International Journal of Computing and Visualization in Science and Engineering Vol 1.3, 2021.https://www.doi.org/10.51375/IJCVSE.2021.1.3

Optimization with Partial Differential Equation Constraints forBio-imaging Applications

Randolph E. Bank · Yiqian Wu · Yiwen Shi · Philip E. Gill · Yingxiao Wang ·Shaoying Lu

Received: 12 January 2018 / Accepted: 15 May 2018© The Author(s) 2018

Abstract Molecular transport and interaction are of funda-mental importance in biology and medicine. The spatiotem-poral diffusion map can reflect the regulation of molecu-lar interactions and their intracellular functions. To con-struct subcellular diffusion maps based on bio-imaging data,we explore a general optimization framework with diffu-sion equation constraints (OPT-PDE). For the solution ofthe spatially piecewise constant and anisotropic diffusion

Bank: The work of this author was supported by the National ScienceFoundation under contracts DMS-1318480 and DMS-1361421.

Wu: The work of this author was supported in part by the NationalScience Foundation under contract DMS-1361421.

Shi: The work of this author was supported in part by the NationalScience Foundation under contract DMS-1361421.

Gill: The work of this author was supported by the National ScienceFoundation under contracts DMS-1318480 and DMS-1361421.

Wang: The work of this author was supported by the National ScienceFoundation under contract DMS-1361421, and the National Instituteof Health under contracts HL121365 and CA204.

Lu: The work of this author was supported by the National ScienceFoundation under contract DMS-1361421, and the National Instituteof Health under contracts HL121365 and CA204.

Bank: Department of Mathematics, University of California, SanDiego, La Jolla, California 92093-0112. Email:rbank@ucsd.eduWu: Department of Bioengineering, Department of Mathematics, Uni-versity of California, San Diego, La Jolla, California 92093-0435.Email:y5wu@ucsd.eduShi: Department of Mathematics, University of California, San Diego,La Jolla, California 92093-0112. Email:yis018@ucsd.eduGill: Department of Mathematics, University of California, San Diego,La Jolla, California 92093-0112. Email:pgill@ucsd.eduWang: Department of Bioengineering, Department of Mathematics,University of California, San Diego, La Jolla, California 92093-0435.Email:yiw015@ucsd.eduLu: Department of Bioengineering, Department of Mathematics, Uni-versity of California, San Diego, La Jolla, California 92093-0435.Email:kalu@ucsd.edu

tensors, we develop an efficient block solver based on ap-plying Newton’s method to the first-order necessary condi-tion for optimality. We characterize the wellposedness ofthe OPT-PDE model problem and the convergence prop-erties of the solver. We also demonstrate the general util-ity of the solver in recovering spatially heterogeneous andanisotropic diffusion maps with computer-simulated bio-images. The results indicate that the solver can accuratelyrecover piecewise-constant isotropic and anisotropic diffu-sion coefficients, while exhibiting efficient convergence androbustness. This work highlights the power of the OPT-PDEmodel and the solver in recovering the diffusion map fromimaging data, and demonstrates that it has significant impli-cations in bio-imaging analysis.

Keywords Optimization, Partial Differential Equation,Constrained Optimization, OPT-PDE, Finite ElementMethods, Diffusion Coefficients

Mathematics Subject Classification (2010) 65N30,65N15, 65N50

1 Introduction

Molecular diffusion and transport is of fundamental impor-tance in cell biology [9,20]. Diffusion and transport of agiven molecule vary at different subcellular locations andcan represent its local activity and function. Therefore, thespatiotemporal diffusion map can reflect the regulation ofmolecular interactions and their intracellular functions [35,8,13]. However, the construction of subcellular diffusionmaps based on image data remains a tremendous challenge[25,7].

Traditional diffusion analysis approaches by fluores-cence recovery after photobleaching (FRAP) suffers from

2 Randolph E. Bank et al.

low spatiotemporal resolution, with significant limitationson the experimental photobleaching protocols [1,21,23]. Toovercome some of these limitations, fluorescence correla-tion spectroscopy (FCS) methods have been used to analyzethe diffusion process in live cells [7,40]. Nevertheless, thespatial resolutions remain relatively low, and FCS methodscan only handle images of low molecular density, and es-timate the diffusion rate of a subpopulation of intracellu-lar molecules. [7]. In contrast, model-based methods suchas finite difference or finite element methods have the ad-vantage of being theoretically well-defined, with high spa-tiotemporal resolution and less limitation on molecular den-sity. These methods have been applied to study spatially ho-mogeneous apparent molecular diffusion in live cells and tis-sues by the authors and others, respectively [25,36]. In thiswork, we develop a general optimization framework withdiffusion equation constraints (OPT-PDE) and finite elementdiscretization, with the goal of solving spatiotemporal het-erogeneous and anisotropic diffusion maps based on live-cell FRAP images.

OPT-PDE models are extremely versatile, with applica-tions ranging from biology, geophysics, aerodynamics, andfinance [15,14,38]. However, the formulation and solutionof OPT-PDE model problems are usually individualized anddependent on the specific applications involved. Therefore,it is important to develop the well-posedness theoreticalframework for each application. The analytic optimality the-ory for OPT-PDE problems is usually posed in terms of op-erators defined on reflective Banach spaces with Lp normsand inner products [16,12]. Thus, functional analysis the-ory on Banach spaces, the existence and stability results ofthe PDE constraints, the weak smoothness of the objectivefunctional, and the weak compactness of the Banach spaceof the state and control functions, are essential to guaranteethe existence of optimal solutions and the optimality condi-tions [16,12,6].

The optimization problem in our application is to min-imize a quadratic objective functional subject to a PDEwith an unknown piecewise constant diffusion map. Withtime-space discretization of the PDE, good estimates of thefirst and second derivatives of the discretized problem canbe computed with ease. Therefore, we utilize the moderndiscretize-then-optimize approach to solve the optimizationmodel, with all the finite element unknowns of the PDE asvariables [17,3]. As a result, the discrete optimization prob-lem has a large number of variables and constraints that can-not be handled efficiently by conventional general-purposeoptimization algorithms and software [26,11]. In order todeal with this difficulty, the proposed optimization algorithmis specifically designed to exploit the special structure inthe derivatives of the differential-equation constraints. TheOPT-PDE model was solved at the outer layer by Newton’smethod applied to the first-order necessary condition for op-

timality. At the inner layer, the large-scale linear system wassolved by block elimination, while the sub-block linear sys-tem derived from the PDE was iteratively solved by an effi-cient preconditioner [16,2].

In this work, we formulate a general OPT-PDE modelproblem for our applications and examine the theoreticalfoundation of the well-posedness of the model problem.We develop a new OPT-PDE solver algorithm for inverselyrecovering spatially heterogeneous diffusion tensors fromconcentration maps, and implement different biologicallymotivated test problems generated by simulation. With thenumerical solution of these test problems, we demonstratethat the proposed algorithm converges efficiently and re-covers the underlying diffusion tensors with high accuracy.Thus, this general OPT-PDE model and solution frameworkcan provide a new and important approach to recover spa-tially different diffusion coefficients and diffusion tensorsfrom bio-images.

