Research Statement

Alexander V. Mamonov

My research interests are in analytical and computational aspects of forward and inverse problems forpartial differential equations (PDE) with applications in geophysical exploration and medical imaging. Outof these two broad areas the following problems are of particular interest for me.

1. Model order reduction techniques with applications to geophysics: novel numerical methods for seismicfull waveform inversion (FWI), non-linear true amplitude seismic migration algorithms, efficient high-performance parallel solvers for seismic wave propagation, controlled source electro-magnetic method(CSEM).

2. Various modalities of medical imaging: electrical impedance tomography (EIT), quantitative photoa-coustic tomography (QPAT), image processing and pattern recognition (endoscopy).

Since my research is driven by applications, I strive to maintain interdisciplinary collaborations with theresearchers from both industry and academia. While my current work is concentrated on the topics listedabove, I am constantly looking for new areas of research and collaboration.

1 Reduced order models for geophysicsThe research described in this section was started with my colleagues at Schlumberger-Doll Research Center.I still maintain a strong collaboration with them and the projects described below are the main focus of mycurrent work.

1.1 Non-linear preconditioners for the full waveform inversion

The problem of seismic full waveform inversion is an optimization problem conventionally formulated in theoutput least squares (OLS) form

minimizec

‖F ? − F (c)‖22, (1)

and solved for a seismic model c (say, the subsurface sound speed) by minimizing the misfit between theseismic data F ? measured on or near the surface and the prediction of the forward model F (c). The objectivefunctional of (1) is notoriously non-convex with multiple local minima, an effect known as the cycle skipping.The linearization to (1) is also prone to other issues such as the effects of multiple reflections, etc.

We propose to replace the conventional formulation (1) with another one

minimizec

‖Q(F ?)−Q(F (c))‖22, (2)

where the non-linear preconditioner Q is chosen to convexify the objective of (2) which improves the con-vergence and the quality of the solution. The preconditioner Q is constructed from a reduced order model(ROM) that interpolates the measured seismic data. This approach can also be applied to other types ofinverse problems as shown in the following sections.

For the wave equationutt = A(c)u, u(0) = B, ut(0) = 0, (3)

assuming that the sources and receivers are collocated, the seismic data is F (t; c) = B∗ cos(t√−A(c))B.

Here the spatial PDE operator A(c) can be acoustic or elastic. The dependency on c is omitted hereafter.When sampled at time intervals τ the data becomes

Fk = F (kτ) = B∗ cos(kτ√−A)B = B∗ cos(k arccos(cos τ

√−A))B = B∗Tk(P )B, (4)

where the propagator operator is P = cos(τ√−A) and Tk are Chebyshev polynomials of the first kind. If

the matrix of solution snapshots uk = u(kτ) is denoted by U then we can write a ROM for the propagator

P = (U∗U)−1/2(U∗PU)(U∗U)−1/2 = V ∗PV, (5)

where V = U(U∗U)−1/2 is an orthonormal basis for the subspace of solution snapshots.

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. True sound speed (in km/s)

. ROM-preconditioned FWI solution

. Conventional FWI (OLS NLCG) solution

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.21

1.5

2

2.5

Well log at x(80)=2.40 km, rel err: 0.114621

InitTrueROM FWI

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.21

1.5

2

2.5

Well log at x(80)=2.40 km, rel err: 0.176601

InitTrueOLS NLCG

Figure 1: Comparison of the conventional OLS NLCG FWI (1) and the ROM-preconditioned FWI (2) after10 iterations for a section of Marmousi model. The sources/receivers are black ×, the distances are in km.In the vertical sounds speed logs on the right the true sound speed is black , the initial guess is blue 5 andthe FWI solution is red solid line.

The ROM (5) fits the measured seismic data exactly

Fk = B∗Tk(P )B = B∗Tk(P )B, B = V ∗B, (6)

and can be obtained entirely from the data via simple trigonometric identities

uTi uj =1

2(Fi+j + Fi−j), uTi Puj =

1

4(Fj+i+1 + Fj−i+1 + Fj+i−1 + Fj−i−1). (7)

After the ROM is constructed, we can unitarily transform it to the block tridiagonal form using a blockLanczos procedure. These blocks constitute the non-linear preconditioner Q from (2). Now instead ofminimizing the data misfit we minimize the misfit of the quantities that resemble the PDE operator itself.Since the dependence A(c) of the PDE operator on the unknown coefficient c is linear, the ROM-preconditinedoptimization problem (2) becomes close to quadratic and thus nearly convex. This constitutes a significantadvantage over the conventional formulation (1) in terms of robustness and speed of convergence. Also, theorthogonalization of the solution snapshots in (5) removes the multiple reflections. This contrasts sharplywith traditional Born-like approximations that produce an artifact for each multiple reflection.

