SCHOLARSHIP STATEMENT AKHTAR A. KHAN

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1 SCHOLARSHIP STATEMENT AKHTAR A. KHAN My research interests lie broadly in the theory, computation, and applications of inverse problems. Besides this, I am also actively engaged in set-valued optimization, quasi-variational inequalities, optimal control, and uncertainty quantification. In the following, I briefly discuss some of my main research contributions in these areas. 1. Elliptic Inverse Problems: Analysis, Computation, and Applications. Applied models frequently lead to partial differential equations involving parameters attributed to physical characteristics of the model. The direct problem in this setting is to solve the partial differential equation. By contrast, an inverse problem seeks for the identification of the parameters when a measurement of a solution of the partial differential equation is available. For clarification, consider the boundary value problem (BVP) (q u) = f in, u = 0 on, (1.1) where is a sufficiently smooth domain in R 2 or R 3 and is its boundary. The above BVP models interesting real-world problems and has been studied in great detail. For instance, here u = u(x) may represent the steady-state temperature at a given point x of a body; then q would be a variable thermal conductivity coefficient, and f the external heat source. This system also models underground steady state aquifers in which the parameter q is the aquifer transmissivity coefficient, u is the hydraulic head, and f is the recharge. The inverse problem in the context of the above BVP is to estimate the coefficient q from a measurement z of the solution u. A number of methods to the aforementioned inverse problem have been proposed in the literature; most involve either regarding (1.1) as a hyperbolic PDE in q or posing an optimization problem whose solution is an estimate of q. The approach of reformulating (1.1) as an optimization problem is divided into two possibilities, namely either formulating the problem as an unconstrained optimization problem or treating it as a constrained optimization problem, in which the PDE itself is the constraint. Among the optimization-based techniques the output least-squares (OLS) method is among the most widely used methods. The output least-squares approach minimizes q u(q) z 2, (1.2) where z is the data and u(q) solves the variational form of (1.1) given by q u v = fv, for all v H0(). 1 (1.3) A known deficiency of the OLS functional is that it is, in general, nonconvex. There are other functionals that have been used. For example, the equation error method (cf. [1, 2]), consists of minimizing the functional q div(q z) + f 2 H 1 () where H 1 () is the dual of H 1 0() and z is the data. Chen and Zou [8] developed an augmented Lagrangian algorithm to solve the OLS problem by treating the PDE as an explicit constraint. 1

2 Knowles [38] proposed minimizing a coefficient-dependent norm q q (u(q) z) (u(q) z), (1.4) where z is the data and u(a) solves (1.3). Knowles [38] established that the above functional is convex. Some related developments are given in [4, 5, 13, 39]. It is convenient to investigate the inverse problem of parameter identification in an abstract setting allowing for more general PDEs. Let B be a Banach space and let A be a nonempty, closed, and convex subset of B. Given a Hilbert space V, let T : B V V R be a trilinear form with T (a, u, v) symmetric in u, v, and let m be a bounded linear functional on V. Assume there are constants α > 0 and β > 0 such that the following conditions hold: T (a, u, v) β a B u V v V, for all u, v V, a B, (1.5) T (a, u, u) α u 2 V, for all u V, a A. (1.6) Consider the variational problem: Given a A, find u = u(a) V with T (a, u, v) = m(v), for every v V. (1.7) Due to the imposed conditions, it follows from the Riesz representation theorem that for every a A, the variational problem (1.7) admits a unique solution u(a). In this abstract setting, the inverse problem of identifying parameter now seeks a in (1.7) from a measurement z of u. In collaboration with Mark Gockenbach, in [17], I proposed the following modified OLS functional (MOLS) J(a) = 1 T (a, u(a) z, u(a) z) (1.8) 2 where z is the data (the measurement of u) and u(a) solves (1.7). This functional generalizes (1.4). In [17], we established that (1.8) is convex and used it to estimate the Lamé moduli in the equations of isotropic elasticity. The first observation necessary for the convexity of the MOLS is that for each