arXiv subjects

George Biros

Publications and source records attributed to George Biros.

At least 55 records · Page 3Linked to original sources

arXiv2019

CLAIRE: A distributed-memory solver for constrained large deformation diffeomorphic image registration

With this work, we release CLAIRE, a distributed-memory implementation of an effective solver for constrained large deformation diffeomorphic image registration problems in three dimensions. We consider an optimal control formulation. We invert for a stationary velocity field that parameterizes the deformation map. Our solver is based on a globalized, preconditioned, inexact reduced space Gauss--Newton--Krylov scheme. We exploit state-of-the-art techniques in scientific computing to develop an effective solver that scales to thousands of distributed memory nodes on high-end clusters. We present the formulation, discuss algorithmic features, describe the software package, and introduce an improved preconditioner for the reduced space Hessian to speed up the convergence of our solver. We test registration performance on synthetic and real data. We demonstrate registration accuracy on several neuroimaging datasets. We compare the performance of our scheme against different flavors of the Demons algorithm for diffeomorphic image registration. We study convergence of our preconditioner and our overall algorithm. We report scalability results on state-of-the-art supercomputing platforms. We demonstrate that we can solve registration problems for clinically relevant data sizes in two to four minutes on a standard compute node with 20 cores, attaining excellent data fidelity. With the present work we achieve a speedup of (on average) 5$\times$ with a peak performance of up to 17$\times$ compared to our former work.

math.OC↗

arXiv2019

BIMC: The Bayesian Inverse Monte Carlo method for goal-oriented uncertainty quantification. Part II

In Part I (arXiv:1911.00619) of this article, we proposed an importance sampling algorithm to compute rare-event probabilities in forward uncertainty quantification problems. The algorithm, which we termed the "Bayesian Inverse Monte Carlo (BIMC) method", was shown to be optimal for problems in which the input-output operator is nearly linear. But applying the original BIMC to highly nonlinear systems can lead to several different failure modes. In this paper, we modify the BIMC method to extend its applicability to a wider class of systems. The modified algorithm, which we call "Adaptive-BIMC (A-BIMC)", has two stages. In the first stage, we solve a sequence of optimization problems to roughly identify those regions of parameter space which trigger the rare-event. In the second stage, we use the stage one results to construct a mixture of Gaussians that can be then used in an importance sampling algorithm to estimate rare event probability. We propose using a local surrogate that minimizes costly forward solves. The effectiveness of A-BIMC is demonstrated via several synthetic examples. Yet again, the modified algorithm is prone to failure. We systematically identify conditions under which it fails to lead to an effective importance sampling distribution.

stat.CO↗

arXiv2019

BIMC: The Bayesian Inverse Monte Carlo method for goal-oriented uncertainty quantification. Part I

We consider the problem of estimating rare event probabilities, focusing on systems whose evolution is governed by differential equations with uncertain input parameters. If the system dynamics is expensive to compute, standard sampling algorithms such as the Monte Carlo method may require infeasible running times to accurately evaluate these probabilities. We propose an importance sampling scheme (which we call BIMC) that relies on solving an auxiliary, fictitious Bayesian inverse problem. The solution of the inverse problem yields a posterior PDF, a local Gaussian approximation to which serves as the importance sampling density. We apply BIMC to several problems and demonstrate that it can lead to computational savings of several orders of magnitude over the Monte Carlo method. We delineate conditions under which BIMC is optimal, as well as conditions when it can fail to yield an effective IS density.

stat.CO↗

arXiv2019

A domain decomposition preconditioning for the integral equation formulation of the inverse scattering problem

We propose domain decomposition preconditioners for the solution of an integral equation formulation of forward and inverse acoustic scattering problems with point scatterers. We study both forward and inverse problems and propose preconditioning techniques to accelerate the iterative solvers. For the forward scattering problem, we extend the domain decomposition based preconditioning techniques presented for partial differential equations in {\em "A restricted additive Schwarz preconditioner for general sparse linear systems", SIAM Journal on Scientific Computing, 21 (1999), pp. 792--797}, to integral equations. We combine this domain decomposition preconditioner with a low-rank correction, which is easy to construct, forming a new preconditioner. For the inverse scattering problem, we use the forward problem preconditioner as a building block for constructing a preconditioner for the Gauss-Newton Hessian. We present numerical results that demonstrate the performance of both preconditioning strategies.

