APPLICATION OF A CONSTRAINED OPTIMIZATION TECHNIQUE TO THE IMAGING OF HETEROGENEOUS OBJECTS USING DIFFUSION THEORY. A Thesis MATTHEW RYAN STERNAT

Transcription

1 APPLICATION OF A CONSTRAINED OPTIMIZATION TECHNIQUE TO THE IMAGING OF HETEROGENEOUS OBJECTS USING DIFFUSION THEORY A Thesis b MATTHEW RYAN STERNAT Submitted to the Office of Graduate Studies of Teas A&M Universit in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE December 9 Major Subject: Nuclear Engineering

2 APPLICATION OF A CONSTRAINED OPTIMIZATION TECHNIQUE TO THE IMAGING OF HETEROGENEOUS OBJECTS USING DIFFUSION THEORY A Thesis b MATTHEW RYAN STERNAT Submitted to the Office of Graduate Studies of Teas A&M Universit in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Approved b: Chair of Committee, Committee Members, Head of Department, Jean C. Ragusa Wolfgang Bangerth William S. Charlton Ramond Juzaitis December 9 Major Subject: Nuclear Engineering

3 iii ABSTRACT Application of a Constrained Optimization Technique to the Imaging of Heterogeneous Objects Using Diffusion Theor. (December 9) Matthew Ran Sternat, B.S., Teas A&M Universit Chair of Advisor Committee: Dr. Jean C. Ragusa The problem of inferring or reconstructing the material properties (cross sections) of a domain through noninvasive techniques, methods using onl input and output at the domain boundar, is attempted using the governing laws of neutron diffusion theor as an optimization constraint. A standard Lagrangian was formed consisting of the objective function and the constraints to satisf, which was minimized through optimization using a line search method. The chosen line search method was Newton s method with the Armijo algorithm applied for step length control. A Gaussian elimination procedure was applied to form the Schur complement of the sstem, which resulted in greater computational efficienc. In the one energ group and multi-group models, the limits of parameter reconstruction with respect to maimum reconstruction depth, resolution, and number of eperiments were established. The maimum reconstruction depth for one-group absorption cross section or multi-group removal cross section were onl approimatel -7 characteristic lengths deep. After this reconstruction depth limit, features in the center of a domain begin to diminish independent of the number of eperiments. When a small domain was considered and size held constant, the maimum reconstruction resolution for one group absorption or multi-group removal cross section is approimatel one fourth of a characteristic length. When finer resolution then this is considered, there is simpl not enough information to recover that man region s cross sections independent of number of eperiments or flu to cross-section mesh

4 iv refinement. When reconstructing fission cross sections, the one group case is identical to absorption so onl the multi-group is considered, then the problem at hand becomes more ill-posed. A corresponding change in fission cross section from a change in boundar flu is much greater then change in removal cross section pushing convergence criteria to its limits. Due to a more ill-posed problem, the maimum reconstruction depth for multi-group fission cross sections is 5 characteristic lengths, which is significantl shorter than the removal limit. To better simulate actual detector readings, random signal noise and biased noise were added to the snthetic measured solutions produced b the forward models. The magnitude of this noise and biased noise is modified and a dependenc of the maimum magnitude of this noise versus the size of a domain was established. As epected, the results showed that as a domain becomes larger its reconstruction abilit is lowered which worsens upon the addition of noise and biased noise.

5 To m father and mother, Louis Sternat Jr. and Patricia Sternat v

6 vi ACKNOWLEDGMENTS I would like to acknowledge m father and mother, Louis Sternat Jr. and Patricia Sternat, for their guidance, love and support through this journe. The alwas allowed me to pursue m passion for science. I would also like to acknowledge the rest of m famil and friends for allowing me to become the person I am toda. I further acknowledge Dr. Jean C. Ragusa for his advice, advisor, and allowing me to work on this project and Dr. Wolfgang Bangerth for help with optimization methods and techniques.

7 vii TABLE OF CONTENTS CHAPTER Page I INTRODUCTION A. Objective B. Imaging C. Optimization and Inverse Problem Solving D. Thesis Overview II OPTIMIZATION METHODS A. Optimization Classifications B. Optimalit Conditions C. Line Search Methods Steepest Descent Newton s Method D. Convergence Criteria E. Step-Length Selection Control and Algorithms F. Schur Complement Method III INVERSE DIFFUSION MODELS A. Neutron Diffusion Theor B. Finite Element Diffusion Solver Finite Element Meshes Finite Element Methods C. Optimization Functional Misfit: To Minimize Lagrangian Functional D. Hessian Sstem E. Implementation of Schur Complement F. Etension to Multiple Eperiments Optimalit Conditions Hessian Sstem Schur Complement Modification G. Multigroup Analsis Multigroup Diffusion Theor Cross-Section Data for Various Materials