In Figure 1 we compare the performance of the conventional and ROM-preconditioned FWI approacheson a standard Marmousi model. We observe that our proposed method does a much better job recoveringthe small features of the model as well as the overall contrast. Seismic inversion with ROMs is an areaof active research for me. There are many aspects of the problem that we are working on including thepractically important case of non-collocated sources and receivers, efficient computation of gradients for theoptimization objective (2), incorporation of realistic source models and more.

1.2 Non-linear true amplitude seismic migration

The projection-based ROMs discussed in the previous section can also be used to derive novel seismicmigration algorithms. Using the first two terms of cosine Taylor expansion we define the reduced ordermodel for the spatial PDE operator A as

A =2

τ2(P − I) ≈ V ∗AV, (8)

where P , A and V are exactly the same as in the previous section.

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Figure 2: Non-linear true amplitude seismic migration for the full Marmousi model. The 90 sources/receiversare black ×. Top: true sound speed c. Middle: smooth kinematic model c0. Bottom: migration imagec0 + αδc.

If the basis functions V were known, we could estimate the unknown PDE operator A from the backpro-jection relation

A ≈ V AV ∗. (9)

Since the basis functions V cannot be determined from the measured data Fk, we use a standard idea inseismic migration of using a known background kinematic model c0. For this known kinematic model wecan compute the quantities A0, V0 and approximate V ≈ V0.

If the spatial PDE operator A depends linearly on c, then c ∝ diag(A), thus

δc = c− c0 ∝ diag(A−A0) ≈ diag(V0(A− A0)V ∗0 ), (10)

where A is known from the measured seismic data and A0, V0 are known from the fixed kinematic model.Then the formula

c = c0 + αδc, (11)

for a scalar step length α gives the true amplitude migration image which means that it does not onlyrecover the locations of the model discontinuities (reflectors) but also their amplitudes. Note that unlike theconventional migration methods like Kirchhoff or the reversed time migration (RTM) our approach (10) is

non-linear, since the dependency of P and thus A on the seismic data (5) is non-linear. As mentioned in theprevious section, this automatically eliminates the multiple reflections.

In Figure 2 the preliminary results of the non-linear true amplitude migration are shown. The observedrecovery of both the reflector locations and their amplitudes is very good. There many possible improvementsto the method that I am currently working on. This includes rewriting the method in the first orderformulation which should reduce the dependency of the image on the kinematic model quality.

1.3 Parallel solvers for seismic wave propagation

In [9] we outlined a method for numerical time-domain seismic wave propagation based on the model orderreduction approach referred to as the S-fraction multiscale spectral finite volume method (SFMSFV). Themethod is built with high-performance computing (HPC) implementation in mind that implies a high levelof parallelism and greatly reduced communication requirements compared to the conventional high-orderfinite-difference time-domain (FDTD) methods.

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(a) Sound speed (b) Fine grid seismogram (c) SFMSFV seismogram

Figure 3: Numerical example of SFMSFV method performance. The 2D slice of the sound speed (a) isoverlayed with the sub-domain boundaries (yellow lines). The source location is yellow ×. The scaledseismograms (b) and (c) are shown for the receivers located on the top boundary of the domain. The relativeerror between the SFMSFV seismogram (c) and the reference fine grid solution (b) is 2.7%.

Our approach is inherently multiscale, with a reference fine grid model being split into sub-domains asshown in Figure 3(a). On each sub-domain we discretize the wave equation on a fine reference grid

Au+ ω2u = 0, (12)

with A being the spatial PDE operator and u the wavefield. The coupling between the adjacent sub-domainsis completely defined by the frequency-dependent Neumann-to-Dirichlet (NtD) map

M(ω) = B∗ (A+ ω2I)−1

B, (13)

where F are the basis functions for the wavefield on the boundary of the sub-domain.Then for each sub-domain the coarse scale reduced order models (ROMs) can be precomputed off-line in

a parallel manner. Thus, the fine grid operators A, B in (12)–(13) are replaced by their coarse counterparts

A, B which can be obtained by projection

A = V ∗AV, B = V ∗B. (14)

The projection subspace with the orthonormal basis V is chosen so that the ROMs approximate the NtD mapswith high (spectral) accuracy and are used to couple the adjacent sub-domains on the shared boundaries.