2 Optimization model with partial differential equationconstraints

2.1 The Model Problem

An OPT-PDE model was formulated to estimate diffusionmaps based on fluorescence images. The time-evolution offluorescence images of live cells are usually recorded bymicroscope to represent the spatiotemporal distribution ofmolecular concentration u(x, t). The concentration maps atgiven time points t1 and t2 are known functions u1(x) andu2(x) of space. It is assumed that the time-evolution of se-quential fluorescence images is caused by the diffusion offluorescent molecules with an unknown diffusion coefficientmap d(x), which is a piecewise constant tensor function ofspace. The system of differential equations is given by

∂u(x, t)∂ t

= ∇ · (d(x)∇u(x, t))

u(x, t1) = u1(x)

u(x, t2) = u2(x),

for all x ∈Ω ⊂ R2 or R3, and t ∈ [t1, t2].The inverse problem for the diffusion map d(x) is for-

mulated as a PDE-constrained optimization problem, wherethe objective function includes the L2 norms of u− u2 andd − d0. The initial value problem is imposed as an equal-ity constraint. This gives a problem in which the objectivefunction

Φ(u(·, t2),d) =∫

Ω

(u−u2)2 + γ(d−d0)

2dx (Eq 2.1)

is minimized subject to the constraints u(x, t1) = u1, and

∇ · (d(x)∇u) =∂u∂ t

, for x ∈Ω and t1 < t ≤ t2, (Eq 2.2)

Optimization with Partial Differential Equation Constraints for Bio-imaging Applications 3

with the homogeneous Neumann boundary conditions

d(x)∇u ·~n = 0, for x ∈ ∂Ω , and t1 < t ≤ t2.

Here the unit vector ~n is the outward normal on the bound-ary ∂Ω . The diffusion coefficient map function is piecewiseconstant and bounded above and below by the box con-straint,

0 < dmin ≤ d(x)≤ dmax. (Eq 2.3)

In the objective function (Eq 2.1), the Tikhonov regulariza-tion term γ(d−d0)

2 is needed to ensure the well-posednessof the optimization problem, and to allow the efficient solu-tion of the linear system defined by the numerical discretiza-tion [37,33]. The scalar γ is chosen as a small constant oforder 10−5 to provide the needed regularization while notforcing the diffusion coefficient to the initial guess d0. A dis-advantage of the regularization term is that it can limit theaccuracy of the solution by forcing the diffusion coefficientsto converge to a point near d0. However, a more accuratesolution is possible if a sequence of problems are solvedin which the solution of one problem is used for the regu-larization term d0 of the next. This method with modifiedTikhonov regularization terms is similar to the proximal-point method for nonlinear optimization (see, e.g., [29,34]).The benefits of this approach are illustrated in Section 4.

The PDE constraint (Eq 2.2) is approximated in time us-ing the backward difference

∇ · (d∇u) =u−u1

∆ t,

or, equivalently,

−∇ · (d∇u)+(∆ t)−1u = (∆ t)−1u1.

This gives the weak form of the PDE constraint as: find u ∈H1(Ω) such that

a(u,v)≡∫

Ω

d∇u ·∇v+(∆ t)−1uvdx

=

∫Ω

(∆ t)−1u1vdx≡((∆ t)−1u1,v

)for all v∈H1(Ω). Let Sh⊂H1(Ω) denote the finite elementspace consisting of continuous piecewise linear polynomialsassociated with a shape regular triangulation Th of Ω . Thediscrete bilinear form can be written as: find uh ∈ Sh suchthat

a(uh,vh) = ((∆ t)−1u1,vh) for all vh ∈ Sh. (Eq 2.4)

Several observations may be made regarding the dis-cretization process. Although the test examples are gener-ated through simulation, in practice the data comes from livecell images, in which case the solutions u1, u2, the shapeof the domain Ω as well as its internal geometric details

will be limited by the image resolution. As these data arederived from point (pixel) values of the images, the use oflow-order finite elements is natural and appropriate in thissetting. The use of simple backward difference for the timediscretization is also appropriate for the same reason. Here∆ t is a constant bounded below and away from zero suchthat (∆ t)−1 remains bounded above. Practical problems willlikely have more than two data points in time; in anticipationof this, we note that the objective function can be general-ized by summing over multiple time steps. The time dis-cretization can also be extended to cover the case of multi-ple time steps. However, in this study, our main goal is tovalidate the convergence and efficiency of the overall opti-mization approach, and thus we present a simplified versionof the time evolution problem.

2.2 The well-posedness of the PDE

As the bilinear form a(·, ·) is symmetric, uniformly boundedabove and below (i.e., uniformly elliptic, see (Eq 2.3)),the Riesz representation theorem gives the existence anduniqueness of the solution [16,6]. In this case, the right-handside function f (x) = (∆ t)−1u1(x) is in L2(Ω). The theoryallows that the function f (x) be discontinuous at an internalinterface between subdomains within Ω [16,6].

Elliptic problems with discontinuous piecewise constantdiffusion coefficients are also called Laplace-interface prob-lems or transmission problems. The solutions of these prob-lems possess characteristic singularities, with decreased reg-ularity and low-order approximation errors in finite elementapplications [31]. The discontinuity of the diffusion coeffi-cients implies that the solution u will not have the piecewiseregularity H2(Ωi) implied by the piecewise L2 property ofthe right-hand side.

In the numerical examples of Section 4, the domain Ω

contains several subdomains, with discontinuous piecewiseconstant diffusion coefficients. Therefore, Ω = ∪K

k=1Ωk,and the diffusion map function d(x) = dk for x ∈ Ωk, andk = 1,2, . . . ,K. The dk’s are either positive constants in R,or symmetric positive-definite diffusion tensor matrices inR2×2. The weak form of the elliptic problem is to find u,such that for every test function v, the trilinear form satisfiesT (d,u,v) = 0 [12], where

T (d,u,v)≡K∑

k=1

∫Ωk

∇u · (dk∇v)dx+∫

Ω

1∆ t

uvdx

−∫

Ω

1∆ t

u1vdx = 0. (Eq 2.5)

Similar to the case of continuous constant diffusion coeffi-cients, if dk is uniformly bounded above and below (see (Eq2.3)), or if dk is a symmetric tensor matrix with eigenval-ues that are uniformly bounded above and below, the weak

4 Randolph E. Bank et al.

forms of elliptic problems with discontinuous diffusion co-efficients have a unique solution [12,6].

The regularity properties for elliptic problems with dis-continuous coefficients have been studied by Petzoldt inhis thesis and two related papers [31,30,32]. The major-ity of regularity results depend on the so-called “quasi-monotonicity condition” [31,10], which requires that thediffusion coefficients around each singular point be tracedin a monotonic order. In particular, the solution u is inH1+1/4(Ωi) if and only if quasi-monotonicity is satisfied,with the bound shown to be optimal [31,30,19]. All thepiecewise constant diffusion coefficients in our numericalexamples are quasi-monotone.

The ratio between the upper and lower bounds of the dif-fusion coefficients may be regarded as a measure of problem“stiffness”. If this ratio is bounded above and below by theuniform elliptic condition on the PDE operator, an iterativemultigrid solver can achieve a convergence rate independentof the jumps in the diffusion coefficients [39,28,42], whichis the case for the PDE solver used here.

2.3 The PDE-constrained optimization problem

Based on (Eq 2.1), the optimization problem is formulatedwith a PDE equality constraint, i.e.,

min(u,d)∈Uccs×Dccs

Φ(u,d) =∫

Ω

(u−u2)2 + γ(d−d0)

2dxdy,

subject to e(u,d)≡ ∇ · (d∇u)− u−u1

∆ t= 0, (Eq 2.6)

where Uccs = H1(Ω), and Dccs = d ∈ L∞(Ω), and 0 <

dmin ≤ d ≤ dmax.In weak form, the PDE constraint is the trilinear func-

tional given by (Eq 2.5). The discrete OPT-PDE problem isthen

min(uh,dh)∈RN×RN

Φ(uh,dh) = (uh−uh2)

T M(uh−uh2)

+ γ(dh−dh0)

T M(dh−dh0)

subject to T (d,uh,vh)≡

(K∑

k=1

dkAk

)uh +

M(uh−uh1)

∆ t= 0.