8 viii CHAPTER Page IV RESULTS A. Eample : Misfit Plots B. Eample : Comparison of Convergence Between Steepest Descent Method and Newton s Method for a Homogeneous Problem C. Eample : Multiple Region Single Eperiment Results.. 9 D. Multiple Eperiment Results Eample : Reconstruction Resolution Testing with Increasing Number of Eperiments on a cm cm Domain Eample 5: Reconstruction Resolution Testing with Increasing Number of Eperiments on a cm cm Domain Eample : Effects on Reconstruction When the Domain Size is Increased Using and Eperiments 5. Eample 7: Dual Strong Absorbers Embedded in a Large Highl Scattering Domain E. Addition of Signal Noise and Bias Addition of Random Signal Noise a. Eample : Signal noise added on a homogenous domain b. Eample 9: Signal noise added to bars of various materials c. Eample : Reconstruction testing with signal noise on a centered strong absorber inside various size domains Addition of Signal Bias a. Eample : Reconstructing with a positive signal bias of a centered strong absorber b. Eample : Reconstructing with a negative signal bias on a centered strong absorber F. Muli-group Results Multi-group Misfit Plots a. Eample : Multigroup misfit plots of absorption cross sections onl b. Eample : Multigroup misfit plots of fission cross sections onl

9 i CHAPTER Page c. Eample 5: Multigroup misfit plots of mied parameters Multigroup Reconstruction Results a. Eample : Reconstruction of a thermal strong absorber b. Eample 7: Reconstruction of centered fissile material c. Eample : Maimum reconstruction depth testing for νσ f d. Eample 9: Mied parameter reconstructions.. 7 e. Eample : Variation of incident neutron and measurement energ V CONCLUSIONS REFERENCES VITA

10 LIST OF TABLES TABLE Page III-I Cross-section data for various materials at. MeV III-II Cross-section data for various materials at fission spectrum average. III-III Cross-section data for HEU in borated polethlene at. MeV.. III-IV Cross-section data for HEU in borated polethlene at fission spectrum average III-V Cross-section data for HEU in borated polethlene at thermal energ IV-I E. : Convergence comparison between two line search methods.. IV-II E. : Cross-section data IV-III E. 7: Cross-section data for dual strong absorbers embedded in a large highl scattering domain IV-IV E. 9: Cross-section data for bars of various material IV-V E. 7: Domain parameters of centered fissile material IV-VI E. : Domain parameters of two region fissile material IV-VII E. : Domain parameters for a multigroup centered strong absorber 7

11 i LIST OF FIGURES FIGURE Page I- Eample of incoming and outgoing particle currents II- Eample of a constrained objective function II- Eample of an unconstrained objective function II- Iso-contour plot showing an objective function and a constraint... 9 II- Steepest descent direction II-5 Newton s method vs. steepest descent direction III- Eample of the finite element meshes for the diffusion problem.... IV- Two-region domain of Eample IV- E. : Various cases of beam incidence IV- E. : Misfit plot for case a.) IV- E. : Misfit plot for case b.) IV-5 E. : Misfit plot for case c.) IV- IV-7 E. : Convergence of misfit and Lagrangian for steepest descent and Newton s methods Eample : Convergence of dual strong absorbers in a homogenous domain IV- E. : True cross section with position dependence IV-9 E. : Reconstructed cross section with position dependence..... IV- E. : Error in cross section reconstruction with position dependence IV- E. : Effects on reconstruction resolution while increasing the number of eperiments on a cm cm domain

12 ii FIGURE Page IV- IV- IV- IV-5 E. : Error in reconstructions resolution testing while increasing the number of eperiments on a cm cm domain E. 5: Reconstruction resolution testing of centered strong absorber in a cm cm domain E. 5: Error in reconstructions for resolution testing of centered strong absorber in cm cm domain E. : Effects on reconstructions when domain size is increased using eight eperiments IV- E. : Error in reconstructions when domain size is increased using eight eperiments IV-7 E. : Effects on reconstructions when domain size is increased using eperiments IV- E. : Error in reconstructions when domain size is increased using eperiments IV-9 IV- E. 7: Reconstruction of two strong absorbers in a large highl scattering domain E. 7: Error in reconstruction of two strong absorbers in a large highl scattering domain IV- E. : Reconstruction with signal noise of a homogeneous domain.. 5 IV- E. : Error in reconstruction with signal noise of a homogeneous domain IV- E. 9: Reconstruction with signal noise of multiple materials IV- E. 9: Error in reconstruction with signal noise of multiple materials 5 IV-5 E. Case : Reconstruction with signal noise of a centered strong absorber cm IV- E. Case : Error in reconstruction with signal noise of a centered strong absorber cm

13 iii FIGURE Page IV-7 E. Case : Reconstruction with signal noise of a centered strong absorber cm IV- E. Case : Error in reconstruction with signal noise of a centered strong absorber cm IV-9 E. Case : Reconstruction with signal noise of a centered strong absorber cm IV- E. Case : Error in reconstruction with signal noise of a centered strong absorber cm IV- IV- IV- IV- IV-5 IV- E. : Reconstruction with a positive signal bias of a centered strong absorber E. : Error in reconstruction with a positive signal bias of a centered strong absorber E. : Reconstruction with a negative signal bias of a centered strong absorber E. : Error in reconstruction with a negative signal bias of a centered strong absorber E. : Misfit surface plot of Σ r, and Σ r, for group homogeneous region while varing source energ E. : Misfit surface plot of Σ f, and Σ f, for group homogeneous region IV-7 E. 5: Misfit surface plot of Σ r, and Σ f, for and group homogeneous regions IV- E. 5: Mied parameter misfit surface plots for energ groups.. IV-9 E. : Multigroup reconstructions of a thermal centered absorber. 5 IV- E. : Error in multigroup recon. of a thermal centered absorber. IV- E. 7: Σ f, and Σ f, reconstructions of centered fissile material... 7