The on-line part of the method is an explicit time stepping with the coupled ROMs. To lower the on-line computation cost the reduced order spatial operator is sparsified by transforming to a matrix Stieltjescontinued fraction (S-fraction) form. The on-line communication costs are also reduced due to the ROMNtD map approximation properties. Another source of performance improvement is the time step length.Properly chosen ROMs substantially improve the Courant-Friedrichs-Lewy (CFL) condition. This allowsthe CFL time step to approach the Nyquist limit, which is typically unattainable with conventional schemesthat have the CFL time step much smaller than the Nyquist sampling rate.

The performance of the SFMSFV method is illustrated in Figure 3 for an acoustic wave equation in3D. Another advantage of the method is that the sub-domain boundaries need not to be aligned with thecoefficient discontinuities for optimal accuracy. Also, the online time stepping solver does not depend on theparticular choice of equation (acoustic, elastic, anisotropic), only the ROM construction does. One of ourimmediate goals is a high performance implementation of the anisotropic elastic case utilizing the GPUs formaximum performance.

1.4 Reduced order models for the controlled source electro-magnetic method

Historically the model order reduction techniques have been developed best for the parabolic (stable) systems.Thus, our first work on applications of ROMs to inversion [6] was for the CSEM problem which considers aparabolic approximation of Maxwell’s equations. Instead of recovering a coefficient of a wave equation (3)in CSEM we are interested in recovering the resistivity r in

ut = A(r)u, u(0) = B, (15)

where A(r) = ∇ · (r∇) from the data Y (t) = B∗u. As before, B here is a matrix of sources and receivers.

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Figure 4: CSEM reconstructions after a single Gauss-Newton iteration starting from a uniform initial guessr ≡ 1. Sources/receivers are black ×. Top: true resistivities. Bottom: reconstructions.

Unlike the wave case, where we work directly with the time domain data, in the parabolic setting weconstruct a ROM that approximates the frequency-domain transfer function

G(ω) = B∗(ωI −A)−1B ≈ B∗(ωI − A)−1B, (16)

that is also of projection type A = V ∗AV , B = V ∗B for an appropriate choice of the subspace with basisV . Another distinction from the wave case is that the parabolic inverse problem is severely ill-posed. Thus,the size of the reduced order model has to be strictly constrained so that it can be recovered stable from themeasured data.

A preconditioned optimization formulation similar to (2) can be obtained for CSEM. Since the size ofthe reduced order model is small it becomes computationally feasible to use Gauss-Newton method to solvethe optimization problem. In FWI we could only use the non-linear conjugate gradients (NLCG) methodthat does not require the computation of the Jacobian.

The 2D CSEM reconstructions obtained with the procedure outlined above are given in Figure 4 for theinitial guess r ≡ 1. Although only a single Gauss-Newton iteration is performed, the obtained solution isof very good quality. This confirms that the non-linear preconditioner Q acts as an approximate inverse onthe forward mapping and thus the minimization problem becomes almost quadratic. The example in Figure4 is shown for the collocated sources and receivers. Similarly to the FWI inversion, the non-collocated caseremains a topic of my future research.

2 Medical imagingA second area of application of inverse problems that I am interested in involves various modalities of medicalimaging. This includes the electrical impedance tomography (EIT) [3, 4, 5, 7], a hybrid modality known asthe quantitative photoacoustic tomography (QPAT) [11] and an image processing and classification problemin capsule endoscopy [10]. Some of these research directions are discussed below.

2.1 Resistor networks for electrical impedance tomography with partial data

In electrical impedance tomography with partial measurements the unknown conductivity coefficient σ of anelliptic PDE

∇ · (σ∇u) = 0, in Ω, (17)

is to be determined from the partial knowledge of the Dirichlet-to-Neumann (DtN) map

Λσφ = σ∇u · ν, on ∂Ω. (18)

For example, the DtN map can be restricted to a segment of the boundary BA ⊂ ∂Ω, called the accessibleboundary, for excitations with suppφ ⊆ BA.

Similarly to the CSEM problem EIT inversion is severely ill-posed and thus is well suited for the numericalapproach based on model reduction. The discrete reduced models appropriate for EIT are the resistornetworks that correspond to a finite volume discretization of (17). In the case of two spatial dimensions atheory of inverse problems for networks with circular planar graphs was developed in [8]. In such problemsone recovers the conductances γk of resistors in a network from the knowledge of a discrete analogue of aDtN map. For the discrete problem to have a unique solution the graphs of the networks must have theproperty of being critical [8].