(Eq 2.7)

Here N is the number of nodes in the triangulation, whichrepresents the degrees of freedom in the discrete linear sys-tem, and K is the number of non-overlapping subdomainsΩk ⊂ Ω with distinct diffusion coefficients. M is the finiteelement mass matrix, and Ak is the stiffness matrix corre-sponding to the subdomain Ωk.

Based on the theory of invertible functions in Banachspaces [16,18,41], this optimization problem has a uniquesolution. In particular, it has been shown that the solution

of an elliptic boundary-value problem is an infinitely dif-ferentiable functional of the diffusion coefficients d, witha bounded d-derivative, and subsequently a bounded d-inverse [12]. When the objective function for the optimiza-tion is chosen as an energy norm that is dependent on thediffusion coefficients, it is a smooth and convex functionalof the coefficients [12,43,22]. With this objective functionthe discontinuous coefficients are estimated from measuredinterior fluorescent intensity or molecular concentrations, byapproximating the optimal solution in both continuous anddiscrete settings [12].

For the optimization of an objective function with PDEconstraints in the setting of our bioimaging applications, theminimization problem

min(u,d)∈Uccs×Dccs

Φ(u,d), such that e(u,d) = 0,

has an optimal solution if three conditions are satisfied [16].

1. Dccs is bounded, convex, and closed. Uccs is convex andclosed.

2. Φ(u,d) is sequentially weakly lower semi-continuous.3. The PDE constraint e(u,d) = 0 is continuous under

weak convergence and has a bounded d-inverse opera-tor e−1

d : d ∈ Dccs 7→ u(d) ∈Uccs.

For the continuous problem, Uccs is the set of concentrationfunctions, which is a closed subset of the Hilbert space H1

with norm and inner product. Dccs is the set of diffusion co-efficients, which is a bounded, convex, and closed subset ofL2. Both Uccs and Dccs are Hilbert spaces and thus reflex-ive Banach spaces. It follows that condition (1) holds. Inaddition, Φ is quadratic in both u and d, so it is obviouslycontinuous, and hence condition (2) is satisfied. Conditions(1) and (2) guarantee that Φ has an achievable infimum Φ∗,i.e., there is a minimizing sequence (uk,dk) ⊂Uccs×Dccs,such that Φ(uk,dk)Φ∗. As the convex and closed set Dccsis bounded, the sequence dk has a weakly convergent sub-sequence dki, such that dki d∗. The equality constraint,e(u,d) has a bounded d-inverse, which implies that condi-tion (3) holds. It follows that if uki = e−1

u (·,dki) and u∗ =e−1

u (·,d∗) then it can be shown that uki u∗. In addition,the objective function is quadratic, i.e., Φ(uki ,dki) Φ∗.Therefore, our continuous OPT-PDE problem has an opti-mal solution (u∗,d∗).

Similarly, for the discrete problem, Uccs is the set of dis-crete concentration, which is the finite element discretiza-tion of a bounded, convex, and closed subset of the Hilbertspace H1, or RN . Dccs is the set of discrete diffusion coef-ficients, which is a convex, and closed subset of RN . Thisimplies that both Uccs and Dccs are reflexive. In addition,Z = RN . Φ is quadratic in both uh and dh, so it is obviouslyweakly continuous. Because of the well-posedness and the

Optimization with Partial Differential Equation Constraints for Bio-imaging Applications 5

stability of the discrete PDE, the condition (3) is also sat-isfied. Therefore, the discrete OPT-PDE model problem hasan optimal solution (uh,∗,dh,∗).

In summary, these results for the continuous and discreteproblems imply that the solution of an optimization problemwith a PDE constraint exists when the objective function Φ

is weakly continuous, the domain of the constraint operatoris a compact subset of Banach space, and the equality con-straint contains a well-posed PDE with bounded d-inverse.

An alternative implementation of the objective functionin the discrete OPT-PDE model is to normalize the diffusioncoefficients by their magnitude in such a way that the con-vergence of the optimization problem and the error of solu-tions are independent of the values of diffusion coefficients.In this case, the discrete objective function becomes

Φh(uh,dh) =12(uh−uh

2)T M(uh−uh

2)

+γ

2

K∑k=1

(dk−d0,k

d0,k

)2

Area(Ωk), (Eq 2.8)

and the corresponding discrete OPT-PDE problem remainswell-posed.

3 Description of the algorithm

The discrete OPT-PDE problem given by (Eq 2.7) can besolved by minimizing the Lagrangian function [26,2,5].

L(u,d,v) =12(uh−uh

2)T M(uh−uh

2)

+12

γ(dh−dh0)

T M(dh−dh0)

+uh,T

(K∑

k=1

dkAk

)vh +∆ t−1(uh−uh

1)T Mvh,

where Ak is the subdomain stiffness matrix. First, we con-sider the simple case where dk is a scalar in each subdomainΩk, k = 1, 2, . . . , K.

The Lagrangian function is minimized numerically byapplying Newton’s method to the first-order conditions [26,2]. As the PDEs are solved to an accuracy that is finer thanthe convergence tolerance of the Newton iterations, we ex-pect the usual theoretical rate of convergence of Newton’smethod for a system of nonlinear equations [27]. The gra-dient of the Lagrangian is J = [ ∂L

∂u ,∂L∂v ,

∂L∂d ]

T . The quadraticfunctional of the control variable dh,

q(dh−dh0)≡

12

γ(dh−dh0)

T M(dh−dh0),

can be simplified when dh = [d1,d2, . . . ,dK ]T is a discrete

piecewise constant function in space. In particular,

q(dh−dh0) =

K∑k=1

12

γS(Mk)(dk−d0)2,

where S(Mk) is the sum of all entries in the subdomain massmatrix Mk, for k = 1, 2, . . . , K.

It follows that the gradient of the Lagrangian is

∂L∂d

=

γS(M1)(d1−d0)+uh,T A1vh

...γS(MK)(dK−d0)+uh,T AKvh

.The first-order necessary condition implies that the gradientof the Lagrangian functional is zero, i.e.,

M(uh−uh2)+Avh = 0

Auh−Muh1/∆ t = 0

∂L/∂d = 0

where A =∑K

k=1 dkAk + M/∆ t. The KKT matrix H (see,e.g., [26]) is given by

H =

M A CuA 0 Cv

CTu CT

v G

,where the submatrices Cu = [A1vh · · ·AKvh],Cv = [A1uh · · ·AKuh], and the diagonal matrixG = γ diag[S(M1), . . . ,S(MK)]. Observe that Cu and Cvmay be rank deficient if the discrete Laplacian of uh and vh

are both zero in a certain subregion. In this case, a nonzeroγ is needed to guarantee the non-singularity of the KKTmatrix and the well-posedness of the optimization problem.This notion will be further illustrated in the numericalresults section.

In the general case, each dk is a symmetric tensor ma-

trix dk =

[d11,k d12,kd12,k d22,k

]in the subdomain Ωk. In this case, the

Lagrangian function is

L(u,d,v) =12(uh−uh

2)M(uh−uh2)+q(dh−dh

0)

+uh,T ( K∑k=1

A(dk))vh +∆ t−1(u−uh

1)T Mvh.

The quadratic function q(dh−dh0) is

q(dh−dh0) =

14

γ(dh11−dh

11,0)T M(dh

11−dh11,0)

+12

γ(dh12−dh

12,0)T M(dh

12−dh12,0)

+14

γ(dh22−dh

22,0)T M(dh

22−dh22,0).

