selective decay properties due to the presence of several ideal invariants, with different decay rates. Conservation of cross-helicity leads to aligned states, while conservation of magnetic helicity yields force-free magnetic configurations, and the existence of an inverse cascade of the magnetic helicity. Turbulence in fluid dynamics. Turbulent flows consist of complex, interacting three dimensional eddies of various sizes down to the Kolmogorov microscale, η = O(Re 3/4 ) in 3d. A direct numerical simulation of the persistent eddies in a 3d flow thus requires roughly O(Re +9/4 ) mesh points in space per time step. Therefore, direct numerical simulation of turbulent flows is often not computationally economical or even feasible. On the other hand, the largest structures in the flow (containing most of the flow s energy) are responsible for much of the mixing and most of the flow s momentum transport. This led to various numerical regularizations [4, 11, 17 22, 26]; one of these is Large Eddy Simulation (LES). It is based on the idea that the flow can be represented by a collection of scales with different sizes, and instead of trying to approximate all of them down to the smallest one, one defines a filter width δ > 0 and computes only the scales of size bigger than δ (large scales), while the effect of the small scales on the large scales is modelled. This reduces the number of degrees of freedom in a simulation and represents accurately the large structures in the flow. If the LES model does not dissipate enough energy, there can be an accumulation of energy around the smallest resolved scales (i.e., wiggles in the computed velocity). The energy dissipation rates in various LES models are adjusted in various ways, such as using mixed models (i.e., adding eddy viscosity) and picking the parameters introduced to match the model s time averaged energy dissipation rate to that of homogeneous, isotropic turbulence. Parameter free (not mixed) models have many advantages and understanding their important statistics, such as their energy dissipation rate, is critical to advancing their reliability. Approximate deconvolution models are used, with success, in many simulations of turbulent flows [17, 18, 23]. They are among the most accurate of turbulence models, and one of the few turbulence models for which a mathematical confirmation of their effectiveness is known. Magnetohydrodynamic turbulence. In many of the electrically and magnetically conducting fluids, turbulent MHD (magnetohydrodynamics [1]) flows are typical [12 15, 18, 19, 24, 36]. The difficulties of accurately 3

4 modelling and simulating turbulent flows are magnified many times over in the MHD case. They are evinced by the more complex dynamics of the flow due to the coupling of Navier-Stokes and Maxwell equations via the Lorentz force and Ohm s law. The MHD turbulent flow has many more parameter regimes and more uncertainties about basic physical mechanisms than turbulent flow of a non conducting fluid. There are self-organization processes in MHD turbulence that have no hydrodynamic counterpart, originating from the existence of several ideal invariants with different decay rates, Alfvèn waves. The conservation of crosshelicity leads to highly aligned states, while the conservation of magnetic helicity yields the formation of force-free magnetic configurations, a inverse cascade of the magnetic helicity, and excitation of increasingly larger magnetic scales. There are several other important flow statistics, among which the time averaged energy, the magnetic and the cross helicity dissipation rates [18, 19, 23]. The experience in turbulence models suggests that one critical factor for a reliable model is that it contains sufficient model dissipation for the required scale truncation without over dissipation of critical structures and important dynamics. The added complexities, length scales, dynamics and the extra variables needed for MHD flows make accurate simulation of MHD turbulence still more challenging. Because of these (and other) factors, the development of accurate and reliable models for MHD turbulence is correspondingly more important than for ordinary turbulent flows and correspondingly less well developed. With my collaborators I investigate the mathematical properties of several models for the simulation of the large eddies in turbulent viscous, incompressible flows, electrically conducting flows and new numerical models that permit long-time simulations. This includes the mathematical and the numerical analysis of new models for turbulent flows and MHD flows [17 19, 23], e.g., stability and error estimation [20, 34], timesteeping schemes [20, 25, 36], spatial filtering and spectral eddy-viscosity models [11, 22], modular linear and nonlinear filters [4, 20, 21], commutators, filtering through the wall [26], computational algorithms, develop new filters specific to MHD turbulence. Optimal control, parameter estimation, uncertainty quantification. Predictions based on computational simulations are central to not only science and engineering, but also to risk assessment and decision making in economic, public policy and many other venues. In all cases, the information used by engineers for design purposes and by those formulating decisions is uncertain in the sense one cannot unequivocally state that it is correct; the best one can hope to do is to quantify