14 iv FIGURE Page IV- E. 7: Error in Σ f, and Σ f, reconstructions of centered fissile material IV- E. : Σ f, and Σ f, reconstructions of a two zone fissile step IV- E. : Error in Σ f, and Σ f, reconstructions of two zone fissile step 9 IV-5 E. 9: Mied parameter reconstructions for a group problem... 7 IV- E. 9: Error for mied parameter reconstructions for a group problem IV-7 E. 9: Mied parameter reconstructions for a group problem... 7 IV- E. 9: Error for mied parameter reconstructions for a group problem IV-9 E. : Reconstruction of Σ r, with incident neutrons onl in group one and measuring onl in group IV-5 E. : Reconstruction of Σ r, with incident neutrons onl in group one and measuring onl in group

15 CHAPTER I INTRODUCTION A. Objective In the field of nuclear and global securit, smuggling of special nuclear materials b transportation in containers on boats poses strong threat. To prevent this possible smuggling pathwa, a detection sstem must be implemented that will have the abilit to detect high enriched uranium (HEU) where current detection sstems cannot. Due to self-shielding and long half-lives, uranium can be hard to detect through conventional methods, especiall in large scale sstems such as cargo containers. There are approimatel, ships docking at the United States per ear currentl and efficient detection methods must be implemented. As of the 9/ Commission Act of 7, foreign seaports must scan percent of the cargo entering the United States b. A possible method of detection would be an active neutron imaging technique which would involve incident beams of neutrons upon the cargo container and neutron detectors surrounding the container. Using these detector readings and a constrained optimization technique, reconstructions of the material properties inside a container could be performed to determine the contents. We propose to address this parameter identification b posing it as an optimal control problem where a cost function is to be minimized. This cost function is defined as the difference between the boundar detector measurements and the boundar neutron flues computed from the inferred material properties inside the cargo. While man sets of material parameters ma The journal model is Nuclear Science and Engineering.

16 have the abilit to reconstruct the outer detector readings, constraints upon these must be applied to limit the number of solutions. The valid constraint used in this work will involve the governing conservation law of neutron phsics in the container, thereb limiting the solution of material parameters to a realistic or phsical case. This is an optimal control problem because the difference between the computed iterative solution at the boundar and the neutron detector readings must be minimized while satisfing the neutron transport equation or an approimation of it. The equations derived from the optimization process are nonlinear, naturall requiring a descent method to solve them. This problem is ill-posed because small changes in the material properties can often lead to large changes in the neutron flues at the boundaries. Application of iterative methods cannot guarantee convergence for an realistic initial guess due to the ill-posedness of this nonlinear problem. B. Imaging Active neutral particle imaging techniques involve illuminating a domain with beams of particles of known intensit and taking measurements around the domain of the boundar outflow in an orientation shown in Figure I-. Active neutral particle imagining is performed to reconstruct information of the inside of the domain. The location, energ, and angle of incidence of the incoming particles can be varied and more information can be gathered. With multiple eperiments of incoming beams around the domain, the material properties reconstruction satisfing all eperiments at once can ield improved reconstruction.

17 Fig. I-. Eample of incoming and outgoing particle currents An eample of this is optical tomograph, where a nonlinear sstem containing nonlinear combinations of the parameters intended to be reconstructed and the state variables is formed b the equations that define how light is transmitted and scattered through an object and often have no analtical solution. B observing the light eiting the tissues, a reconstruction of the absorption and scattering coefficients inside the sample is performed. These problems are solved iterativel using forward models to solve for the outgoing currents based on an initial guess on the interaction coefficients directl, and nonlinear optimization techniques to update the interaction coefficients. 5 This algorithm process is repeated until the iterative solution converges with the observed light eiting the tissues. This is ver similar to the problem of special nuclear material (SNM) smuggling, but instead of biological matter, containers that can be up to man optical thicknesses deep are to be imaged using neutrons. Another eample of neutral particle imaging is in large ports for object detection. There are sstems that use photons that operate in the -9 MeV range to image large

18 cargo containers. Most of currentl implemented cargo imaging uses either -ras or gamma ras. The -ra sstems are commonl used to ensure containers are empt without opening them or to determine contents of smaller containers where gamma ras are not needed. These tpes of sstems are capable of producing images of large containers and trucks with spatial resolution of 9mm for the gamma sstems and mm for -ra sstems. While these tpes of sstems can produce an image of the internal contents of a container, the cannot b themselves determine if fissile material is present. This is where a multigroup neutron imaging sstem would have the greatest impact. If a sstem were able to reconstruct fission cross sections to determine whether fissile material were present accuratel, greater detection probabilit of smuggled HEU could be achieved. Neutron imaging varies from gamma or -ra imaging in the wa the interact with matter quite differentl then -ras do, having a high interaction probabilit with hdrogen and much lower attenuation in heavier elements such as lead. While -ra interaction probabilit is directl proportional to the atomic number of the material, neutron interaction is isotope-dependent causing both radiograph mechanisms to ecel in different media tpes. 7 Common eamples of neutron radiograph include nuclear fuel surves, multi-phase flow imaging, and eplosive device imaging. In the case at hand, HEU could easil be shielded from -ras causing methods involving - ras or radiation emitted from the material itself to be ineffective. Neutron interaction probabilities are energ-dependent, where neutrons of tpical source energ have high scattering interaction probabilit in man materials, limiting the abilit of larger scale imaging.

