Sava Dediu-Research Interests My research interests include inverse problems for partial differential equations (in particular inverse problems for wave propagation phenomena), sensitivity analysis for dynamical systems, and model-oriented experimental design. Inverse problems are defined, as the term itself indicates, as the inverse of direct or forward problems. They are encountered typically in situations where one makes indirect observations of a quantity of interest. Common examples include computerized tomography, image deconvolution, inverse scattering problems, inverse boundary value problems, parameter estimation problems, etc. A particularly important category of inverse problems are those for wave propagation phenomena, where the aim is to detect/identify unknown objects from indirect, incomplete or noisy measurements through the use of acoustic, electromagnetic or elastic waves. The applications of inverse problems in practice are ubiquitous and can be found in medical imaging, nondestructive testing, seismic exploration, underwater imaging, microscopy, atmospheric sciences, astronomy as well as life sciences. I became interested in inverse problems as a graduate student at Rensselaer Polytechnic Institute, where I studied the scattering of acoustic waves and numerical techniques to simulate wave propagation and where I completed my dissertation with a thesis on recovering inhomogeneities in waveguides. As a postdoctoral research associate at the Statistical and Applied Mathematical Sciences Institute (SAMSI) and Center for Research in Scientific Computation (CRSC), North Carolina State University, I have become interested in sensitivity analysis for dynamical systems and model-oriented experimental design. The motivation for sensitivity analysis is driven by the interdisciplinary character of the scientific research today when mathematical models based on systems of algebraic or differential equations are commonly used. If the sensitivity of the model outcome to the various parameters can be determined, then we have a chance of learning how to control the system to optimize the outputs or to avoid undesirable system outcomes. However, due to an increasing need to better describe social, physical or biological reality, and supported by a tremendous improvement in computing technology, in recent years these models have became more complex and more sophisticated, and the number of variables and parameters utilized have increased considerably. This naturally has motivated new questions of determining the relative importance of parameters, the effect on the model output of variation in parameters, the uncertainty in the model results due to the uncertainty of parameters, etc. Answers are not obvious for large, complex models and the associated problems provide significant new challenges for scientists. During my appointment at SAMSI and CRSC, I started working on two research projects involving sensitivity analysis. One required developing a robust theoretical and computational framework for the sensitivity of a general nonlinear dynamical system depending on parameters in a convex subset of a topological vector space, and the other one required modeling the impact of disturbances in fragile systems such as agricultural production networks. In addition to sensitivity analysis, I have become interested in model-oriented experimental design while investigating the relevance (with respect to the information content) of data measurements for the identification of certain parameters in a Verhulst-Pearl logistic growth model and a recently developed agricultural production network model. The motivation for studying experimental design is twofold and comes from both the scientific and the financial aspects of the problem. On one hand, as part of model validation and verification, one typically needs to estimate model parameters from data measurements as accurately as possible, and a related question of paramount interest is related to sampling; specifically, at which time points the measurements are most informative in the estimation of a given parameter. On the other hand, quite often the process of data collection in an experiment is an expensive one, requiring time, equipment, qualified personal and other financial means. An optimal design can potentially indicate to the experimentalist how to use these resources in the most efficient way. Recovering Inhomogeneities in a Waveguide using Eigensystem Decomposition My thesis research consisted in determining the index of refraction in a compact region of a medium in a waveguide with known background, from knowledge of the far-field scattered acoustic waves. This inverse problem was motivated by ocean acoustics in shallow waters, where one wants to localize and identify inhomogeneities and objects (submarines, mines, sand banks, navigational obstacles, etc.) submerged in the continental shelves of the oceans. Here, the depth of the water stays almost constant for large distances in the range and the approximation with a rectangular waveguide is very good. The mathematical model for this problem is based on the Helmholtz equation in a two-dimensional waveguide slab, infinite in the horizontal direction, with Dirichtet and Neumann boundary conditions respectively at the top and bottom, and radiation conditions at infinity for the two openings. Due to the particular geometry of the waveguide, which supports only a finite number of propagating modes, and where a scattered acoustic wave bounces back and forth between the surface and the bottom, the problem of recovering inhomogeneities in a waveguide has a different set of challenges than the corresponding problem in free space. As a research assistant under the guidance of Professor Joyce McLaughlin at Rensselaer Polytechnic Institute, I developed a new method to recover inhomogeneities in waveguides from measurements of the far-field scattered acoustic waves. This method takes advantage of the spectral properties of the far-field matrix when one uses propagating modes of the waveguide as incident fields. The problem