the uncertainty in the information, e.g., by providing confidence intervals or error bars. One of the central objectives of the scientific community is to investigate and resolve several important mathematical, algorithmic and practical issues related to the efficient, accurate, and robust computational determination of quantities of interest, i.e., the information used by engineers and decision makers, that are determined from solutions of (stochastic) partial differential equations. Optimal control theory is a mathematical technique used to determine control variables that maximize a performance criterion subject to constraints (state equations). It continues to be an active area of research with applications in many areas. Parameter estimation and data assimilation problems can be formulated in this context by seeking model parameters corresponding to state variables that best approximate observed data in some sense, for example least squares. When a cost functional measuring the discrepancy between solutions and target functions is formulated, constrained optimization techniques are employed (one-shot, sensitivity or adjoint-based methods) to construct an iterative algorithm for estimating model parameters. The curriculum starts with formulation of the inverse problem in an abstract infinite dimensional space framework, then continues with use of functional, convex / non-convex analysis and control theory to prove the existence of optimal solutions (i.e. solutions that minimize the cost functional and satisfy the state equation), and the derivation of the necessary conditions of optimality for the continuous control problem constrained by partial differential equations. The computational procedure emerges with the mathematical and numerical analysis of continuous, semi-discrete and fully-discrete space-time discretizations of the deterministic and stochastic partial differential equations with distributed parameters, associated with forward and inverse problems (optimal control and parameter identification), convergence, error estimates, the development of numerical algorithms and execution of computations. The inverse problems are intrinsically ill-posed in the Hadamard sense, meaning that a solution might not exist, or if it exists is not unique, or does not depend continuously on data. Even if the continuous inverse problem is well posed, when discretization in time and/or space is performed, the approximating inverse problem could become ill-posed. This means lack of smoothness in the solutions with respect to the parameters, which commands regularization techniques and extra-effort (such as the Brezzi-Rappaz-Raviart theory) to prove error estimates and 4

5 the convergence and of the regularized (discrete) problem to the primary (continuous) problem [13, 16]. I successfully solved optimal control and parameter identification problems associated with reaction-diffusion equations and pattern formation [5, 7, 10], the time-periodic elliptic equations [31], wave equation [30], the Euler-Bernoulli equation [32], the Boussinesq equations [33], the Navier-Stokes equations [3, 26], and the magnetohydrodynamic (MHD) equations [12 15]. Motivation for stochastic models. Many applications (especially those predicting future events) are affected by a relatively large amount of uncertainty in the input data such as model coefficients, forcing terms, boundary conditions, geometry, etc. An example includes forecasting financial markets where this may depend on the number of economic factors, number of underlying assets or the number of time points/time steps, human behaviors, etc. Important examples include the enhancement of reliability of smart energy grids, development of renewable energy technologies, vulnerability analysis of water and power supplies, understanding complex biological networks, climate change estimation and design and licensing of current and future nuclear energy reactors. The model itself may contain an incomplete description of parameters, processes or fields (not possible or too costly to measure). There may be small, unresolved scales in the model that act as a kind of background noise (i.e. macro behavior from micro structure). All these and many others introduce uncertainty in mathematical models. Stochastic models give quantitative information about uncertainty. In practice it is necessary to address the following types of uncertainties: aleatoric and epistemic. The aleatoric (random) uncertainty is due to the intrinsic/irreducible variability in the system, e.g. turbulent fluctuations of a flow field around an airplane wing, permeability in an aquifer, etc. The epistemic uncertainty is due to incomplete knowledge, and can be reduced by additional experimentation, improvements in measuring devices, etc., e.g. mechanic properties of many bio-materials, polymeric fluids, highly heterogeneous or composite materials, the action of wind or seismic vibrations on civil structures, etc. Uncertainty quantification (UQ) attempts to quantitatively assess the impact of uncertain data on simulation outputs. In the stochastic context, the forward problem consists in determining, for known probability distribution functions (PDF) of coefficients, the PDF of the (uncertain) outputs of the system. The inverse problem consists in, given a set of measurements that correspond to some statistical quantities of interest (e.g. moments, inverse