This definition of q(dh − dh0) is consistent with the scalar

case where dk =

[d11,k 0

0 d11,k

].

6 Randolph E. Bank et al.

The subregion stiffness matrix Ak(dk) can be written as alinear combination of three terms corresponding to the 2nd-order derivatives of u in the Laplace operator, i.e.,

Ak(dk) = d11,kA11,k +d12,kA12,k +d22,kA22,k,

where A11,k, A12,k, and A22,k are stiffness matrices corre-sponding to the respective diffusion tensors[

1 00 0

],

[0 11 0

], and

[0 00 1

].

It follows that, the partial derivatives of q and Ak are

∂q∂di j,k

=

12 γM(dh

i j,k−dhi j,k,0), when i = j;

γM(dhi j,k−dh

i j,k,0), when i 6= j,

and

∂Ak(dk)

∂di j,k= Ai j,k.

This gives the first-order derivatives of the Lagrangian as

∂L∂d

=

...12 γS(Mk)(d11,k−d11,k,0)+uh,T A11,kvh

γS(Mk)(d12,k−d12,k,0)+uh,T A12,kvh

12 γS(Mk)(d22,k−d22,k,0)+uh,T A22,kvh

...

.

The block sub-matrices in the KKT matrix H have 3Kcolumns, with

Cu =[· · · A11,kvh, A12,kvh, A22,kvh · · ·

],

Cv =[· · · A11,kuh, A12,kuh, A22,kuh · · ·

],

and

G = diag[· · · 1

2 γS(Mk), γS(Mk),12 γS(Mk) · · ·

].

In the numerical examples, the diffusion tensor is a 2× 2diagonal matrix with a known rotation matrix Q, i.e., dk =

QT[

d11,k 00 d22,k

]Q. In this case, the matrices Cu and Cv have

two columns corresponding to each such diffusion tensor ina subdomain, and the diagonal matrix G has two correspond-ing diagonal entries.

For both scalar and tensor diffusion, the Newton direc-tion p= [δu, δv, δd]T at the jth step can be calculated usingthe linear system involving the KKT matrix and the negativegradient of the Lagrangian evaluated at the jth iterate [26].

H p =−J(u j,v j,d j) =−

M(u j−uh2)+Av j

Au j−Muh1/∆ t)

(∂L/∂d) j

≡−Ju

JvJd

.

To simplify the notation, we drop the Newton iteration indexj. At each Newton step, the Newton equations may be ex-pressed in terms of the KKT equations [26], which are givenby the block 3×3 linear system of the formM A Cu

A 0 CvCT

u CTv G

δuδvδd

=−

JuJvJd

. (Eq 3.1)

(Normally such KKT systems are expressed in block 2× 2form with the zero block as the lower-right diagonal.) In thiscase, we wish to exploit the special sparse structure of themass matrix M and stiffness matrix A, with A being posi-tive definite. The corresponding equation can be solved effi-ciently by many standard iterative methods. In addition, theblock matrices Cu and Cv are rectangular sub-matrices con-sisting of column vectors, and G is a low-dimensional di-agonal matrix. Each block system is solved using a variantof block Gaussian elimination that exploits existing efficientsolvers for the linear systems with matrix A. Starting withblock factorization, the matrixM A Cu

A 0 CvCt

u Ctv G

=

I 0 00 A 00 0 I

M I 0I 0 0

Ctu Ct

v G

I 0 Cv0 I Cu0 0 I

I 0 00 A 00 0 I

,(Eq 3.2)

where

ACv =Cv,

Cu =Cu−MCv,

G = G−CtuCv−Ct

vCu.

For each subdomain scalar diffusion coefficient, the compu-tation of Cv requires the solution of one linear system with A,while each 2×2 diffusion tensor requires the solution of twoor three such linear systems. At the right-hand side of (Eq3.2), each of the two block-diagonal matrices requires thesolution of one linear system with A. It follows that a totalof Ks+2Kr+3Kt +2 elliptic PDE systems must to be solvedat each Newton step, where Ks, Kr, and Kt are the numberof scalars, tensors with a known rotation, and those with anunknown rotation respectively, such that Ks +Kr +Kt = K.The block lower-triangular system requires the solution ofone dense linear system with the low-dimensional symmet-ric matrix G.

The overall solution procedure given below is similarto an earlier version of the algorithm described and imple-mented in the PDE solver package PLTMG [2]. As the linearsystems involving A may be solved approximately by itera-tion, we introduce the matrix Cv ≈ Cv, which is generally theapproximate Cv saved from the previous Newton step. Thisallows the matrix Cv to be updated iteratively at each New-ton step. The matrix Cv is initially set to zero, and updatedat every Newton step.

Optimization with Partial Differential Equation Constraints for Bio-imaging Applications 7

The first step is to solve

AJv = Jv,

AW =Cv−ACv,

Cv← Cv +W,

Cv = Cv.

All the linear systems involving A are solved approximatelyusing an incomplete LU factorizing (ILU) preconditionedconjugate-gradient method. Then we form

Ju = Ju−MJv,

Cu =Cu−MCv,

which requires sparse matrix multiplications with M. Thenext step is to compute δd using the Schur complement

G = G−CtuCv−Ct

vCu,

Gδd =−(Jd−CtuJv−Ct

vJu).

Finally, δu and δv are formed from

δu =−(Jv +Cvδd),

Aδv =−(Ju +Cuδd).

The second equation requires the linear solver for A. Thissolver is implemented in PLTMG, which is applied to threenumerical examples in the next section.

4 Numerical Results

In order to illustrate the numerical stability and computa-tional efficiency of the OPT-PDE approach, the solver wasapplied to four examples arising from three different scenar-ios of molecular diffusion in live cells, i.e., spot diffusion,layered diffusion, and filament tensor diffusion. In each ofthe examples, the forward numerical simulation functionsin our Fluocell software package were used to generate in-put concentration maps of known diffusion maps [25,24].These concentration maps and their finite element triangu-lations were read into PLTMG as input for the OPT-PDEmodel. The diffusion coefficients were then estimated bysolving the optimization problem described in (Eq 2.8) us-ing the OPT-PDE algorithm of PLTMG, which implementsthe method described in Section 3. The computed diffusioncoefficients were compared with the known diffusion mapto determine the accuracy and efficiency of the OPT-PDEsolver. The PLTMG OPT-PDE solver was run on an AppleMac Pro Laptop computer with an intel i5 dual-core proces-sor (2.9 GHz) and 8GB RAM.

a b

Figure 1

Fig. 4.1: The cell model and the photobleaching patternused in the simulation. (a) The basic geometry of the cellmodel (white) used for the spot diffusion and filament tensordiffusion simulation. (b) shows the subcellular region to bephotobleached in dark gray.

In the forward simulation, a certain region in an ellipticcell model is designated as the photobleached region (seeFigure 4.1), which represents the subcellular region withlower initial concentration at time t = 0s. A diffusion mapon the cell model is designated to be piecewise constantwith assigned values on a given subdomain structure (e.g.,Figure 4.2a). A computer simulation was then performedto mimic the fluorescence recovery after photobleaching(FRAP) process via diffusion (Figure 4.2b) [25].

The OPT-PDE algorithm has a three levels of iterations.At the top level, a sequence of “rounds” are performed inwhich the optimization problem is solved with the regular-ization term d−d0 defined with d0 the solution of the prob-lem from the previous round. The next level of iterations arethose of Newton’s method applied to the discretized first-order optimality conditions. At the lowest level, the itera-tions are those of the method used to solve the KKT equa-tions (Eq 3.1).