of finding the unknown inhomogeneity reduces to a problem of solving a finite system of linear equations whose coefficients depend on the background medium and the eigenvalues and eigenvectors of the far-field matrix. It is worth pointing out, that unlike other inversion algorithms, in the numerical implementation of this method one never solves a constrained optimization problem. Solving a constrained optimization problem in practice is usually carried out by iterative algorithms which are often time-consuming procedures. Instead, by using the analytic derivation of the optimality condition for this problem, one obtains a representation for the linearized solution simply by solving a finite system of linear equations. In addition, by pre-computing parts of the coefficients of this system which depend only on the background medium, this method results in a extremely fast (real-time) algorithm. In order to obtain synthetic data and to apply the inverse algorithm described above, I needed first to solve the direct problem, that is, to solve the Helmholtz equation in the infinite waveguide slab with incident propagating fields coming from the sides. Since the inverse problem problem is ill-posed, it is crucial to have a very good solver for the direct problem which to provide reliable, accurate synthetic data. Although currently there are very efficient numerical solvers for the Helmholtz equation in bounded domains, there is still a challenge to solve the Helmholtz equation numerically in infinite domains, since it requires a careful treatment in the implementation of the so-called Dirichlet-to-Neumann (DtN) boundary conditions. These nonlocal boundary boundary conditions, also called transparent boundary conditions, are essential in reformulating a scattering problem on an unbounded domain as an equivalent problem on a bounded one. I wrote a fourth order accurate finite difference solver for the direct problem, and by implementing the inverse algorithm with the synthetic data obtained, I was able to reconstruct various inhomogeneities present in the waveguide. It was shown that even with as few as seven propagating modes a good recovery of the size and shape of the inhomogeneity is obtained; multiple inhomogeneities can be well recovered and, as expected, the recovery improves when the number of propagating modes increases. The exact formulation of the problem along with the description of the method and its applications and results were published in in Inverse Problems. An additional existence and uniqueness result for the corresponding direct problem will be submitted for publication soon. My current research as a postdoctoral fellow under the guidance of Professor H. T. Banks at the Center for Research in Scientific Computation (CRSC), North Carolina State University is focused on sensitivity analysis for dynamical systems and model-oriented experimental design. Traditionally, sensitivity analysis refers to a procedure used in simulation studies (direct problems) where one needs to evaluate the effects of parameter variations on the time course of model outputs and to identify the parameters or the initial conditions to which the model is most/least sensitive. Equations for the sensitivity of a system with respect to finite dimensional vector parameters are used in standard methodologies for optimization and inverse problems including least squares, maximum likelihood, computation of standard errors in statistics, etc., as well as in model discrimination/model selection related quantities (dispersion matrix, Fisher information matrix, etc). In later years however, due to an increasing interest in incorporating uncertainty into models and in ascertaining the sensitivity of parameter estimates with respect to data measurements, this concept broadened its semantic significantly. The need to employ dynamics with probabilistic structures has recently received increased emphasis in important applications in biology, physics, and materials science. In particular, systems with probability measures or distributions embedded in the dynamics (problems involving aggregate dynamics) represent an important class of problems where the parameter sets are infinite dimensional, and for which performing a sensitivity analysis leads to new theoretical and computational challenges.  One of the first research projects I worked on at CRSC, involved studying the sensitivity of a general nonlinear dynamical system depending on parameters in a convex subset of a topological vector space. More specifically, my colleagues and I investigated the sensitivity with respect to $P$ of systems of the form \dot{x}(t) = F(t, x(t), P) where $P\in \mathcal{M}$ is a probability distribution or measure, which may be discrete, absolutely continuous or a combination of both. The motivation for this work comes from the fact that in many applications of practical interest, the parameter space or set $\mathcal{M}$ is not a Banach space (or even a linear space) and the sensitivity analysis previously developed is therefore not applicable. One of the most important example which falls in this category involves the set $\mathcal{P}(S)$ (or $\mathcal{M}$ a convex subset of $\mathcal{P}(S)$), the general metric space of probability measures defined on a set $S$, taken with the Prohorov metric, when $\mathcal{P}(S)$ is viewed as a subset of $X = (C^*_B(S); \mathcal{T}_{w^*})$, the dual of the bounded continuous functions on $S$ with the weak-star topology. Performing a sensitivity analysis with respect to $P$ for the above system raises a new set of both theoretical and computational challenges since the topological vector space structure of $X$ (and the underlying convex subset $\mathcal{M}\subset X$) does not allow the usual framework where one can define and use a Frechet derivative of $x$ with respect to $P$. However, using the convexity assumption for $\mathcal{M}$, my colleagues and I showed how one can perform a sensitivity analysis for this problem by means of \emph{directional derivatives} of $x$ only. To illustrate the practical applicability of this