CDF, etc.) corresponding to the solution of the stochastic partial differential equation (SPDE), determining statistical moments of the input random fields. One approach is by minimization of appropriately formulated cost functionals. In the recent work [16] we introduce for the first time several identification objectives that either minimize the expectation of a tracking cost functional or minimize the difference of desired statistical quantities in the appropriate L p norm, where the distributed parameters/control can both deterministic or stochastic. For a given objective we prove the existence of an optimal solution, establish the validity of the Lagrange multiplier rule and obtain a stochastic optimality system of equations. The modeling process may describe the solution in terms of high dimensional spaces, particularly in the case when the input data (coefficients, forcing terms, boundary conditions, geometry, etc) are affected by a large amount of uncertainty. For higher accuracy, the computer simulation must increase the number of random variables (dimensions), and expend more effort approximating the quantity of interest in each individual dimension. Hence, we introduce a novel stochastic parameter identification algorithm that integrates an adjoint-based deterministic algorithm with the sparse grid stochastic collocation FEM approach. This allows for decoupled, moderately high dimensional, parameterized computations of the stochastic optimality system, where at each collocation point, deterministic analysis and techniques can be utilized. The advantage of our approach is that it allows for the optimal identification of statistical moments (mean value, variance, covariance, etc.) or even the whole probability distribution of the input random fields, given the probability distribution of some responses of the system (quantities of physical interest). We rigorously derive error estimates, for the fully discrete problems, and use them to compare the efficiency of the method with several other techniques. Numerical examples in [16] illustrate the theoretical results and demonstrate the distinctions between the various stochastic identification objectives and the efficacy of our approach. I am collaborating with researchers at Florida State University, Oak Ridge National Laboratory, Clemson University, University of Guelph and Oxford University. My work was partially supported by grants from the Air Force Office of Scientific Research and the University of Pittsburgh. 5

6 3. List of selected publications. A list of selected published work and ongoing projects is given below, with brief descriptions of my research program and collaborations A generalized stochastic collocation approach to constrained optimization for random data identification problems. [16] Collaboration with Max Gunzburger (Florida State University) and Clayton Webster (Oak Ridge National Laboratory) led to a paper submitted to the International Journal of Uncertainty Quantification. In this work we introduce and analyze a novel general methodology for quantifying uncertainty and robust computational determination of quantities of interest. I am particularly proud of this work, and we anticipate the technique developed in here to have a significant impact on a number of interesting physical problems and change the way UQ is performed going forward. Abstract. We present a scalable, parallel mechanism for stochastic identification/control for problems constrained by partial differential equations with random input data. Several identification objectives will be discussed that either minimize the expectation of a tracking cost functional or minimize the difference of desired statistical quantities in the appropriate L p norm, and the distributed parameters/control can both deterministic or stochastic. Given an objective we prove the existence of an optimal solution, establish the validity of the Lagrange multiplier rule and obtain a stochastic optimality system of equations. The modeling process may describe the solution in terms of high dimensional spaces, particularly in the case when the input data (coefficients, forcing terms, boundary conditions, geometry, etc) are affected by a large amount of uncertainty. For higher accuracy, the computer simulation must increase the number of random variables (dimensions), and expend more effort approximating the quantity of interest in each individual dimension. Hence, we introduce a novel stochastic parameter identification algorithm that integrates an adjoint-based deterministic algorithm with the sparse grid stochastic collocation FEM approach. This allows for decoupled, moderately high dimensional, parameterized computations of the stochastic optimality system, where at each collocation point, deterministic analysis and techniques can be utilized. The advantage of our approach is that it allows for the optimal identification of statistical moments (mean value, variance, covariance, etc.) or even the whole probability distribution of the input random fields, given the probability distribution of some responses of the system (quantities of physical interest). Our rigorously derived error estimates, for the fully discrete problems, will be described and used to compare the efficiency of the method with several other techniques. Numerical examples illustrate the theoretical results and demonstrate the distinctions