19 5 C. Optimization and Inverse Problem Solving The majorit of inverse problems or imaging techniques involve an optimization process in which a function is minimized or maimized b iterating the functions variables, often subject to constraints. The most commonl used methods to solve problems of an tpe involve iterative algorithms. In the optimization process, the optimum of a given function is obtained b solving the optimalit conditions using an optimization algorithm. There is no universal optimization algorithm but instead a collection of algorithms in which each is valid for specific problem tpes. An eample of application of inverse transport is the determination of interface locations in a multilaer domain of unknown dimensions. In this specific eample, source gamma-ras were passed through a domain and observed at boundaries, then the location of the interfaces is solved for using optimization methods. 9 This is similar to the problem at hand ecept that instead of the material properties being known and the interface locations reconstructed, the material properties are unknown but reconstructed and assumed piecewise constant over a mesh. D. Thesis Overview The net chapter provides an in depth look at optimization methods from a mathematical standpoint. This chapter provides a complete step b step approach to optimization problems including specific methods. Chapter III contains the development and implementation of the presented optimization methods to the inverse problem using diffusion theor. Chapter IV presents the results of reconstructions of various domains. Man tests were performed in order to have an understanding of the workable space with respect to domain size, mesh size, number of eperiments, and measurement location.

20 CHAPTER II OPTIMIZATION METHODS The goal of an optimization problem is to find the combination of parameters that optimize a given quantit subject to some restrictions or constraints. The parameters that ma be changed in the process of optimization are called control or decision variables while the restrictions on parameters are known as constraints. Mathematicall speaking, optimization is the minimization or maimization of an objective function defined b a problem statement and is subject to constraints on its variables. Often a vector is formed that consists of the unknowns or parameters, f if the objective function, a scalar function of, that we want to maimize or minimize, and a series of constraint functions, c i, which are scalar functions of that define constraints the unknown vector must satisf. Using this notation, the optimization problem can be written as shown in Equation.. min f() subject to c Rn i() =, i ξ c i (), i I (.) where c i can be an equalit or inequalit constraint and ξ and I are the sets of equalit and inequalit constraints. This chapter provides an overview of optimization methods in general, starting with section A on optimization problem classifications.section B provides the optimalit conditions. The section of these optimalit conditions is then detailed in section C using steepest descent and Newton s method. This chapter will then cover convergence criteria D, step-length control E and conclude with the Schur complement method emploed to reduce the sstem s dimensions F.

21 7 A. Optimization Classifications In deterministic optimization methods, first and second derivatives of the objective function, f(), need to be computed. These problems are classified b the tpe of their control variables and nature of their objective functions which are usuall linear, quadratic, or full nonlinear. In certain cases, this function can be discontinuous and ma contain integers and binar variables; these problems can onl be optimized using discrete optimization methods for which derivatives are not defined. Other classes of problems, where the components of the given function are allowed to be real numbers can be optimized continuousl. These continuous functions are normall easier to solve because the are often smooth and twice differentiable. When a problem is considered, it is classified b the nature of its objective function where some problems have constraints upon their variables and some do not. Problems that involve constrained variables are optimized using constrained optimization. Sometimes these constraints pla a important role in determining the solution and an eample of a constrained objective function can be seen in Figure II-. Fig. II-. Eample of a constrained objective function For instance, in a budgetar problem, if the global solution lies outside the limits due to budgetar constraints, a local solution that lies within these constraints will be the best solution. Whereas in full unconstrained optimization, there are no limits

22 on an of the variables and the global minimum is the true function minimum as shown in Figure II-. Fig. II-. Eample of an unconstrained objective function Sometimes, even if there are minor constraints on a problems variables, if the do not interfere with the optimization algorithms, unconstrained optimization can be applied. B. Optimalit Conditions To find the minimum of f(), conditions are applied to find where f =. When constraints are upon f(), for eample c() =, then a Lagrangian functional is introduced such as in Equation.. min f() subject to c() = L(, λ) = f + λ c() (.) Rn where λ is a Lagrange multiplier. A saddle point in L is found where L =, which is L = = f + λ c (.) L λ = = c() (.) where the first equation implies that f c and in the second equation the con-

23 9 straint arises c() =. These derivatives of the Lagrangian form a set of Karush- Kuhn-Tucker conditions or optimalit conditions to be satisfied. Figure II- shows a simple iso-contour plot of f() with a c() = line solution. Fig. II-. Iso-contour plot showing an objective function and a constraint C. Line Search Methods In a line search method, algorithms choose a direction p k and search along this direction from the current iterate for a new iterate that is closer to the optimalit conditions. There are various methods that can be used to determine a line search direction along with man algorithms to determine how far in that direction to go. The goal of this optimization problem is the minimization of f() while satisfing an given conditions. At this minimum j L() = where j is an field variable in L(). Just as in an iterative method, an initial guess is made and at this iterate L( k ). We will now describe two such techniques: the steepest descent and Newton s descent and then provide an eample of a step-length control algorithm.. Steepest Descent An obvious direction is the steepest descent. The steepest descent direction follows the opposite direction of the gradient, or the direction perpendicular to the iso-contours. For eample in a simple two dimensional optimization scheme, this would be ver

24 similar to a ball in a valle rolling to the bottom. This can be seen in Figure II-, where X is the global minimum. The gradient of f is perpendicular to the iso-contour of L. Fig. II-. Steepest descent direction In this steepest descent method, the descent direction is p k = L k as shown in Equation.5. k+ = k + αp k = k α L k (.5) The steepest descent algorithm consists of the following:. Initialization: set k=, set convergence criteria ϵ, choose k.. If L k < ϵ then eit, otherwise continue.. Compute p k = L k. Determine step length α k (see Section.). 5. Compute new update according to Eq..5.. k k + and go to.