4.1 Spot Diffusion

Under the assumption that intra-cellular molecules diffusemore slowly in certain subcellular regions, smaller diffu-sion coefficients were assigned to four small elliptic or cir-cular sub-cellular regions within the cell body. The cor-responding diffusion map and finite element triangulationwere generated accordingly (Figure 4.2a), with the photo-bleach pattern shown in (Figure 4.1b). The forward diffusionsimulator of Fluocell was used to simulate FRAP, with thesimulated concentration u1, u2, and scaled change of con-centration u2 − u1 shown in (Figure 4.2b). The change ofconcentration u2 − u1, was scaled by multiplying the fac-tor s = (maxx(u1,u2)−minx(u1,u2))

−1, such that its mag-nitude is less or equal to 1. The new solver implementedin PLTMG takes the forward simulation results as an inputand the concentration and diffusion maps are estimated bymultiple rounds of OPT-PDE solving (Figure 4.3)[2]. The

8 Randolph E. Bank et al.

regularization constant is γ = 10−5 for all three test prob-lems. Alternatively, it is possible to adjust the values of γ ateach round of OPT-PDE solve, which may potentially speedup convergence, although this approach was not adopted inour solver. For all the test problems, the lower and upperbounds for the diffusion coefficients are dmin = 0.1 µm2/sand dmax = 500 µm2/s.

For the first round of OPT-PDE solves, the initial guessof the diffusion coefficients were 10 µm2/s for all subcellu-lar regions, which was implemented into the solver as d1,0and d2,0 in the regularization term of the objective function.It took four steps for Newton’s method to converge to theoptimal solution, with a Newton increment larger than orequal to 10−2 and the residual decreasing until it was smallerthan 10−5 (Figure 4.3(a)). The PDE was discretized usingfinite element method, and the resulting linear system in-volving the stiffness matrix A was solved using incompleteLU factorization [2,4]. There were 3882 mesh nodes, and ittook 1.23 s to solve each linear system, with majority (about69.2%) of time spent on assembling the stiffness and massmatrices (Figure 4.3b). As there were two different scalardiffusion coefficients, four solves of the PDE were neededfor each Newton’s iteration. The computing time necessaryto solve a single PDE is estimated to be 0.18s, and the over-all computing time is 5.15s, which is a 29-fold increase overthe cost of a single solve. The diffusion coefficients con-verged to d1 = 27.4 µm2/s and d2 = 6.79 µm2/s at the endof the 1st round of the OPT-PDE solver (see Figure 4.3c).

In the 2nd round of the OPT-PDE solver, the estimatesd1,0 and d2,0 were set to the solutions from the first OPT-PDE solve, which provide an improved estimation of theoptimal diffusion coefficients (Eq 2.8). By iteratively im-proving the regularization term in the objective function, theOPT-PDE solver converges at round 6, when the incrementsin the diffusion coefficients were less than 1% of their ini-tial values (see Figure 4.3c). The solutions of diffusion co-efficients are recorded as d∗1 and d∗2 . The convergence pro-cess of the relative diffusion coefficients d1/d∗1 and d2/d∗2are shown in Figure 4.3d. These results show that the opti-mized solution of d is accurate, and that both diffusion co-efficients converged with similar efficiency, as intended bythe normalized regularization terms in (Eq 2.8). In addition,we can estimate the optimal Lagrange multiplier vh and thescaled solution of u− u1 (Figures 4.3e and 4.3f). The solu-tion u− u1 was scaled in the same way as the input changeof concentration, u2− u1, as shown in Figure 4.2b. The re-sults given in Figures 4.2b and 4.3f) indicate that the scaledu−u1 approximates the scaled u2−u1 very well.

In order to demonstrate the need for the regularizationterm, a model problem was constructed where the KKT ma-trix is singular, or the discrete Laplacian of the solution iszero in certain local regions. In this case, a nonzero regu-larization term is necessary to ensure that the discrete op-

timization problem is well-posed. As in the forward simu-lation results shown in Figure 4.3, the diffusion map, tri-angulation, u1, u2, and scaled u1 − u2 are shown in Fig-ure 4.4. If the regularization constant γ is zero and theinitial estimate of the diffusion coefficients was 10 µm2/s,Newton’s method was unable to compute a solution. How-ever, for γ = 10−5, the solver converged successfully tod1 = 29 µm2/s, d2 = 5 µm2/s. In this case d3 converged tothe initial estimate 10 µm2/s because of the lack of infor-mation provided by the diffusion equation (see Figure 4.5).These results show that the regularization term is necessaryto ensure that the optimization problem is well-posed whenthe KKT matrix is rank deficient. Regularization can alsobe used to improve the rate of convergence and robustnessof the solver when an optimization problem is well-posed,or when the block Gaussian elimination is too expensive tocompute (see, e.g., [29]). Therefore, the proposed theoreti-cal and numerical framework can serve as a foundation forsolving a wider class of optimization problems with PDEconstraints.

4.2 Layered Diffusion

In the second example, the cell model was divided into fourlayers in the axial direction, with four different scalar diffu-sion coefficients (see Figure 4.6a). The photobleach patternwas defined to allow the bleached region encompass all fourlayers (Figure 4.6b). Again, forward simulation of the dif-fusion process was performed to generate the concentrationmaps u1 and u2 at different time points after photobleach-ing, as well as the scaled change of concentration u2− u1(Figure 4.6c). These simulation results, but not the diffusionmap, were then used as input for the OPT-PDE solver inPLTMG to estimate the diffusion map given in Figure 4.7.

Similar to the first numerical example, PLTMG took theforward simulation results as input and estimated the con-centration and diffusion maps by multiple rounds of OPT-PDE solving, with γ = 10−5 and adaptive regularizationterms in the objective functions. For the initial round ofOPT-PDE solve, the initial guess of the diffusion coefficientswere again 10 µm2/s for all subcellular regions. There were6820 mesh nodes, and it took 2.38 s to solve each PDE prob-lem (Figure 4.7b). The computing time necessary to solve asingle PDE is estimated to be 0.38s, and the overall comput-ing time is 6.59s, which is a 17-fold increase over the costof a single solve. For each Newton iteration, six solves ofthe PDE were needed to compute the four diffusion coeffi-cients. Each Newton iteration required 4 steps to convergeto the optimal solution of the objective (Figure 4.7(a)). TheOPT-PDE solver converged at round 3 (Figure 4.7c), withthe diffusion coefficients converges to d∗1 , d∗2 , d∗3 and d∗4 .The convergence of the relative diffusion coefficients di/d∗i

Optimization with Partial Differential Equation Constraints for Bio-imaging Applications 9

Figure 2

a b

Diffusion Map5 29 µm2/s

u1 at 1.17 s u2 at 2.17 s Scaled (u2-u1)6.0e40 -0.08 0.12

Fig. 4.2: Forward simulation results. (a) The diffusion map for spot diffusion overlaid with the finite element mesh. Thediffusion coefficient of the large cell body area was assigned to be d1 = 29 µm2/s, while that of the small elliptic regionswas d2 = 5 µm2/s. (b) The left and middle figures depict the simulated concentration maps of the diffusing molecules atdifferent time points after photobleaching. The right figure gives the scaled concentration map u2−u1 used as the input forthe OPT-PDE solver.

a Residue Increment

Newton Step

b

f

0.13

-8.28e-2

5.84e-5

-1.38e-4

e Scaled (u-u1)The Lagrange Multiplier

d

PLTMG 2.9Percentage of Computing Time (1.23 s)

0 2 4 6

1E1

1E0

1E-1

1E-2

1E-3

1E-4

1E-5

Nor

m. D

iff. C

oef.