between the various stochastic identification objectives Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows. [36] My recent work on partitioning methods for magnetohydrodynamics, submitted to Applied Mathematics Letters proves, for the first time, that the full evolutionary MHD equations at high Reynolds numbers, can be discretized and decoupled in an unconditional scheme, allowing for larger computations by halving the number of variables on each processor. Abstract. The MHD flows are governed by the Navier-Stokes equations coupled with the Maxwell equations through coupling terms. We prove the unconditional stability of a partitioned method for the evolutionary full MHD equations, at high magnetic Reynolds number, in the Elsässer variables. The method we analyze is a first order, one step scheme, which consists of implicit discretization of the subproblem terms and explicit discretization of coupling terms. Future research. In collaboration with Alexander Labovsky (Michigan Technological University) we are developing a unconditional second order splitting scheme using defect-correction. My PhD student Hoang Tran (University of Pittsburgh) is working on the analysis of a finite element method for a second-order decoupling Stability of partitioned IMEX methods for systems of evolution equations with skewsymmetric coupling. [35] This work, submitted to Journal of Computational and Applied Mathematics, is a generalization of the methods published in the recently published joint paper with William Layton (University of Pittsburgh) [27]. It gives a novel proof for the stability of a family of partitioned methods for decoupling multi-physics problems, which allow the use of off-the-shelf codes with the cost of only one extra subroutine function. Abstract. Stability is proven for an implicit-explicit, second order, two step method for uncoupling a system of two evolution equations with exactly skew symmetric coupling. The form of the coupling studied arises in spatial discretizations of the Stokes-Darcy problem. The method proposed is an interpolation of the Crank-Nicolson Leap Frog (CNLF) combination with the BDF2-AB2 combination, being stable under the time step condition suggested by linear stability theory for the Leap-Frog scheme and BDF2-AB2. 6

7 3.4. Second order implicit for local effects and explicit for nonlocal effects is unconditionally stable. [34] While 1999 when the first order method in [2] appeared, there have been no higher order extensions, despite the intense interest in higher order methods. The methods I propose in this report, submitted to the Electronic Transactions on Numerical Analysis, generalize the first order method of [2] to a whole family of second-order IMEX schemes, including Crank-Nicolson and BDF2 with linear extrapolations. Abstract. A family of implicit-explicit second order time-stepping methods is analyzed for a system of ODEs motivated by ones arising from spatial discretizations of evolutionary partial differential equations. The methods we consider are implicit in local and stabilizing terms in the underlying PDE and explicit in nonlocal and unstabilizing terms. Unconditional stability and convergence of the numerical scheme are proved by the energy method and by algebraic techniques. Future research. Nan Jiang (University of Pittsburgh, a student of William Layton) is currently applying these results to the Navier-Stokes equations with linear and nonlinear filtering Numerical analysis of modular regularization methods for the BDF2 time discretization of the NSE. [20] This work was done in collaboration with William Layton (University of Pittsburgh), Nathaniel Mays (Wheeling Jesuit University) and Monika Neda (University of Nevada Las Vegas), and has been submitted to ESAIM: Mathematical Modelling and Numerical Analysis. Abstract. We consider an uncoupled, modular regularization algorithm for approximation of the Navier- Stokes equations. The method is: Step 1: Advance the NSE one time step, Step 2: Regularize to obtain the approximation at the new time level. Previous analysis of this approach has been for simple time stepping methods in Step 1 simple stabilizations in Step 2. In this report we extend the mathematical support for uncoupled, modular stabilization to (i) the more complex and better performing BDF2 time discretization in Step 1, and (ii) more general (linear or nonlinear) regularization operators in Step 2. We give a complete stability analysis, derive conditions on the Step 2 regularization operator for which the combination has good stabilization effects, characterize the numerical dissipation induced by Step 2, prove an asymptotic error estimate incorporating the numerical error of the method used in Step 1 and the regularization s consistency error in Step 2 and provide numerical tests Analysis of Long Time Stability and Errors of Two Partitioned Methods for Uncoupling Evolutionary Groundwater - Surface Water Flows. [24] Jointly with William Layton (University of Pittsburgh) and our student Hoang Tran (University of Pittsburgh), we have analyzed the long time stability of two partitioned methods for decoupling a multi-physics problem, namely Stokes-Darcy system. This gives to our proposed methods the mathematical justification for use of highly optimized (for each subphysics problem) of existing legacy codes in a black-box manner. It has been submitted to SIAM Journal