25 This method requires onl first derivatives, tends to get stuck in local minima, and is slowl converging while it iterativel takes steps in the gradient direction to a new solution with lower optimalit condition. The steepest descent direction is updated at ever step indeed b k and its progress is slow as some regions of indefinite curvature are encountered especiall near a solution. The convergence rate of this method is much slower than other higher order methods.. Newton s Method A significantl more efficient higher order method can be derived from where the steepest descent method left off. Newton s method is comprised of second derivatives and is a curve of best fit method. This uses a line search direction other then the steepest descent, and is derived from the second order Talor series approimation of f( k + p) and is shown in Equation.. L( k + p) L k + p T L k + pt L k p = m k (p) (.) An eample of such direction can be shown in Figure II-5. Fig. II-5. Newton s method vs. steepest descent direction

26 Using this second-order Talor series approimation, the vector p that minimizes m k (p) is obtained b setting the derivative of m k (p) to zero leaving Equation.7. L k + L k p k = (.7) where f k = H, the Hessian matri. Then solving for p k ields p k = ( L k ) L k = H L k (.) k+ = k + αp k = k αh L k (.9) This Newtonian search direction tries to quadraticall approimate a curve at iterate k and goes to the minimum of the quadratic fit. For a simple quadratic sstem, the minimum of f() could be met after one step. Due to the nonlinearit and compleit of most sstems, Newton s method often is applied where steepest descent methods will not converge. The steepest descent and Newton s method are both of the form: k+ = k + αp k = k αb f k, (.) with B = I, for steepest descent and B = H, for Newton s method. D. Convergence Criteria The nonlinear sstem in Equation. will converge when the optimalit conditions are satisfactor close to their solution. Some ver nonlinear sstems with random noise and bias will be ver difficult to drive the optimalit conditions close to the true solution. At the optimum, the optimalit conditions will be met but when constraints are present the solution that is closest to the optimalit conditions while

27 satisfing the constraints will be the global solution. For eample, if the true global solution was unreachable due to constraints then the convergence criteria of the optimalit conditions will not be achievable and the closest solution to these optimalit conditions will be the solution. E. Step-Length Selection Control and Algorithms Now that a line search direction has been determined, how far to travel in that direction is established net. When the objective function is not smooth, a full Newtonian direction step ma not lead to a reduction in optimalit condition. Simple algorithms can be used to attempt to ensure the optimalit conditions are lowered. Starting with the general sufficient decrease condition: L( k + αp k ) L( k ) < αλ L( k ) T p k (.) where the descent direction derived from Newton s Method: p k = H L() (.) where λ (, ) is an algorithmic parameter tpicall around. Beginning with α = repeatedl reduce α using an strateg that satisfies the general sufficient decrease condition. α + [β low α c, β high α c ] (.) where < β low < β high <. The choice of β = β low = β high is a simple rule in the Armijo algorithm shown below.. Initialization: set α= and λ (, ), set convergence criteria ϵ, choose k.. If L( k + αp k ) L( k ) < αλ L( k ) T p k, k+ = k + αp k. If not, continue.

28 . Reduce α, return to step. In an eact line search, the special case where λ leads to the eact minimum of L( c + αp k ), is not onl more epensive computationall but can often degrade the performance of the algorithm in general. F. Schur Complement Method The Schur Complement Method is a process of sstem simplification for a sstem involving a Karush-Kuhn-Tucker (KKT) tpe matri. It is a method that involves eliminating variables in a sstem to simplif and often obtain a linear sstem of one variable. One eample of such matri sstem can be shown in Equation.. G A T p = g where: dim(p) >> dim(λ) (.) A λ h This KKT matri has blocks of entries equal to zero and can easil simplified b simple algebra. A tpical KKT matri ma have more rows of blocks, as man multivariate problems have multiple optimalit conditions, but can be simplified in the same manner. After assuming G is positive definite, p is solved for in the first equation in. in terms of λ then substituted in the second equation leaving a sstem of λ alone as shown in Equation.5. λ = A T (g GA h) (.5) This smaller sstem is solved for λ and then the other vector variable p can be directl solved for as shown in Equation.. Gp = A T λ g (.) This method involves a matri inversion of G and A T which often results in signif-

29 5 icantl lower condition number than inverting the full original sstem matri. This method of sstem simplification can be applied to reduce sstem runtime and improve computational efficienc.