work, we performed a sensitivity analysis with respect to a class of mixed probability distributions (which have both an absolutely continuous and a saltus component) for two dynamical systems possessing similarities to those arising in modeling HIV cell dynamics. We also established a rigorous theoretical and computational framework (existence and uniqueness results, continuous dependence of the solution $x$ on the measure $P$, existence of directional derivatives of $x$ with respect to $P$, and derivation of the sensitivity equations they satisfy) to treat this kind of problems. Common examples for real world applications where this theory applies include models with delays and hysteresis arising in HIV infection modeling, electromagnetics, vaccine production modeling, polymeric materials, communication networks, etc., where probabilistic uncertainty is present in a significant way. This work has led to the publication of two papers in Mathematical Biosciences and Engineering and Annual Reviews in Control Modeling the impact of disturbances in fragile systems Another research project in which I was heavily involved concerned modeling the impact of disturbances in fragile systems such as agricultural production networks. This project was part of the 2005-2006 SAMSI Program on National Defence and Homeland Security and required close collaboration with statisticians, scientists from veterinary medicine, immunologists and applied mathematicians. The motivation for this work comes from its potential applicability to a wide range of problems of national interest related to bioterrorism. These include the investigation of the spread of diseases through spatially or structurally distributed dynamic populations (contagious infections through highly mobile and/or age-structured human and animal populations) but also a wide range of perturbations other than disease in supply networks (e.g. a hurricane that throttles inter-farm transportation for a short duration, or loss of capacity at a given node such as a factory being shut down for some reason, etc). The current production methods for livestock follow the just-in-time philosophy of manufacturing industries. Feedstock and animals are grown in different areas. Animals are moved from one farm to another, depending on their age. Unfortunately, shocks propagate rapidly through such systems; an interruption to the feed supply has a much larger impact when farms have minimal surplus supplies in-stock than when they have large reserves. The just-in-time movement of animals between farms serves as another vulnerability. Stopping movement of animals to and from a farm with animals infected by a disease will have effects that quickly spread through the system. Nurseries supplying the farm will have nowhere to send their animals as they grow up. Finishers and slaughterhouses supplied by the farm will have their supply interrupted. My colleagues and I used statistical and mathematical modeling ideas to address the above issues, using the North Carolina swine industry's potential response to Foot and Mouth Disease (FMD) as an example. We focused our attention on the North Carolina swine industry because it is both the second largest swine industry in the United States, and local to us. Our research group considered a simplified swine production network with four aggregated nodes (sows, nurseries, finishers and slaughter houses) and we modeled the evolution of the network as a continuous time discrete state Markov Chain. We also made the assumption that we deal with a \emph{closed network}, which is a realistic approximation when the network is efficient and operates at or near capacity (i.e, when the number of animals removed from the chain are immediately replaced by new production/growth, avoiding significant idle times). The stochastic system obtained typically cannot be solved for a stationary distribution and an empirical approach based on the so-called Gillespie algorithm needs be used to investigate the long term behavior of the system. However, using the Strong Law of Large Numbers for Poisson processes we argue that the approximate large population behavior of this system is equivalent to a deterministic limit model arising from ensemble averaging. This deterministic system is instrumental for our modeling goals, since not only facilitates numerical simulations and parameter estimation, but is also readily available for sensitivity and generalized sensitivity analysis studies which allows us to develop a model that could be used to investigate how small perturbations to the agricultural supply system (such as disease that impairs the network) would affect its overall performance. Having established a basic model for the movement of animals in the agricultural system (e.g. pork industry of North Carolina), we finally incorporate an SIR-type infection into the system and present simulations to illustrate the spread of Foot and Mouth Disease throughout the aggregated agricultural network. This research effort was supported by the National Science Foundation and the results obtained were published in a paper in Mathematical Biosciences and Engineering. Model-oriented experimental design I have recently become interested and started working on model-oriented experimental design for parameter estimation problems in nonlinear dynamical systems. The motivation for this research came from the need to quantify the information content of data measurements in the identification of certain parameters in a Verhulst-Pearl logistic model and an agricultural production model. In general, as part of the validation and verification step in mathematical modeling, one typically needs to estimate model parameters from observations as accurately as possible. Traditionally, this is done by mutually respecting the following order: experimentalists collect a set of data measurements more or less at their will  a mathematical model is chosen  an estimation procedure, such as least squares, maximum likelihood, etc., is utilized to obtain the estimates.  