on Numerical Analysis in 2011 and has been revised 2/2012 in response to positive referee reports. Abstract. The most effective simulations of the multi-physics coupling of groundwater to surface water must involve employing the best groundwater codes and the best surface water codes. Partitioned methods, which solve the coupled problem by successively solving the sub-physics problems, have recently been studied for the Stokes-Darcy coupling with convergence established over bounded time intervals (with constants growing exponentially in t). This report analyzes and tests two such partitioned (non-iterative, domain decomposition) methods for the fully evolutionary Stokes-Darcy problem. Under a modest time step restriction of the form t C where C = C(physical parameters) we prove unconditional asymptotic (over 0 t < ) stability of both partitioned methods. From this we derive an optimal error estimate that is uniform in time over 0 t < Modular Nonlinear Filter Stabilization of Methods for Higher Reynolds Numbers Flow. [21] In this paper, which appeared in Journal of Mathematical Fluid Mechanics, together with William Layton (University of Pittsburgh) and Leo Rebholz (Clemson University), we develop nonlinear filter stabilizations which tune the amount and location of eddy viscosity to the local flow structures and whether the nonlinearity tends to break marginally resolved scales down to smaller scales or let the local structure persist. Indeed, nonlinearity does not break down scales uniformly. Intermittence, nonuniformity, locality and even backscatter are typical features in the energy cascade of high Reynolds number flow problems. We also give a path for further development of models where subsequent improvement (as our understanding of turbulent flows improves) are easy to incorporate into models (with rigorous analytic foundations) and flow simulation codes. Abstract. Stabilization using filters is intended to model and extract the energy lost to resolved scales due to nonlinearity breaking down resolved scales to unresolved scales. This process is highly nonlinear 7

8 and yet current models for it use linear filters to select the eddies that will be damped. In this report we consider for the first time nonlinear filters which select eddies for damping (simulating breakdown) based on knowledge of how nonlinearity acts in real flow problems. The particular form of the nonlinear filter allows for easy incorporation of more knowledge into the filter process and its computational complexity is comparable to calculating a linear filter of similar form. We then analyze nonlinear filter based stabilization for the Navier-Stokes equations. We give a precise analysis of the numerical diffusion and error in this process. Future research. Find the eddy viscosity formula for magnetohydrodynamics flows and coherent structures for which it vanishes by extending Vreman s work [38], and analyze the nonlinear filter based on thus constructed indicator function Improved accuracy in regularization models of incompressible flow via adaptive nonlinear filtering. [4] This work is a result of a collaboration with Abigail Bowers (Clemson University), Leo Rebholz (Clemson University), Aziz Takhirov (University of Pittsburgh, student of William Layton), and it will be published in International Journal for Numerical Methods in Fluids (2012) (appeared online DOI: /fld.2732). Abstract. We study adaptive nonlinear filtering in the Leray regularization model for incompressible, viscous Newtonian flow. The filtering radius is locally adjusted so that resolved flow regions and coherent flow structures are not filtered-out, which is a common problem with these types of models. A numerical method is proposed that is unconditionally stable with respect to timestep, and decouples the problem so that the filtering becomes linear at each timestep and is decoupled from the system. Several numerical examples are given that demonstrate the effectiveness of the method Stability of two IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations. [27] In collaboration with William Layton (University of Pittsburgh) we published in Applied Numerical Mathematics (2012) where we proved the stability of two partitioned methods for decoupling a multi-physics problem. This answered the 1963 unsolved problem of proving the stability of Crank-Nicolson leap-frog method, see e.g. Verwer [37, Remark 3.1, page 6]. This warrants the use of optimized/commercial codes with minimal extra computer time cost. Abstract. Stability is proven for two second order, two step methods for uncoupling a system of two evolution equations with exactly skew symmetric coupling: the Crank-Nicolson Leap Frog (CNLF) combination and the BDF2-AB2 combination. The form of the coupling studied arises in spatial discretizations of the Stokes-Darcy problem. For CNLF we prove stability for the coupled system under the time step condition suggested by linear stability theory for the Leap-Frog scheme. This seems to be a first proof of a widely believed result. For BDF2-AB2 we prove stability under a condition that is better than the one suggested by linear stability theory for the individual methods The Das-Moser commutator closure for filtering through a boundary is well posed. This is joint work with William Layton (U. Pittsburgh), published in Mathematical and Computer Modelling (2011). It is the first (and so