a in the interior of A, the first derivative δu = Du(a)δa is the unique solution of the variational equation: T (a, δu, v) = T (δa, u, v), for every v V, (1.9) Using (1.9), in [17] we obtained the following derivative formulae: DJ(a)δa = 1 T (δa, u(a) + z, u(a) z), 2 (1.10) D 2 J(a)(δa, δa) = T (a, Du(a)δa, Du(a)δa). (1.11) Due to the coercivity (1.6), it follows that D 2 J(a)(δa, δa) α Du(a)δa 2 V, and hence the convexity of (1.8) in the interior of A follows. Much recently, in [26], we devised a rigorous regularization framework for the identification of smooth as well as discontinuous coefficients. This framework subsumes the total variation regularization that has attracted a great deal of attention in identifying 2

3 discontinuous coefficients. We gave new existence results for the regularized optimization problems for the output least-squares and modified OLS objectives. We carried out a detailed study of various stability aspects of the inverse problem under data perturbation. We showed that a sequence of regularized problems, defined through a sequence of noisy data, converges to the regularized problem with the exact data provided that the noisy data converges to the exact data. Furthermore, under suitable assumptions, we showed that the regularized problems converge to the original problem if the regularization parameter converges to zero in an appropriate manner. This result and the adopted technique cannot be directly applied to the OLS formulation. We gave new stability estimates for general inverse problems using the OLS and the MOLS formulations. In essence, these results show that the OLS formulation requires that the regularization parameter should be large enough to ensure stability. On the other hand, for the MOLS formulation, the stability is ensured for any (positive) regularization parameter. We gave a discretization scheme for the continuous inverse problem and proved the convergence of the discrete inverse problem to the continuous one. We collected discrete formulas for the OLS and the MOLS and computed their respective gradient and Hessian. To show the advantage of an abstract framework, we presented several applications of the theoretical results. We also provide computational results for the inverse problem of identifying coefficients in three important classes of partial differential equations. Studies related to MOLS functional and its extensions can be found in [16, 18, 24]. A careful look at our proofs of the above mentioned results reveals that for the convexity of the MOLS, it is essential that the first argument of T be the parameter to be identified. On the other hand, in recent years interesting applications lead to situations when the first argument of T is in fact contains a nonlinear function of the sought parameter (see [42]). Motivated by such applications, in [23], we introduced and analyzed a variant of the MOLS for the inverse problem of identifying parameter that appears nonlinearly in general variational problems. We also derived the first-order and second-order derivative formulas for the new functional and use them to identify the conditions under which the new functional is convex. We now present a numerical example (see [23]) to demonstrate the identification of parameters using the nonlinear analogue of MOLS functional. Our goal is to identify the parameter a in the equation (σ(a) u) = f in. We consider homogeneous Dirichlet boundary condition along the entire boundary. Consider the problem in a two-dimensional domain = [0, 1] [0, 1], the position vector is thus x = (x 1, x 2 ). We choose the parameter a and the load function f as a(x) = x (sin(20x 1 ) + 1) f(x) = (x 2 0.5) 2, and take σ(a) = a 3. The measured solution z is produced by first solving the forward problem with the exact parameter a and then adding random noise. The noise added to the measured solution z is from a uniform random distribution on the range [ α, α] with α = The regularization parameter for these computations is taken as κ = Reconstruction in Figure 1.1 shows the feasibility of the approach. 3