Figure 3

c

Diff

. Coe

f. (µ

m2 /s

)

OPT-PDE Solve

d1

d2

0 1 2 3 4 5 6 0 1 2 3 4 5 6

Assemblym-Graph

69.227.9

Fig. 4.3: The OPT-PDE solution for the spot diffusion problem. At the initial round of the OPT-PDE solve, (a) shows theNewton increments and residuals (logarithmic scale) vs the number of Newton steps; (b) shows the time for each Newtoniteration displayed as a pie graph that gives the percentage of time used by each component of the solver. (c) The optimaldiffusion coefficients after each round of OPT-PDE solves. The exact values of d1 and d2 used in the forward simulationare plotted in black/white stripes as the reference. (d) The normalized diffusion coefficients are shown versus the OPT-PDEsolving rounds. d∗1 and d∗2 are the optimal solutions of diffusion coefficients after convergence. (e) The Lagrange multiplier.(f) The scaled solution of u−u1.

10 Randolph E. Bank et al.

5 29 µm2/s

Figure 4

a b

Diffusion Map0.19- 0.1

u1 at 10.08 s u2 at 20.08 s Scaled (u2-u1)6.0e40

Diffusioncoefficients=29,15,5.Doesnotconvergeforgamma=0,initialguess10,10,10

Fig. 4.4: Forward simulation results. (a) The diffusion map for spot diffusion overlaid with the finite element mesh. Thediffusion coefficient of the large cell body area was assigned to be d1 = 29 µm2/s, while those of the two small ellipticregions at the right side were each d2 = 5 µm2/s. The diffusion coefficients of the two small regions on the left were bothd2 = 15 µm2/s. (b) The left and middle figures give the simulated concentration maps of the diffusing molecules at differenttime points after photobleaching. The right figure gives the scaled concentration map u2−u1 used as the input for the OPT-PDE solver.

a Residue Increment

Newton Step

b

f

0.16

-0.11

1.13e-2

-7.69e-3

e Scaled (u-u1)The Lagrange Multiplier

d

PLTMG 2.9Percentage of Computing Time (0.94 s)

0 2 4 6 8

1E1

1E0

1E-1

1E-2

1E-3

1E-4

1E-5

Nor

m. D

iff. C

oef.

Figure 5

c

Diff

. Coe

f. (µ

m2 /s

)

OPT-PDE Solve

d1

d2

0 1 2. 3 4 0 1 2 3 4 0 1 2 3 4

Assemblym-Graph

60.436.7

d3

Fig. 4.5: The OPT-PDE solution for the spot diffusion problem with necessary Tikohnov regularization. At the initialround of OPT-PDE solve, (a) shows the Newton increments and residuals (logarithmic scale) vs the number of Newtonsteps; (b) shows the computing time of each Newton iteration displayed in the pie graph showing the percentage of timeused by each solver component. (c) The optimal diffusion coefficients after each round of OPT-PDE solve. The exact valuesof d1, d2, and d2 used in forward simulation are plotted in black/white stripes as the reference. (d) The normalized diffusioncoefficients are shown versus the OPT-PDE solving rounds. d∗1 , d∗2 , and d∗3 are the optimal solutions of diffusion coefficientsafter convergence. (e) The Lagrange multiplier. (f) The scaled solution u−u1.

Optimization with Partial Differential Equation Constraints for Bio-imaging Applications 11

0.16- 0.1

u1 at 1.21 s u2 at 2.21 s Scaled (u2-u1)

6.0e40

Figure 6

30

1

a

bDiffusion Map (µm2/s )

c

a b

Diffusion Map (µm2/s ) - 0.1 0.16

u1 at 1.21 s u2 at 2.21 s Scaled (u2-u1)

0 6.0e-4

c30

1

Fig. 4.6: Forward simulation results. (a) The diffusion map for layered diffusion overlaid with the finite element mesh.Moving from the outside to the inside, the diffusion coefficients of the four layers are d1 = 5 µm2/s, d2 = 20 µm2/s, d3 =

30 µm2/s, and d4 = 1 µm2/s. Panel (b) shows the photobleach pattern in gray. (c) The left and middle figures give thesimulated concentration maps of the diffusing molecules at different time points after photobleaching. The right figuredepicts the scaled concentration map u2−u1 used as the input for the OPT-PDE solver.

aResidue Increment

Newton Step

b

f

0.16

-0.10

3.31e-5

-1.42e-5

eThe Lagrange Multiplier

c d

PLTMG 2.2Percentage of Computing Time (2.38 s)

0 2 4 6

1

0

-1

-2

-3

-4

-5

-6

Nor

m. D

iff. C

oef.

Figure 5

Diff

. Coe

f. (µ

m2 /s

)

OPT-PDE Solve0 3 0 3 0 3 0 3

d1

d2

d3

d4

OPT-PDE Solve

Scaled (u-u1)

Assemblym-Graph

57.740.1

Fig. 4.7: The OPT-PDE solution for the spot diffusion problem. At the initial round of OPT-PDE solve, (a) shows theNewton increments and residuals (logarithmic scale) vs. the number of Newton steps; (b) shows the computing time ofeach Newton iteration displayed in the pie graph showing the percentage of time used by each solver component. (c) Theoptimal diffusion coefficients after each round of the OPT-PDE solve. The accurate values of di, i = 1, 2, 3, 4 used in forwardsimulation are plotted in black/white stripes as a reference. (d) The normalized diffusion coefficients are shown vs the OPT-PDE solving rounds. d∗i , i = 1, 2, 3, 4 are the optimal solutions at convergence. (f) The Lagrange multiplier. (g) The scaledsolution of u−u1.

12 Randolph E. Bank et al.

is shown in Figure 4.7d, indicating that all diffusion coef-ficients converged with similar efficiency. In addition, weestimated the optimal Lagrange multiplier vh and confirmedthat the scaled solution of u− u1 approximates the scaledchange of concentration, u2−u1 (Figures 4.6c and 4.7e-f).

4.3 Filament Tensor Diffusion

In cells, it is possible that anisotropic diffusion can occuralong the long axis of certain filamentous structures. Tostudy the applicability of our solver to such scenarios, wedesigned a diffusion map with two independent filaments,where the diffusion coefficients were set to be isotropic inthe general cell model and anisotropic on the filaments (Fig-ure 4.8(a)). The diffusion tensor was calculated using thetransformation matrix based on the known orientations ofthe filaments (Figure 4.8a). The same photobleach patternfrom spot diffusion simulation was used (4.1b). As such, weutilized the known orientation of the filaments to constructthe finite element mesh, simulated locally anisotropic dif-fusion and the resulting concentration maps. These, but notthe diffusion map, were used as input to test the OPT-PDEsolver in PLTMG.