far only) proof of well posedness of the Das-Moser closure. Abstract. When filtering through a wall with constant averaging radius, in addition to the subfilter scale stresses, a non-closed commutator term arises. We consider a proposal of Das and Moser to close the commutator error term by embedding it in an optimization probem. The report [26] shows that this optimization based closure, with a small modification, leads to a well posed problem showing existence of a minimizer. We also derive the associated first order optimality conditions Large eddy simulation for turbulent magnetohydrodynamic flows. [19] This article is a collaboration Alexander Labovsky (Michigan Tech) and has been published in Journal of Mathematical Analysis and Applications (2011). We examine in [19] the mathematical properties of a model for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove existence and uniqueness of weak solutions for the simplest (zeroth) closed MHD model, we prove that the solutions to the LES-MHD equations converge to the solution of the MHD equations in a weak sense as the averaging radii converge to zero, and we derive a bound on the modeling error. Furthermore, we show that the model preserve the properties of the 3D MHD equations. In particular, we prove that the kinetic energy and the magnetic helicity of the model are conserved, while the model s cross helicity is approximately conserved and converges to the cross 8

9 helicity of the MHD equations as the radii δ 1, δ 2 tend to zero. Also, the model is proven to preserve the Alfvén waves, with the velocity converging to that of the MHD, as δ 1, δ 2 tend to zero Analysis of Nonlinear Spectral Eddy-Viscosity Models of Turbulence. [11] This paper a joint work with Max Gunzburger (FSU), Eunjung Lee (Yonsei University), Yuki Saka, and Xiaoming Wang (FSU), has been published in SIAM Journal on Scientific Computing (2010) and studies two types of eddy-viscosities: hyperviscosity and nonlinear viscosity. Abstract. Fluid turbulence is commonly modeled by the Navier-Stokes equations with a large Reynolds number. However, direct numerical simulations are not possible in practice, so that turbulence modeling is introduced. We study artificial spectral viscosity models that render the simulation of turbulence tractable. We show that the models are well posed and have solutions that converge, in certain parameter limits, to solutions of the Navier-Stokes equations. We also show, using the mathematical analyses, how effective choices for the parameters appearing in the models can be made. Finally, we consider temporal discretizations of the models and investigate their stability Bounds on Energy, Magnetic Helicity and Cross Helicity Dissipation Rates of Approximate Deconvolution Models of Turbulence for MHD Flows. [23] This is a joint work with William Layton (U. Pittsburgh) and Myron Sussman (U. Pittsburgh), published in Numerical Functional Analysis and Optimization (2010). Abstract. In [23] we consider a family of high accuracy, approximate deconvolution models of turbulent magnetohydrodynamic flows. For body force driven turbulence, we prove directly from the model s equations of motion the following bounds on the model s time averaged energy dissipation rate ε ADM, time averaged cross helicity dissipation rate γ,adm and magnetic helicity dissipation rate γ mag,adm, where U, B, L are the global velocity scale, global magnetic field scale and length scale, R is a dimensionless constant related to fluid and magnetic Reynolds numbers, S is the reciprocal of the product of fluid density times free-space permeability and δ is the LES filter radius. Theorem. ε ADM γ,adm S L γ mag,adm (U 2 + SB 2 ) 3/2 (1 + L ) 1/2 1R δ2 (1 + 2 L 2 ) Approximate deconvolution models for magnetohydrodynamics. [18] This work is in collaboration with Alexander Labovsky and was published in Numerical Functional Analysis and Optimization (2010). Abstract. In [18] we consider the family of approximate deconvolution models (ADM) for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove existence and uniqueness of solutions, we prove that the solutions to the ADM-MHD equations converge to the solution of the MHD equations in a weak sense as the averaging radii converge to zero, and we derive a bound on the modeling error. We prove that the energy and helicity of the models are conserved, and the models preserve the Alfvén waves. We provide the results of the computational tests, that verify the accuracy and physical fidelity of the models An efficient and robust numerical algorithm for estimating parameters in Turing systems. [5] Collaboration with Marcus R. Garvie (University of Guelph) and Philip Maini (University of Oxford, UK) led to a paper published in the Journal of Computational Physics (2010). For the first time we provide an efficient numerical procedure for estimating parameters in Turing systems. We anticipate the methodology developed in this work to have a significant impact on a number of interesting physical problems. Abstract. One of the central challenges in developmental biology is to understand how spatial patterning arises in embryogenesis. Traditional mathematical biology has addressed this issue through a top-down approach, which assumes that either a gradient in signalling chemical (termed morphogen) is set-up to which cells respond by differentiating in a multi-threshold dependent manner (Wolpert, 1969), or that complex spatial patterns are set up in a self-organising manner which then only require a single threshold for differentiation (Murray, 2002, 2003). The most well-known such model is the Turing model in which spatial patterning emerges in a system of reacting and diffusing chemicals via the phenomenon of diffusion-driven instability (Turing, 1952). This idea was extended by Gierer and Meinhardt (1972) to the general patterning principle of short-range activation, long-range inhibition. Several top-down models have been proposed involving different biological hypotheses (for example, cell-chemotaxis (Keller and Segel, 1971), mechanochemical (Oster, 9