4 Fig. 1.1: Exact and estimated parameters a and a h (top row), error in parameter a (bottom left), and estimated solution u h for the case of parameter map σ. 2. Parameter Identification in Mixed Variational Problems for the Elasticity Imaging Inverse Problem of Tumor Identification. Soft tissue cancers in the interior of the human body reign among the deadliest forms of the disease, making up a majority of the 7.6 million estimated cancer deaths in 2008 [Cancer (Fact Sheet No. 297), WHO, 2013]. The effectiveness of most treatments hinges upon early detection but the process of finding tumors inside the body remains a difficult one. It is known that the stiffness of soft tissue can vary widely based on its molecular makeup and differing macroscopic/microscopic structure and that such changes in stiffness correlate with changes in tissue health. Palpation allows doctors to feel directly for changes in tissue stiffness and can detect such stiff lesions but the practice is subjective and is usually limited to finding exceptionally hard nodules near the skin s surface. Ultrasound can also be used to diagnose tumors further within the body as well as quantify their stiffness, but even hard growths can lack the necessary acoustical properties for effective detection. Elasticity imaging inverse problem extends the practice of palpation, making use of the varying elastic properties of healthy and diseased tissue to identify likely tumors. A relatively small external quasistatic compression force is applied to the tissue and then the tissue s axial displacement field is measured either directly or indirectly through the comparison of an undeformed and deformed image. A tumor can be identified by solving the inverse problem of determining the tissue s underlying elastic properties from this measurement. Although elasticity imaging inverse problem is a well established approach commonly used in a clinical setting, the underlying mathematical formulation is in terms of a non-convex optimization framework and hence it cannot ensure 4

5 that a global solution can be found. It has been a long standing problem whether it is possible to remedy this drawback. In the following, I briefly describe the mathematical setting that plays the key role in elasticity imaging inverse problem: Given the domain as a subset of R 2 or R 3 and = Γ 1 Γ 2 as its boundary, the following system models the response of an isotropic elastic body to the known body forces and boundary traction: σ = f in, (2.1a) σ = 2µɛ(u) + λdivu I, (2.1b) u = g on Γ 1, (2.1c) σn = h on Γ 2. (2.1d) In (2.1), the vector-valued function u = u(x) is the displacement of the elastic body, f is the applied body force, n is the unit outward normal, and ɛ(u) = 1 ) is the 2 ( u+ ut linearized strain tensor. The resulting stress tensor σ in the stress-strain law (2.1b) is obtained under the condition that the elastic body is isotropic and the displacement is sufficiently small so that a linear relationship remains valid. Here µ and λ are the Lamé parameters which quantify the elastic properties of the object. From a mathematical stand point this inverse problem seeks µ from a measurement of the displacement vector u under the assumption that the parameter λ is very large. This is due to the fact that in most of the existing literature on elasticity imaging inverse problem, the human body is modelled as an incompressible elastic object. Although this assumption simplifies the identification process as there is only one parameter µ to identify, it significantly complicates the computational process as the classical finite element methods become quite ineffective due to the so-called locking effect. One of the few techniques to handle this problem is by resorting to mixed finite element formulation. We explain this in the following. For the time being, in (2.1), we set g = 0. For this case, the space of test functions, denoted by V, is given by: V = { v H 1 () H 1 () : v = 0 on Γ 1 }. By using the Green s identity and the boundary conditions (2.1c) and (2.1d), we obtain the following weak form of the elasticity system (2.1): Find ū V such that 2µɛ(ū) ɛ( v) + λ(div ū)(div v) = f v + vh, for every v V. (2.2) Γ 2 The mixed finite elements approach then consists of introducing a pressure term p Q = L 2 () by p = λ(div ū), or equivalently, 1 (div ū)q pq = 0, for every q Q. (2.3) λ The weak form (2.2) then reads: Find ū V such that 2µɛ(ū) ɛ( v) + p(div v) = f v + vh, for every v V. (2.4) Γ 2 The problem of finding ū V satisfying (2.2) has now been reformulated as the problem of finding (ū, p) V Q satisfying the mixed variational problems (2.3) and (2.4). 5