The use of resistor networks for EIT was first proposed in [2], where the circular networks were used for aproblem with full boundary measurements. In [4] we extended this approach to the partial measurement case

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True σ Circular (m = 17) Pyramidal (m = 16) Two-sided (m = 16)

Figure 5: EIT reconstructions with partial boundary measurements using resistor networks. Top row:smooth σ. Bottom row: piecewise constant σ. Electrode locations are ×, accessible boundary BA is thesolid red line. Three resistor network topologies are used: circular, pyramidal and two-sided. The numberof electrodes (boundary network nodes) is m.

by using the theory of extremal quasiconformal (Teichmuller) mappings, including an algorithm to computesuch mappings numerically. I discovered later that a better choice of network topology is the one given bypyramidal graphs. This study was reported in [5], where the criticality of such networks was proved and adirect layer peeling reconstruction algorithm was given. Finally, for the problem with BA consisting of twodisjoint segments of ∂Ω, the two-sided networks were proposed and studied in [3].

In case of resistor networks the non-linear preconditioner Q can be defined as a mapping from themeasured DtN map to resistors γk. The numerical reconstructions obtained with a single step of Gauss-Newton preconditioned with Q are presented in Figure 5 for circular, pyramidal and two-sided networks. EITwith resistor networks still remains a problem if interest to me. In particular, extension to three dimensionsis very important for applications. This will require the study of networks with non-planar graphs, aboutwhich very little is currently known.

2.2 Automated polyp detection in colon capsule endoscopy

My interests in medical imaging are not limited by problems directly involving PDE inversion. For example,while visiting the University of Coimbra in Portugal I studied a medical imaging problem that involves imageprocessing, pattern recognition and classification. In collaboration with the University Hospital in Coimbrawe worked on automated detection of intestinal polyps from the frames captured by a capsule endoscope.The study was reported in [10] and also featured in the MIT Technology Review [1].

In capsule endoscopy the patient swallows a pill equipped with a CCD camera and an LED light. As thepill moves along the intestine, the frames captured by the camera are transmitted wirelessly to the recordingdevice. With a typical time it takes the pill to travel through the small intestine of 6 − 7 hours and thesampling rate of 2− 4 frames per second, the camera may capture up to 50, 000− 100, 000 frames. Havinga skilled medical professional examine the whole video sequence may be economically infeasible. Thus,one would like to automate the procedure of detecting the pathologies by employing an image processingalgorithm.

The most prominent defining feature of intestinal polyps is their shape. Thus, in [10] we proposed analgorithm that acts as a binary classifier labeling the frames as either containing polyps or not, based on thegeometrical analysis of the frame. To separate the features in a frame f , we employ first a mid pass filter

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Original polyp frame f Mid pass filtered frame u Segmentation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1× SPEC: 90.2%, SENS: 81.2%

FPR = 1 - SPEC/100%

TPR

=SENS/100%

Figure 6: Intermediate steps of the polyp detection algorithm. In the segmented image the centers of massare shown with green × and the ellipses of inertia with yellow solid lines. The rightmost plot shows theReceiver Operating Characteristic (ROC) curve, i.e. the plot of the True Positive Rate (Sensitivity) versusthe False Positive Rate (1-Specificity) for various threshold values of the radius of the best fit ball. Thethreshold value corresponding to 90.2% specificity, which yields 81.2% sensitivity is marked with a red cross.

with a cutoff

u = H(w) · w, w =Gσ1∗ f

Gσ2 ∗ f− 1, (19)

where Gσj are Gaussian kernels with variances σ1 < σ2, H is a Heaviside function and multiplication anddivision operations are pixel-wise. From the filtered image u the features are extracted by a thresholdingsegmentation procedure. The shape of each segment is then checked based on the eigenvalues of the tensorof inertia. For the segments passing the check a best fit ball centered at the center of mass of each segmentis computed. The radius of this ball is used as a decision parameter on whether the segment in questionmay possibly be a polyp.

In Figure 6 the work of the algorithm is illustrated. The ROC curve on the right shows that the algorithmachieves over 81.2% sensitivity at 90.2% specificity on a polyp-by-polyp basis when tested on a real dataset with over 18, 000 frames. Endoscopic image processing is still among my research interests and I plan tomaintain my collaboration with my colleagues at Coimbra studying other problems like aberrant crypt fociand bleeding detection.