30 CHAPTER III INVERSE DIFFUSION MODELS Neutron imaging is tpe of non-invasive inverse problem involving incoming and outgoing neutron beams where measurements are made onl on the boundar of a domain. The neutron transport equation defines how neutrons behave in matter through various interaction tpes and can be used in inverse problems. An approimation to the transport model is diffusion theor, which introduces some simplicit for handling the angular dependence of the neutron population. In this thesis, we use the diffusion approimation to model the distribution of particles. In inverse theor, man problems are ill-conditioned or are ill-posed, where a small variation in the input data causes a large change in the results. In inverse diffusion methods, the flu solution to be solved for depends on unknown internal parameters of the domain. Generall, an initial guess is set for the domain parameters, then the flu is solved and the domain parameters are updated using optimization methods. This chapter begins with an introduction to neutron diffusion theor and the application of a finite element method to solve neutron diffusion problems. The implementation of the previous chapters optimization methods applied to inverse diffusion models are described net, first deriving optimalit conditions and then emploing Newton s method to solve them. The optimum control problem is formulated for multiple eperiments in the contet of the multigroup diffusion approimation.

31 7 A. Neutron Diffusion Theor The neutron diffusion equation is derived from the Boltzmann transport equation b integrating over all directions and using the diffusion theor epression for the neutron current derived from Fick s Law. The one-group neutron diffusion equation, shown in Equation. and., is a phase-space dependent equation that relates the neutron scalar flu phase-space distribution across a domain to its nuclear properties. D( r) Φ + (Σ a ( r) νσ f ( r))φ = Q( r) in Ω (.) Φ + D( r) nφ = J inc ( r) in Ω (.) B. Finite Element Diffusion Solver The forward diffusion models used in this problem are solved numericall using finite element methods. The finite element method is a numerical technique for finding approimate solutions of partial differential equations. This method differs from finite difference in such that finite difference methods approimate PDE equations while finite element methods approimate their solutions. Both of these methods discretize the domain into a mesh and the finite element method used in this work approimates the PDE s solution as a piecewise bi-linear function across each mesh cell. In the finite element setting, the diffusion equation becomes: [ A D + A Σ + ] M Ω Φ = AΦ = F (.) with:

32 . A D (i, j) = Ω b i b j (.) If D is constant, it can be factored out and S is known as the stiffness matri A D = D S (.5) with S(i, j) = D b i b j. (.) Ω. A Σ (i, j) = Ω Σb i b j (.7) If Σ is constant, it can be factored out and M is known as the mass matri. A Σ = Σ M (.) with M(i, j) = b i b j. (.9) Ω.. M Ω (i, j) = F (i) = Ω Ω b i b j (.) Qb i + J inc b i (.) Ω 5. Now, Φ is to be understood as a vector containing the flu values Φ i at the nodes.

33 9. Finite Element Meshes There are a variet of element meshes that can be implemented with finite element method. The most common of these are triangular and rectangular elements. After the domain has been broken elements, a set of piecewise polnomials are used for approimation. This must result in a function that is continuous with an integrable or continuous first or second derivative on the entire region. Polnomials of linear tpe in and in Equation. are often used with triangular elements and polnomials of bilinear tpe, shown in Equation. are used with rectangular elements. Φ(, ) = a + b + c (.) Φ(, ) = a + b + c + d. (.) The two dimensional domain is broken up into finite element meshes. This consists of a fine mesh to be used in the flu solver and a coarse mesh that will be the regions where the cross section (taken to be piece-wise constant) are reconstructed. The difference between these two meshes is a refinement which is variable in each dimension of the domain. This refinement is necessar due to the ill-posed problem and lack of information required to solve this inverse diffusion problem. An eample of these meshes and refinement is shown in Figure III-.

34 Fig. III-. Eample of the finite element meshes for the diffusion problem. Finite Element Methods An attractive feature of the finite element method is its abilit to handle complicated geometries with relative ease. This enables much more complicated domains and geometries to be solved where the finite difference method in its basic form is restricted to handle rectangular shapes and simple variations of. One reason for this is the finite element method s relative eas with which the boundar conditions are handled. A lot of problems have boundar conditions involving derivatives and irregularl shaped boundaries which are difficult to handle using finite difference techniques. The finite difference method handles these boundar conditions b approimating the derivative using a difference quotient at the grid points where irregular shaping of the boundar makes the grid point locations difficult. The finite element method handles the boundar conditions in a functional s integral that is being minimized, which is independent of the particular boundar conditions of the problem itself.

35 C. Optimization Functional. Misfit: To Minimize When an iterative solution is considered, a function called the misfit is introduced which represents the iterative solutions s (Φ) distance from the measured solution (z) at the boundar. In the case at hand, a snthetic measured solution is used b evaluating the forward model with the true material parameters. While iterating to satisf the optimalit conditions, this misfit represents the distance of Φ from z at the boundar and should converge to zero as the solution is approached. In the case of the problem, the misfit will be defined b Equation.. misfit = [Φ z], Ω (.) where Ω represents the portion of Ω where measurements are made. In the finite element setting: misfit = [Φ z]t M meas [Φ z], (.5) where M meas (i, j) = b i b j. Ω (.) If the entire boundar is used to measure data, then M meas = M Ω. This misfit is directl used in the Lagrangian.. Lagrangian Functional In constraint optimization problems, a L functional is formed and minimized consisting of two parts, one being the misfit representing distance from the true solution at the boundar and the other being constraints, here the diffusion equation acts as the governing equation, or in other words, L = misfit + constraint. The optimalit