In our investigations, my colleagues and I follow the model-oriented experimental design philosophy, and change the above order. We assume that the only things we know at the beginning are the the physical problem and its underlying mathematical model (assumed to be fixed), but we have the freedom to tell the people who conduct the experiment how many observations to take, and more importantly at what times to take them. This raises three extremely important questions for parameter estimation: How do we choose the sampling time interval How many sample points do we take? How do we distribute them in the sampling interval such that to obtain more accurate results? Given these facts, it makes sense, that once a mathematical model was fixed and an algorithm to perform the inversion (least squares for example) was chosen, to look for a strategy to collect experimental data in an optimal way, such that to take maximum advantage of the structure (with respect to the information content) of the underlying mathematical model. Intuitively the answer to the quest for such strategy should be looked at the information content imbedded in the mathematical model throughout the sampling interval. In order to attack this problem we use an integral form of the least squares algorithm, the corresponding integral versions of the Fisher information matrix and of the standard errors, etc. This work is an ongoing project with Professor H.T. Banks and Stacey L. Ernstberger from the Center for Research in Scientific Computation. One paper analyzing this approach is in preparation and is going to be submitted for publication soon. Due to the fact that in practice the components of the parameter estimates are often correlated, the classical sensitivity functions used alone are not efficient in answering these questions because they do not take into account how model output variations affect parameter estimates in inverse problems. In a recent paper accepted for publication in Journal of Inverse and Ill-Posed Problems, my colleagues and I used a new class of sensitivity functions, called generalized sensitivity functions, recently introduced in the literature by Thomaseth and Cobelli, to graphically illustrate the temporal distribution of the information content with respect to various parameters in the two models mentioned above. We also presented discussions on how to use these functions along with the classical sensitivity functions to improve the parameter estimates in inverse problems and therefore to further validate their utility to experimental design problems. Future research I have a long-term interest in inverse problems, sensitivity analysis, model-oriented experimental design and numerical methods in general, and I plan to continue my research in these directions even after my current postdoctoral appointment. It is my intention to continue exploring different theoretical and numerical methods, and keep up with, as well as contribute to the advances in these fields. In particular, in the area of inverse problems, I plan to investigate the near-field waveguide problem. In that case some of the evanescent modes (i.e. modes corresponding to negative eigenvalues) need to be taken into account. Some of the first evanescent modes, whose eigenvalues are negative but close to zero, carry enough energy to be taken into account for reconstruction. I would like to investigate what role these modes play and when they should be included in the reconstruction algorithm. If one allows the evanescent modes to be taken into account, the far-field matrix will then become an operator from $l^2$ to $l^2$. In this case two important questions arise: is this operator compact?, and does this operator have dense-range in $l^2$? Answering these questions will advance our ability to develop a robust theory for determining the scattering potential, i.e. the unknown inhomogeneity. Also, it would be very interesting to test the stability of our method with respect to the assumed background medium, since in practice the background medium is not usually completely known. Another interesting problem I would like to consider is whether the eigensystem decomposition approach can be extended to the related problem of obstacle scattering in waveguides. In the area of sensitivity analysis for dynamical systems, I plan to focus my attention on the development of a general mathematical and statistical methodology for estimation of uncertainties (in this case represented by probability distributions $P$) in the dynamics of a system. The motivation for this research comes from the fact that in many applications with probability measures embedded in the dynamics, the probability measure $P$ is generally unknown and needs to be estimated from aggregate or population level (as opposed to individual level) observations or data. For the systems of interest here, the goal is to develop a rigorous mathematical/statistical framework for OLS estimators (a theory for the asymptotic properties of the OLS estimators, confidence intervals in a functional setting), when the parameters $P\in \mathcal{P}(S)$ to be estimated are actually probability distributions in an infinite dimensional set. Finally, in the area of experimental design, I plan to extend my investigations on problems concerning the optimal choice of inputs for various inverse problems (optimal distribution of incident fields in inverse scattering problems, optimal time sampling for parameter estimation, etc.) which until recently were not much explored. As an applied mathematician I have a natural tendency towards real world problems and interdisciplinary research. I am very interested in developing collaborations with scientists working in other fields to explore how mathematics, through the process of modeling and its great computational apparatus, can bring innovation and help solving the open problems existing in their fields. On the other hand, I believe that interdisciplinary work can help mathematicians better understand the intricacies of physical or biological applications and challenge them to come up with better models and develop new mathematical techniques.