6 We have studied the elasticity imaging in an abstract framework. Let V and Q be real Hilbert spaces, let B be a real Banach space, let S B be open, and convex, and let A S be closed, and convex. Let a : B V V R be a trilinear form symmetric in the last two arguments, let b : V Q R be a bilinear map, let c : Q Q R be a symmetric bilinear map, and let m : V R be a linear and continuous map. Consider the mixed variational problem: Given l S, find (ū, p) V Q with a(l, ū, v) + b( v, p) = m( v), for every v V, (2.5a) b(ū, q) c(p, q) = 0, for every q Q. (2.5b) Our focus is on the inverse problem of identifying the parameter l A for which a solution (ū, p) of (2.5) is closest to a given measurement ( z, ẑ) of (ū, p). In this setting, the OLS functional reads J O (l) := 1 2 u(l) z 2 Z = 1 2 ū(l) z 2 Z p(l) ẑ 2 Ẑ, with ( z, ẑ) Z := Z Ẑ, where Z is a suitable observation space. The OLS formulation for nonlinear inverse problems of parameter identification is typically nonconvex, and hence it is limited to characterizing local minima. Of course, a natural strategy to circumvent the difficulties associated to the non-convexity of the OLS functional is to introduce an analog of (1.4). As one of my main contributions to elasticity imaging inverse problem, I and my co-authors have recently devised a new convex framework that not only gives a global optimal solution, it provides a fast and reliable computational framework. In [24], the following modified output least-squares (MOLS) was introduced J M (l) := 1 2 a(l, ū(l) z, ū(l) z) + b(ū(l) z, p(l) ẑ) 1 c(p(l) ẑ, p(l) ẑ), 2 where ( z, ẑ) V is the data. In [12], the following energy output least-squares was proposed J E (l) := 1 2 a(l, ū(l) z, ū(l) z) + 1 c(p(l) ẑ, p(l) ẑ), 2 where ( z, ẑ) V is the data. The idea of minimizing the energy of the variational problem was fundamental to the convexity of (1.4). However, since the mixed variational formulation involves a coupled system of equations, the two ways of combining them result in MOLS and EOLS with different features; MOLS preserves convexity but loses positivity whereas EOLS retains positivity but is non-convex in general. In [24, 12] we present a complete theoretical as well as numerical treatment of the elasticity imaging inverse problem. 3. Set-valued Optimization. In the Spring of 2000, I started working on a research project sponsored by German Research Foundation (DFG), devoted to a detailed study of optimization problems involving set-valued maps. In this field, one major difficulty is of differentiating the set-valued maps. We established the usefulness of the notion of epiderivative of set-valued mapping. This concept is a cornerstone of set 6

7 optimization and was coined by Johannes Jahn, who was the main investigator of the project. The basic idea is to use certain tangent cones to define the epiderivatives and use them to state optimality conditions in set optimization. Set optimization subsumes nonsmooth optimization which, due to many real-world applications, was one of the most active branches of optimization before the birth of set-optimization. We have given many useful results in set optimization. We proved new optimality conditions and established new multiplier rules, generalized the Dubovitski-Milutin approach to set optimization, and gave new derivative rules for the epiderivatives. We also introduced new second-order derivatives and gave, probably for the first time, second order optimality conditions in set optimization. In its simplest form, our secondorder optimality condition collapses to the second-order optimality condition available in the calculus text books, emphasizing the nature of its importance (see [28, 27, 29, 30, 20, 21]). In [37], we gave a general second-order sensitivity analysis for set-valued optimization problems and higher-order optimality conditions appeared in [33, 34]. In set-valued optimization, my most significant contribution is the recent monograph entitled Set-Valued Optimization: Theory, methods, and applications, published by Springer (see [36]). This monograph which consumes almost 800 pages, is the first book on this subject and it reports most of the recent developments in this discipline. It contains numerous results, examples, and over six hundred references. 