As in the previous examples, PLTMG took the forwardsimulation results as input and were able to estimate theconcentration and diffusion maps by ten rounds of OPT-PDE solve (Figure 4.9), with adaptive objective functions.The initial guess of the diffusion coefficients were again10 µm2/s for all directions and subcellular regions. In theinitial round of OPT-PDE solve, Newton’s iteration tookfour steps to converge (Figure 4.9(a)). There were 3919mesh nodes, and it took 1.31s to solve each PDE problem(see Figure 4.9b). The computing time necessary to solve asingle PDE is estimated to be 0.21s, and the overall com-puting time is 12.64s, which is a 60-fold increase over thecost of a single solve. Each Newton iteration required fivesolves of the PDE for the three diffusion components. Atthe 10th round of the PDE-OPT solve (Figure 4.9c), the dif-fusion coefficients converges to d∗1 , d∗2 , and d∗3 . The conver-gence of the relative diffusion coefficients di/d∗i is shown inFigure 4.9d, indicating that d1/d∗1 and d3/d∗3 converged ata similar rate while d2/d∗2 was slower to converge. The dif-ferent convergence rates may be caused by the small size ofthe filaments relative to the cell body, as well as the 20-folddifference in the diffusion rate along the filament comparedto the cell body. The optimal Lagrange multiplier vh is givenin Figure 4.9e). The scaled solution u− u1, which approxi-mates the scaled change of concentration u2−u1, is depictedin Figures 4.8c and 4.9f.

Taken together, the numerical simulation and solutionresults demonstrate that the proposed OPT-PDE solver pro-vides a highly accurate, computationally efficient, and nu-

merically robust tool for estimating the spatial diffusionmaps in both isotropic and anisotropic problems.

5 Discussion and Future Work

In this work, we consider the solution of biologically mo-tivated inverse problems associated with spatially heteroge-neous diffusion coefficients based on known concentrationmaps. The problem is formulated as an optimization prob-lem with the diffusion equation as a constraint. The well-posedness of both the PDE and the optimization model isestablished. Newton’s method is used to solve the finite-dimensional optimization problem obtained by discretizingthe PDE with finite elements in space and finite differencesin time. An efficient OPT-PDE solver is formulated that ex-ploits the special algebraic structure of the finite-elementstiffness and mass matrices. Finally, numerical examplesare presented that demonstrate the robust and efficient con-vergence of the solver for three example problems withspatially discontinuous or anisotropic diffusion coefficients.These results indicate that the proposed OPT-PDE approachfor finding the diffusion coefficients is accurate and robust,and it may be used for cellular image-based analysis to re-construct molecular transport parameters.

The ultimate goal of this work is to develop an OPT-PDEanalysis framework that can be used to reveal the spatiotem-poral map of molecular diffusion and transport in live cellswith high resolution. Therefore, a future research directionis to generalize the OPT-PDE solver to imaging problems.The OPT-PDE solver itself can be developed to allow thediffusion map to be a piecewise linear function on the do-main without knowing the specific geometric configurationof the subdomains. The PDE constraint can be a sophisti-cated reaction-diffusion system, from 2D to 3D model prob-lems, or directly from bio-imaging data. In summary, we en-vision significant bio-imaging applications for this class ofOPT-PDE models and a great potential to explore and de-velop new functionality for the solver.

6 References

References

1. Axelrod, D., Koppel, D.E., Schlessinger, J., Elson, E., Webb,W.W.: Mobility measurement by analysis of fluorescence photo-bleaching recovery kinetics. Biophys J 16(9), 1055–69 (1976).URL http://www.ncbi.nlm.nih.gov/pubmed/786399. DOI10.1016/S0006-3495(76)85755-4

2. Bank, R.E.: Pltmg: A software package for solving elliptic partialdifferential equations, user guide 11.0. Technical Report, Depart-ment of Mathematics, University of California, San Diego (2012)

3. Bank, R.E., Gill, P.E., Marcia, R.F.: Interior methods for a class ofelliptic variational inequalities. In: Large-scale PDE-constrainedoptimization (Santa Fe, NM, 2001), Lect. Notes Comput. Sci. Eng.,

Optimization with Partial Differential Equation Constraints for Bio-imaging Applications 13

Figure 6

a b

Diffusion Map5 100 µm2/s 0.19- 0.1

u1 at 10.08 s u2 at 20.08 s Scaled (u2-u1)6.0e40

Fig. 4.8: Forward simulation results for filament tensor diffusion. (a) The diffusion map with the magnitudes of the tensordiffusion vectors overlaid with the finite element mesh. The diffusion coefficients were set to be d1 = 5 µm2/s in the generalcell area, d2 = 100 µm2/s along the filaments, and d3 = 5 µm2/s perpendicular to the filaments. (b) The left and middlefigures give the simulated concentration maps of the diffusing molecules at different time points after photo-bleaching. Theright figure gives the scaled concentration map u2−u1 used as the input for the OPT-PDE solver.

a

Residue Increment

Newton Step

b

d

0.19

- 0.1

2.07e-5

-2.38e-5

e The Lagrange Multiplier

c

f

PLTMG 2.5Percentage of computing Time (1.31 s)

0 2 4 6

1

0

-1

-2

-3

-4

-5

-6

Con

verg

ence

Figure 7

Diff

. Coe

f. (µ

m2 /s

)

OPT-PDE Solve

d1

d2

d3

0 4 8 0 4 8 0 4 8

Scaled (u-u1)

Assemblym-Graph

68.728.7

Fig. 4.9: The OPT-PDE solution for the filament tensor diffusion problem. At the initial round of OPT-PDE solve, (a)shows the Newton increments and residuals (logarithmic scale) vs the number of Newton steps; (b) shows the computingtime of each Newton iteration displayed in the pie graph showing the percentage of time used by each solver component. (c)The optimal diffusion coefficients after each round of OPT-PDE solve. The accurate values of di, i = 1, 2, 3 used in forwardsimulation are plotted in black/white stripes as a reference. (d) The normalized diffusion coefficients are shown versus theOPT-PDE solving rounds. d∗i , i = 1, 2, 3 are the optimal values after convergence. (f) The Lagrange multiplier. (g) The scaledsolution u−u1.

14 Randolph E. Bank et al.

vol. 30, pp. 218–235. Springer, Berlin (2003). DOI 10.1007/978-3-642-55508-4 13

4. Bank, R.E., Smith, R.K.: An algebraic multilevel multigraph al-gorithm. SIAM J. Sci. Comput. 23(5), 1572–1592 (electronic)(2002). DOI 10.1137/S1064827500381045

5. Biros, G., Ghattas, O.: Parallel Lagrange-Newton-Krylov-Schurmethods for PDE-constrained optimization. I. The Krylov-Schursolver. SIAM J. Sci. Comput. 27(2), 687–713 (electronic) (2005).DOI 10.1137/S106482750241565X

6. Brenner, S.C., Scott, L.R.: The mathematical theory of finiteelement methods, Texts in Applied Mathematics, vol. 15, sec-ond edn. Springer-Verlag, New York (2002). DOI 10.1007/978-1-4757-3658-8

7. Capoulade, J., Wachsmuth, M., Hufnagel, L., Knop, M.: Quan-titative fluorescence imaging of protein diffusion and interac-tion in living cells. Nat Biotechnol 29(9), 835–9 (2011). URLhttp://www.ncbi.nlm.nih.gov/pubmed/21822256. DOI 10.1038/nbt.1928

8. Cebecauer, M., Spitaler, M., Serge, A., Magee, A.I.: Signallingcomplexes and clusters: functional advantages and methodologi-cal hurdles. J Cell Sci 123(Pt 3), 309–20 (2010). URL http://

www.ncbi.nlm.nih.gov/pubmed/20130139. DOI 10.1242/jcs.061739

9. Crank, J.: The Mathematics of Diffusion, Second edn. ClarendonPress, Oxford (1975)

10. Dryja, M., Sarkis, M.V., Widlund, O.B.: Multilevel Schwarz meth-ods for elliptic problems with discontinuous coefficients in threedimensions. Numer. Math. 72(3), 313–348 (1996). DOI 10.1007/s002110050172

11. Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithmfor large-scale constrained optimization. SIAM Rev. 47(1), 99–131 (electronic) (2005). DOI 10.1137/S0036144504446096