36 conditions are derived from first derivatives of this Lagrangian with respect to each field variable and will be minimized in an iterative manner. The field variables are: Φ, λ, and Σ where Φ is the neutron flu, λ is the Lagrange multiplier or adjoint flu, and Σ is the set of piecewise continuous cross sections for the domain. Constraints must be applied as several solution sets ma satisf the misfit conditions and application of constraints helps in selecting these solutions. The governing phsics of the domain act as a constraint in the problem at hand and ma help select a solution that phsicall realistic. Application of the neutron diffusion equation here will be the constraint of choice, but additional constraints ma be implemented involving phsical limits: Σ a < D, Σ a >, Σ f < Σ a and ensuring a domain remains subcritical (k eff < ). Most all problems objective functions are smooth enough to where the additional constraints are not needed. The goal of this optimization problem is to find the saddle point in a Lagrangian functional L. If onl Σ is to be determined, then L(Φ, λ, Σ) = [Φ z]t M meas [Φ z] + λ T {[A D + A Σ + M Ω ] } Φ F. (.7) The KKT optimalit conditions arise as the derivatives of the Lagrangian with respect to each field variable and must be satisfied as the solution is approached. When L is the optimum, each of these optimalit conditions will be satisfied. [ L Φ = M meas [Φ z] + A D + A Σ + T Ω] M λ =, (.) [ L λ = A D + A Σ + ] M Ω Φ F =, (.9) L Σ = λt Σ AΦ =. (.)

37 There are several features embedded in the presented optimalit conditions, such as in Eq..9 the constraint to the optimization problem arises as the diffusion residual must approach zero. In Eq.. the adjoint diffusion term arises with the misfit as a forcing term which too must approach zero as the method nears the solution. The Lagrange multiplier, λ, has a clear meaning as the adjoint flu. Equations.-. form a nonlinear sstem of equations to be satisfied. These KKT optimalit conditions form a nonlinear sstem of equations, therefore Newton s method is emploed. D. Hessian Sstem Upon implementing Newton s method of optimization, the Hessian matri must be formed. This Hessian matri is the Jacobian matri of the KKT optimalit conditions which is composed of second derivatives of L. The derivatives of the optimalit condition are taken with respect to each field variable and put together to form a matri. L Φ = M meas (.) L λ = (.) L Σ = (.) L Σ Φ = ΣA T Φ (.) L Σ λ = ΣAΦ (.5) [ L λ Φ = A D + A Σ + ] M Ω (.) Note that M T = M and [ A D + A Σ + M Ω] T = [ AD + A Σ + M Ω] in the case of one-group diffusion approimation. To simplif the sstem, the notation

38 [ AD + A Σ + M Ω ] = A will be used, ielding the Hessian sstem below in Equation.7. M meas A T Σ Aλ δφ M meas [Φ z] + A T λ A Σ AΦ δλ = AΦ F λ T Σ A Φ T Σ A δσ λ T Σ AΦ (.7) where δφ, δλ, and δσ are updates and give the Newton iterate: Hδ k = F ( k ) (.) k+ = δ k + k. (.9) E. Implementation of Schur Complement This Hessian sstem can be simplified to reduce run time b a Gauss elimination of δφ and δλ to arrive at a sstem with onl δσ. The main matri that is inverted in the Schur complement solution has a lower condition number then the straight forward Hessian sstem. The second row of the above Hessian sstem is solved first for δφ in terms of δσ and constants. AδΦ + Σ AΦδΣ = AΦ + F (.) δφ = A ( AΦ + F Σ AΦδΣ) (.) The first row of the above Hessian sstem is solved for δλ in terms of δφ and δσ. M meas δφ + A T δσ + Σ AλδΣ = M meas [Φ z] A T λ (.) δλ = A T ( M meas [Φ z] A T λ M meas δφ Σ AλδΣ ) (.)

39 5 δλ = A T M meas [Φ z] A T λ M meas [A ( AΦ + F Σ AΦδΣ)] Σ AλδΣ (.) These solutions for δφ and δλ can be plugged back into the third row of the above Hessian sstem to solve for δσ. Starting with the third row from Equation.7: then filling Φ and λ solutions: λ T MδΦ + Φ T Mδλ = λ T MΦ (.5) Φ T Σ A A T M meas [Φ z] A T λ M meas [A ( AΦ + F Σ AΦδΣ)] Σ AλδΣ + λ T Σ A ( A ( AΦ + F Σ AΦδΣ) ) = λ T Σ AΦ (.) then grouping terms with and without δσ: [ λ T Σ AA ( Σ AΦ) + Φ T Σ A [ A T ( M meas A ( Σ AΦ) + Σ Aλ )]] [δσ] = λ T Σ AΦ λ T Σ AA ( AΦ + F ) Φ T Σ AA ( Mmeas [Φ z] A T λ M meas A ( AΦ + F ) ) (.7) The operator created on the left hand side of Eq..7 is the Schur complement for the sstem and will be called S. The right hand side will be called U for simple notation. S = λ T Σ AA (MΦ) + Φ T Σ A [ A T ( M meas A ( Σ AΦ) + Σ Aλ )] (.)