10 1983), neural models (Ermentrout) etc) but they all obey this underlying patterning principle. As a result, all these models produce very similar patterns and therefore it makes sense to study the simplest such model, which is the Turing model. The first step towards determining if an observed pattern is a consequence of activator-inhibitor self-organisation is to investigate if such a model system can produce the pattern. While this is by no means a proof that such a pattern in nature is generated by this type of model, it is a necessary starting point. To our knowledge, no systemic procedure exists to investigate this - while simple periodic patterns of stripes and spots can be investigated using linear stabily analysis, more complex patterns are not amenable to such analysis. The goal of in [5] is to propose methods from control theory to address this problem. We introduce a general methodology for parameter identification in reaction-diffusion systems that display pattern formation via the mechanism of diffusion-driven instability. The methodology is illustrated with the Schnakenberg and Gierer-Meinhardt reaction-diffusion systems using PDE constrained optimization techniques. A quadratic cost functional that measures the discrepancy between morphogen concentrations and target concentrations is minimized with the reaction diffusion system as a constraint. The numerical optimal control procedure efficiently and accurately estimates key parameters in the systems concerned Theory of the NS-ω model: A complement to the NS-α model. [22] This is a collaboration with William Layton (U. Pittsburgh) and Iuliana Stanculescu (Nova Southeastern University) that appeared in Communications on Pure and Applied Analysis (2011). Abstract. In [22] we study a new regularization of the Navier-Stokes equations, the NS-ω model. This model has similarities to the NS-model but its structure is more amenable to be used as a basis for numerical simulations of turbulent flows. In this report we present the model and prove existence and uniqueness of strong solutions as well as convergence (modulo a subsequence) to a weak solution of the Navier-Stokes equations as the averaging radius decreases to zero. We then apply turbulence phenomenology to the model to obtain insight into its predictions Mathematical Architecture of Approximate Deconvolution Models of Turbulence. [17] This is a collaboration with Alexander Labovschii, William Layton, Carolina Manica, Monika Neda, Leo Rebholzand and Iuliana Stanculescu, and appeared in Quality and Reliability of Large-Eddy Simulations, by J. Meyers, B. Geurts and P. Sagaut (Eds.) ERCOFTAC Series, Springer, (2008). Abstract. Report [17] presents the mathematical foundation of approximate deconvolution LES models together with the model phenomenology downstream of the theory Analysis of an optimal control problem for the three-dimensional coupled modified Navier-Stokes and Maxwell equations. [15] This is joint work with Max Gunzburger (FSU) that was published in Journal of Mathematical Analysis and Applications (2007). Abstract. In [15] we consider the mathematical formulation and the analysis of an optimal control problem associated with the tracking of the velocity and the magnetic field of a viscous, incompressible, electrically conducting fluid in a bounded three-dimensional domain through the adjustment of distributed controls The velocity tracking problem for MHD flows with magnetic field distributed controls. [12] This is a collaboration with Max Gunzburger (FSU) and Janet Peterson (FSU) which was published in International Journal of Pure and Applied Mathematics (2008). Abstract. In [12] we studied the optimal control problem associated with the 2-D magneto-hydrodynamic system (the Navier-Stokes equation coupled with the Maxwell equation). In a numerical simulation we showed that the flow can be driven from a clockwise spinning initial velocity data to a counterclockwise target using only magnetic field controls The identification of space-time distributed parameters in reaction-diffusion systems. [10] This is joint work with Marcus R. Garvie. Abstract. We consider a general methodology for parameter identification in systems of reactiondiffusion equations. To demonstrate the method we focus on the classic Gierer-Meinhardt reaction-diffusion system. The original Gierer-Meinhardt model [A. Gierer and H. Meinhardt, Kybernetik, 12 (1972), pp ] was formulated with constant parameters and has been used as a prototype system for investigating pattern formation in developmental biology. In our paper [10], the parameters are extended in space and time and used as distributed control variables. In non-dimensional form, the RD system has the following 10