12. Gockenbach, M.S., Khan, A.A.: An abstract framework for ellip-tic inverse problems. I. An output least-squares approach. Math.Mech. Solids 12(3), 259–276 (2007). URL http://dx.doi.org/

10.1177/1081286505055758. DOI 10.1177/108128650505575813. Groves, J.T., Kuriyan, J.: Molecular mechanisms in signal trans-

duction at the membrane. Nat Struct Mol Biol 17(6), 659–65 (2010). URL http://www.ncbi.nlm.nih.gov/pubmed/

20495561. DOI 10.1038/nsmb.184414. Haber, E., Hanson, L.: Model problems in PDE-constrained opti-

mization. Report, Emory University (2007)15. Herzog, R., Kunisch, K.: Algorithms for PDE-constrained opti-

mization. GAMM-Mitt. 33(2), 163–176 (2010)16. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with

PDE constraints, Mathematical Modelling: Theory and Applica-tions, vol. 23. Springer, New York (2009)

17. Hoppe, R.H.W., Petrova, S.I., Schulz, V.: Primal-dual Newton-type interior-point method for topology optimization. J. Op-tim. Theory Appl. 114(3), 545–571 (2002). DOI 10.1023/A:1016070928600

18. Jittorntrum, K.: An implicit function theorem. J. Optim. TheoryAppl. 25(4), 575–577 (1978). DOI 10.1007/BF00933522

19. Kellogg, R.B.: On the Poisson equation with intersecting inter-faces. Applicable Anal. 4, 101–129 (1974/75). Collection of arti-cles dedicated to Nikolai Ivanovich Muskhelishvili

20. Kinkhabwala, A., Bastiaens, P.I.: Spatial aspects of intracellularinformation processing. Curr Opin Genet Dev 20(1), 31–40 (2010)

21. Klonis, N., Rug, M., Harper, I., Wickham, M., Cowman, A., Tilley,L.: Fluorescence photobleaching analysis for the study of cellulardynamics. Eur Biophys J 31(1), 36–51 (2002). URL http://

www.ncbi.nlm.nih.gov/pubmed/12046896

22. Knowles, I.: Parameter identification for elliptic problems. J.Comput. Appl. Math. 131(1-2), 175–194 (2001). URL http://

dx.doi.org/10.1016/S0377-0427(00)00275-2. DOI 10.1016/S0377-0427(00)00275-2

23. Lippincott-Schwartz, J., Altan-Bonnet, N., Patterson, G.H.: Pho-tobleaching and photoactivation: following protein dynamics inliving cells. Nat Cell Biol Suppl, S7–14 (2003). URL http:

//www.ncbi.nlm.nih.gov/pubmed/14562845

24. Lu, S., Kim, T.J., Chen, C.E., Ouyang, M., Seong, J., Liao, X.,Wang, Y.: Computational analysis of the spatiotemporal coordina-tion of polarized pi3k and rac1 activities in micro-patterned livecells. PLoS One 6(6), e21,293 (2011). DOI 10.1371/journal.pone.0021293

25. Lu, S., Ouyang, M., Seong, J., Zhang, J., Chien, S.,Wang, Y.: The spatiotemporal pattern of src activationat lipid rafts revealed by diffusion-corrected fret imag-ing. PLoS Comput Biol 4(7), e1000,127 (2008). URLhttp://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=

Retrieve&db=PubMed&dopt=Citation&list_uids=18711637

26. Nocedal, J., Wright, S.J.: Numerical optimization, second edn.Springer Series in Operations Research and Financial Engineer-ing. Springer, New York (2006)

27. Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinearequations in several variables. Society for Industrial and AppliedMathematics (SIAM), Philadelphia, PA (2000). Reprint of the1970 original

28. Oswald, P.: On the robustness of the BPX-preconditioner with re-spect to jumps in the coefficients. Math. Comp. 68(226), 633–650(1999). DOI 10.1090/S0025-5718-99-01041-8

29. Parikh, N., Boyd, S.: Proximal algorithms. Foundations andTrends in Optimization 1(3), 127–239 (2014). URL http://dx.

doi.org/10.1561/2400000003. DOI 10.1561/240000000330. Petzoldt, M.: Regularity and error estimators for elliptic problems

with discontinuous coefficients. Ph.D. thesis, Weierstrass Institutefor Applied Analysis and Stochastics (2001)

31. Petzoldt, M.: Regularity results for Laplace interface problems intwo dimensions. Z. Anal. Anwendungen 20(2), 431–455 (2001).DOI 10.4171/ZAA/1024

32. Petzoldt, M.: A posteriori error estimators for elliptic equationswith discontinuous coefficients. Adv. Comput. Math. 16(1), 47–75 (2002). DOI 10.1023/A:1014221125034

33. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Nu-merical recipes, third edn. Cambridge University Press, Cam-bridge (2007). The art of scientific computing

34. Rockafellar, R.T.: Augmented Lagrangians and applications of theproximal point algorithm in convex programming. Mathematics ofoperations research 1(2), 97–116 (1976)

35. Scott, J.D., Pawson, T.: Cell signaling in space and time:where proteins come together and when they’re apart. Science326(5957), 1220–4 (2009). URL http://www.ncbi.nlm.nih.

gov/pubmed/19965465

36. Sniekers, Y.H., van Donkelaar, C.C.: Determining diffusion co-efficients in inhomogeneous tissues using fluorescence recoveryafter photobleaching. Biophys J 89(2), 1302–7 (2005). URLhttp://www.ncbi.nlm.nih.gov/pubmed/15894637. DOI 10.1529/biophysj.104.053652

37. Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., Yagola, A.G.:Numerical methods for the solution of ill-posed problems, Mathe-matics and its Applications, vol. 328. Kluwer Academic Publish-ers Group, Dordrecht (1995). Translated from the 1990 Russianoriginal by R. A. M. Hoksbergen and revised by the authors, DOI10.1007/978-94-015-8480-7

38. Van Beusekom, A.E., Parker, R.L., Bank, R.E., Gill, P.E., Con-stable, S.: The 2-d magnetotelluric inverse problem solved withoptimization. Geophysical Journal International 184(2), 639–650(2011). DOI 10.1111/j.1365-246X.2010.04895.x

39. Xu, J.: Iterative methods by space decomposition and subspacecorrection. SIAM Rev. 34(4), 581–613 (1992). DOI 10.1137/1034116

Optimization with Partial Differential Equation Constraints for Bio-imaging Applications 15

40. Yu, S.R., Burkhardt, M., Nowak, M., Ries, J., Petrasek, Z.,Scholpp, S., Schwille, P., Brand, M.: Fgf8 morphogen gradi-ent forms by a source-sink mechanism with freely diffusingmolecules. Nature 461(7263), 533–6 (2009). URL http:

//www.ncbi.nlm.nih.gov/pubmed/19741606. DOI 10.1038/nature08391

41. Zeidler, E.: Nonlinear functional analysis and its applications.I. Springer-Verlag, New York (1986). Fixed-point theorems,Translated from the German by Peter R. Wadsack, DOI 10.1007/978-1-4612-4838-5

42. Zheng, H., Wu, J.: Convergence analysis on multigrid methods forelliptic problems with large jumps in coefficients. IMA J. Numer.Anal. 35(4), 1888–1912 (2015). DOI 10.1093/imanum/dru055

43. Zou, J.: Numerical methods for elliptic inverse problems.Int. J. Comput. Math. 70(2), 211–232 (1998). URL http:

//dx.doi.org/10.1080/00207169808804747. DOI 10.1080/00207169808804747