3 Research programThe following topics summarize my short- and medium-term research program. The list is by no meansexhaustive as I am constantly looking for possible new directions of research.

1. Reduced order models for inversion continue to be a big focus of my research. In my immediate plansis to study the non-symmetric ROMs to handle the cases of non-collocated sources and receivers.It requires the study of the projection ROMs with distinct left and right projection subspaces, two-sided Lanczos iterations, etc. This will bring the ROM-based inversion techniques to many morerealistic measurement settings. Another area of investigation is the numerically efficient formulas forthe differentiation of the ROMs. This is required to make our inversion approach computationallycompetitive with the existing conventional FWI solvers.

2. Many possible improvements to the non-linear true amplitude migration algorithm are possible. Thisincludes switching to the first order PDE formulation, a technique which may also be beneficial forthe ROM-preconditioned FWI. It should make the migration image less dependent on the choice ofthe kinematic velocity model. Also, one may use the migration images to update the kinematic modeland thus turn the procedure into an iterative inversion scheme. Unlike the optimization-based FWIinversion such scheme is derivative-free, i.e. it does not require the computation of the gradient.

3. I also plan to work on further theoretical development and the high performance parallel implemen-tation of the ROM-based wave propagation forward solver. The implementation should make the on-line time stepping component completely independent of the particular PDE solved (acoustic, elastic,isotropic, anisotropic). so that a single solver can be used for a wide variety of geophysical applications.

4. Finally, various modalities of medical imaging remain among my research interests. In particular, I aminterested in 3D EIT. One possible approach is to use the planar networks to fit the subsets of the 3D

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DtN data to construct the non-linear preconditioner Q. Alternatively, one may consider the projectionROM formulation, in which the DtN map is viewed as the forward problem operator projected on thesolutions corresponding to the excitations of each electrode. Such setting may allow a construction of abackprojection imaging algorithm similar to the non-linear true amplitude seismic migration approach.Working on the image processing and classification algorithms in collaboration with my colleagues atthe University of Coimbra and the University of Texas at Austin is also among my goals.

References

[1] The Algorithm That Automatically Detects Polyps in Images from Camera Pills, May, 2013. MITTechnology Review: www.technologyreview.com/view/514786.

[2] L. Borcea, V. Druskin, and F. Guevara Vasquez. Electrical impedance tomography with resistor net-works. Inverse Problems, 24(3):035013, 2008.

[3] L. Borcea, V. Druskin, F. Guevara Vasquez, and A.V. Mamonov. Resistor network approaches toelectrical impedance tomography, volume Inverse Problems and Applications: Inside Out II. CambridgeUniversity Press, 2012.

[4] L. Borcea, V. Druskin, and A.V. Mamonov. Circular resistor networks for electrical impedance tomog-raphy with partial boundary measurements. Inverse Problems, 26(4):045010, 2010.

[5] L. Borcea, V. Druskin, A.V. Mamonov, and F. Guevara Vasquez. Pyramidal resistor networks for elec-trical impedance tomography with partial boundary measurements. Inverse Problems, 26(10):105009,2010.

[6] L. Borcea, V. Druskin, A.V. Mamonov, and M. Zaslavsky. A model reduction approach to numericalinversion for a parabolic partial differential equation. Inverse Problems, 30(12):125011, 2014.

[7] L. Borcea, A.V. Mamonov, and F. Guevara Vasquez. Study of noise effects in electrical impedancetomography with resistor networks. Inverse Problems and Imaging, 7(2):417–443, 2013.

[8] E.B. Curtis, D. Ingerman, and J.A. Morrow. Circular planar graphs and resistor networks. Linearalgebra and its applications, 283(1):115–150, 1998.

[9] V. Druskin, A.V. Mamonov, and M. Zaslavsky. S-fraction multiscale finite-volume method for spectrallyaccurate wave propagation, 2014. arXiv:1406.6923 [math.NA].

[10] A.V. Mamonov, I.N. Figueiredo, P.N. Figueiredo, and Y.-H. R. Tsai. Automated polyp detection incolon capsule endoscopy. IEEE Transactions on Medical Imaging, 33(7):1488–1502, 2014.

[11] A.V. Mamonov and K. Ren. Quantitative photoacoustic imaging in radiative transport regime. Com-munications in Mathematical Sciences, 12(2):201–234, 2014.

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