4. Optimal Control Problems with Pointwise State Constraints. In recent years, I have focused on deriving new error estimates for the conical regularization for optimal control of partial differential equations with pointwise state constraints. Conical regularization, developed in [25, 32, 22], provides a unified framework for optimization problems for which a Slater-type constraint qualification fails to hold due to the empty interior of the ordering cone associated to the inequality constraints. The failure of a Slater-type constraint qualification is a common hurdle in numerous branches of applied mathematics including optimal control, inverse problems, nonsmooth optimization, and variational inequalities, among others. Research in [25, 32, 22] gave convincing evidence that the conical regularization has a potential to be a valuable theoretical and computational tool in different disciplines suffering due to the lack of a Slater-type constraint qualification. There has been extensive research on control problems, most of which focused on control-constrained problems, see [9, 11]. In recent years, however, the focus has shifted to optimal control problems with pointwise state constraints that frequently appear in applications (see [6, 15]). For instance, while optimizing the process of hot steel profiles by spraying water on their surfaces, it is necessary to control the temperature differences to avoid cracks, which leads to pointwise state constraints, see [14]. A similar situation arises in the production of bulk single crystals, where the temperature in the growth apparatus must be kept between prescribed bounds, see [40]. Moreover, local hyperthermia in cancer treatment, pointwise state constraints are imposed to ensure that only the tumor cells gets heated and not the healthy ones from a close vicinity, see [10]. To illustrate the obstacles in a treatment of pointwise state constraints, we consider the control problem (Q) Minimize 1 u z d ) 2 (y 2 dx + κ (u u d ) 2 dx, 2 subject to y u (x) + y u (x) = u(x) in, n y u = 0 on, y u (x) w(x), a.e. in, 7

8 where R d (d = 2, 3) is an open, bounded, and convex domain with boundary, κ > 0 is a fixed parameter, z d, u d, and w are given elements, and n is the outward normal derivative to. If the control u is taken in L 2 (), then the state y u is in H 2 (). Since H 2 () C( ), the state can be viewed as a continuous function. This fact has an impact on deriving optimality conditions for (Q). In particular, a Lagrange multiplier exists due to the following well-known Slater constraint qualification: there exists ū L 2 () such that yū w int C + ( ), (4.1) The subtlety, however, is that the multiplier now belongs to the dual of C + ( ), which turns out to be the space of Radon measures (see [3, 7]). This causes low-regularity in the control and has an adverse effect on both the analytical level when deriving optimality conditions for the control problem and on the numerical level when performing discretization. In fact, the appearance of measures is inevitable in such studies. Examples of (Q) where the multipliers are Radon measures are given in [25]. To give another interpretation of the difficulties in control problems with pointwise state constraints, we consider an abstract problem. Let U, Y, and H be Hilbert spaces equipped with the norms U, Y and H. Let C Y be a closed and convex cone which is not necessarily solid, that is, it may have an empty interior. Let C + be the positive dual of C given by C + := {λ Y : λ (c) 0, c C}. Given linear and bounded maps S : U H and G : U Y, and a parameter κ > 0, we consider: (P ) Minimize J(u) := 1 2 Su z d H + κ 2 u u d 2 U, subject to G(u) C w, u U. Problem (P ) represents a wide variety of PDE constrained optimization and optimal control problems. Notice that for (P ), the existence of Lagrange multipliers and reliable and efficient computational schemes can only be developed through a careful selection of the state space U and the control space Y. We note that (Q) is a special case of (P ) with U = H = L 2 (), S : L 2 () L 2 () and G : L 2 () Y. For the control-to-state map (u y u ), there are two natural choices of the range space, namely, Y = L 2 () and Y = C( ). However, as already noted above, to ensure the existence of Lagrange multipliers for the L 2 -controls, the choice Y = C( ) seems natural, as the Slater constraint qualification (4.1) holds. The main drawback is that the multipliers are Radon measures, making the numerical treatment of the optimality conditions quite challenging. On the other hand, for the choice Y = L 2 (), the main technical hindrance is that the cone of positive functions L 2 +() has an empty interior. Consequently, no general Karush-Kuhn-Tucker theory is available. From the mathematical programming point of view, the lack of regularity can be attributed to the fact that both the ordering cone C in (P ), and correspondingly L 2 +() in (Q), are not solid. It has been long recognized that the inequality constraints defined by an ordering cone that has an empty interior are a major obstacle in a satisfactory treatment of optimization and optimal control problems. In recent years, regularization methods have been quite popular for control problems of type (Q) to handle the low regularity of the multipliers. These methods associate, to the original problem, a family of perturbed problems with more regular Lagrange multipliers. The methods that fall in this category are the penalty method (see [19]), the Lavrentiev regularization (see [41]), and the conical regularization (see [25, 32, 22]). 8