40 δσ = S U (.9) This method for the one group case creates a sstem for δσ onl which is significantl smaller in size then the original sstem shortening iteration runtime. F. Etension to Multiple Eperiments To also enable greater reconstruction abilities, multiple eperiments can be performed over a domain, each eperiment involving different source locations. Ever eperiment has a unique flu and adjoint solution to reconstruct the same cross sections to better the likelihood of success. The most logical choices are to break the boundar into halves, quarters, eighths, and siteenths. The optimalit conditions and misfit will be reduced for each of these eperiment s flu solutions while optimizing the same set of parameters for the domain. This will provide much more data and enable greater reconstruction abilit then a single eperiment. Man runs will be done with this code to test the limits of reconstruction with respect to various elements of the domain such as mesh size, number of eperiments, variable refinement, and domain size.. Optimalit Conditions The Lagrangian with multiple eperiments will be a simple summation over the Lagrangian for each eperiment. This Lagrangian for a total of I eperiments is: L (Φ, λ, Σ) = I i= [Φ i z i ] T M meas [Φ i z i ] + I λ T i [A iφ i F i ] (.) i= Similar to the single eperiment case, the optimalit conditions to be satisfied are derived from the first derivatives with respect to Φ i and λ i for each eperiment

41 7 and the same Σ as before. L Φ i = M meas [Φ i z i ] + A T i λ i i (.) L λ i = A i Φ i F i i (.) L I Σ = λ T i Σ AΦ i (.) i= This creates more conditions for the sstem to be satisfied enabling greater reconstruction abilit.. Hessian Sstem The Hessian sstem for multiple eperiments is similar to the single eperiment case. The corresponding second derivatives for the Hessian sstem were derived for each eperiment forming a new Hessian matri. L Φ i L λ i = M meas,i (.) = (.5) L Σ = (.) L Σ Φ i = Σ A T λ i (.7) L Σ λ i = Σ AΦ i (.) L I λ Φ = A D + A Σ + M Ω (.9) i= where M meas,i is the mass matri corresponding to eperiment i s measurement location. This multiple eperiment Hessian sstem has identical equations for Φ and λ but with an equation for each eperiment. These eperiments all operate over the same

42 set of cross sections, therefore the final equation has terms from ever eperiment. M meas, A T Σ Aλ δφ M meas, [Φ z ] + A T λ M meas, A T Σ Aλ δφ M meas, [Φ z ] + A T λ. A.. Σ AΦ δλ A Φ F = A Σ AΦ δλ A Φ F I λ T Σ A λ T Σ A Φ T Σ A Φ T Σ A δσ λ i Σ AΦ i i= (.5). Schur Complement Modification Ever time another eperiment is considered, another flu and adjoint correpsonding to that eperiment will provide additional matri equations in the Schur Complement. This final equation for δσ can be simplified and epressed as: [ I ] [ I ] δσ = S i U i i= i= (.5) where S i is the Schur complement and U i is the corresponding right hand side from Equations G. Multigroup Analsis The final modifications to the code account for multiple energ groups of neutrons. This multi-group code will allow reconstruction of multigroup cross sections. This model includes reconstruction of fission cross sections, fission spectrum, group removal cross section, and intergroup scattering cross sections. The optimalit conditions are again derived including first derivatives and the Hessian involving the second derivatives taken with respect to each of the new variables. It is supposed that this

43 9 model would enable greater acquisitions of realistic data that can be used to detect materials inside these multiple optical thickness thick objects. Multigroup diffusion theor has more meaningful application to the problem at hand. The incident neutron beams can be classified due to their energ and with material constants for uranium, or other fissile material, such as χ g, the fission cross sections for each group can be reconstructed to determine if fissile material is present in the cargo container. One eample of a test case would be if the incident neutrons were onl in the slow energ groups, but neutrons in fast energ groups were detected. Due to the nature that neutrons onl have a reasonable probabilit to upscatter in the thermal Mawellian range, those neutrons must have been born in the domain. That would be a greater chance of determination of SNM.. Multigroup Diffusion Theor In neutron diffusion theor, neutrons can be classified b their energ and broken into groups. Due to scattering and fission, neutrons are able to be redistributed in energ based on the magnitude of their cross sections and fission spectrum, χ g. In neutron diffusion theor equations of each group of neutrons can be formed with scattering terms that represent timerate densities of group to group scattering events. D g Φ g + Σ r,g Φ = χ g G g = G G νσ f,g Φ g + Σ s,g gφ g (.5) g =,g g Where Σ r,g = Σ a,g + Σ s,g g Φ g or removal from the group g due to absorption g =,g g and outscatter. An eample, the group diffusion operator is given below: A = D + Σ r, χ νσ f, Σ s, χ νσ f, (.5) Σ s, χ νσ f, D + Σ r, χ νσ f,

44 Everthing remains unchanged for the optimization problem, ecept there are more parameters (Φ g, λ g, and Σ g ) for multiple energ groups creating the same AΦ = F sstem. The diffusion operator is no longer smmetric, A T A, because of scattering and fission.. Cross-Section Data for Various Materials To gain a greater understanding of the reconstruction length-scale with respect to different materials, macroscopic cross sections for various materials are computed at fission spectrum average and MeV energies. Using these cross sections, diffusion coefficients and diffusion lengths can be compared for different materials that ma be present in a container. The macroscopic cross sections for various materials can be seen in Tables III-I - III-II. 5 The steel composition used consisted of: 5.% iron,.5% aluminum,.% chromium, and.% carbon. If fissile material were present, it ma be shielded with a strong absorbing material such as borated polethlene. Enriched boron is assumed in these computations at 9% B- and assuming uranium enriched to % U-5. The thermal, fission spectrum average, and. MeV macroscopic cross sections are computed and shown in Tables III-III - III-V. 5