9 We remark that whereas the Lavrentiev regularization is designed for specific PDEs, the conical regularization provides a general framework that is by no means restricted to the control problems with pointwise state constraints. This particular case, however, is of importance as pointwise state constraints appear in numerous applied models. 5. Quasi Variational Inequalities. During the last several years, I have been quite interested in quasi-variational inequalities. In recent years the theory of quasivariational inequalities has emerged as one of the most promising domains of applied mathematics. Quasi-variational inequalities not only subsume variational inequalities and nonlinear partial differential inequations, they also provide a unified framework for studying general boundary value problems with complicated, possibly unilateral, boundary conditions. This field offers us a powerful mathematical apparatus for investigating a wide range of problems arising in diverse domains such as mechanics, economics, finance, optimization, optimal control, and others. Recently, in [31], we studied a quite general class of evolutionary quasi-variational inequalities. Our first result in [31] deals with evolutionary variational inequalities involving different types of pseudo-monotone operators. We also gave a new existence result for elliptic variational inequalities with generalized pseudo-monotone maps. Moreover, we also give another new existence result for variational inequalities for pseudomonotone maps. Another existence result embarked on elliptic variational inequalities driven by maximal monotone operators. Another major contribution of our work is to obtain criteria for solvability of general quasi-variational inequalities treating in a unifying way elliptic and evolutionary problems. We provided examples and applications of abstract results. In [35] we stabilized an ill-posed quasi-variational inequality with contaminated data by employing the elliptic regularization. We showed that under suitable conditions, the sequence of bounded regularized solutions converges strongly to a solution of the original quasi-variational inequality. Moreover, the conditions that ensure the boundedness of regularized solutions, become sufficient solvability conditions. Our results reflected that the regularization theory is quite strong for quasi-variational inequalities with set-valued monotone maps but restrictive for generalized pseudo-monotone maps. We applied our results to ill-posed variational inequalities, hemi-variational inequalities, inverse problems, and split feasibility problem, among others. REFERENCES [1] R. Acar, Identification of the coefficient in elliptic equations, SIAM J. Control Optim. 31 (5) (1993) [2] M. F. Al-Jamal, M. S. Gockenbach, Stability and error estimates for an equation error method for elliptic equations, Inverse Problems 28 (9) (2012) , 15. [3] J.-J. Alibert, J.-P. Raymond, Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls, Numer. Funct. Anal. Optim. 18 (3-4) (1997) [4] H. Ammari, P. Garapon, F. Jouve, Separation of scales in elasticity imaging: a numerical study, J. Comput. Math. 28 (3) (2010) [5] H. T. Banks, K. Kunisch, Estimation techniques for distributed parameter systems, vol. 1 of Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, [6] S. C. Brenner, L.-Y. Sung, Y. Zhang, Post-processing procedures for an elliptic distributed optimal control problem with pointwise state constraints, Appl. Numer. Math. 95 (2015) [7] E. Casas, Control of an elliptic problem with pointwise state constraints, SIAM J. Control Optim. 24 (6) (1986) [8] Z. Chen, J. Zou, An augmented Lagrangian method for identifying discontinuous parameters in elliptic systems, SIAM J. Control Optim. 37 (3) (1999) [9] F. Clarke, Y. Ledyaev, M. d. R. de Pinho, An extension of the Schwarzkopf multiplier rule in optimal control, SIAM J. Control Optim. 49 (2) (2011)

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