math.NA↗

arXiv2019

Image-Driven Biophysical Tumor Growth Model Calibration

We present a novel formulation for the calibration of a biophysical tumor growth model from a single-time snapshot, MRI scan of a glioblastoma patient. Tumor growth models are typically nonlinear parabolic partial differential equations (PDEs). Thus, we have to generate a second snapshot to be able to extract significant information from a single patient snapshot. We create this two-snapshot scenario as follows. We use an atlas (an average of several scans of healthy individuals) as a substitute for an earlier, pretumor, MRI scan of the patient. Then, using the patient scan and the atlas, we combine image-registration algorithms and parameter estimation algorithms to achieve a better estimate of the healthy patient scan and the tumor growth parameters that are consistent with the data. Our scheme is based on our recent work (Scheufele et al, "Biophysically constrained diffeomorphic image registration, Tumor growth, Atlas registration, Adjoint-based methods, Parallel algorithms", CMAME, 2018), but apply a different and novel scheme where the tumor growth simulation in contrast to the previous work is executed in the patient brain domain and not in the atlas domain yielding more meaningful patient-specific results. As a basis, we use a PDE-constrained optimization framework. We derive a modified Picard-iteration-type solution strategy in which we alternate between registration and tumor parameter estimation in a new way. In addition, we consider an $\ell_1$ sparsity constraint on the initial condition for the tumor and integrate it with the new joint inversion scheme. We solve the subproblems with a reduced-space, inexact Gauss-Newton-Krylov/quasi-Newton methods. We present results using real brain data with synthetic tumor data that show that the new scheme reconstructs the tumor parameters in a more accurate and reliable way compared to our earlier scheme.

q-bio.QM↗

arXiv2019

Where did the tumor start? An inverse solver with sparse localization for tumor growth models

We present a numerical scheme for solving an inverse problem for parameter estimation in tumor growth models for glioblastomas, a form of aggressive primary brain tumor. The growth model is a reaction-diffusion partial differential equation (PDE) for the tumor concentration. We use a PDE-constrained optimization formulation for the inverse problem. The unknown parameters are the reaction coefficient (proliferation), the diffusion coefficient (infiltration), and the initial condition field for the tumor PDE. Segmentation of Magnetic Resonance Imaging (MRI) scans from a single time snapshot drive the inverse problem where segmented tumor regions serve as partial observations of the tumor concentration. The precise time relative to tumor initiation is unknown, which poses an additional difficulty for inversion. We perform a frozen-coefficient spectral analysis and show that the inverse problem is severely ill-posed. We introduce a biophysically motivated regularization on the tumor initial condition. In particular, we assume that the tumor starts at a few locations (enforced with a sparsity constraint) and that the initial condition magnitude in the maximum norm equals one. We solve the resulting optimization problem using an inexact quasi-Newton method combined with a compressive sampling algorithm for the sparsity constraint. Our implementation uses PETSc and AccFFT libraries. We conduct numerical experiments on synthetic and clinical images to highlight the improved performance of our solver over an existing solver that uses a two-norm regularization for the calibration parameters. The existing solver is unable to localize the initial condition. Our new solver can localize the initial condition and recover infiltration and proliferation. In clinical datasets (for which the ground truth is unknown), our solver results in qualitatively different solutions compared to the existing solver.

physics.med-ph↗

arXiv2019

ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs

Residual neural networks can be viewed as the forward Euler discretization of an Ordinary Differential Equation (ODE) with a unit time step. This has recently motivated researchers to explore other discretization approaches and train ODE based networks. However, an important challenge of neural ODEs is their prohibitive memory cost during gradient backpropogation. Recently a method proposed in [8], claimed that this memory overhead can be reduced from O(LN_t), where N_t is the number of time steps, down to O(L) by solving forward ODE backwards in time, where L is the depth of the network. However, we will show that this approach may lead to several problems: (i) it may be numerically unstable for ReLU/non-ReLU activations and general convolution operators, and (ii) the proposed optimize-then-discretize approach may lead to divergent training due to inconsistent gradients for small time step sizes. We discuss the underlying problems, and to address them we propose ANODE, an Adjoint based Neural ODE framework which avoids the numerical instability related problems noted above, and provides unconditionally accurate gradients. ANODE has a memory footprint of O(L) + O(N_t), with the same computational cost as reversing ODE solve. We furthermore, discuss a memory efficient algorithm which can further reduce this footprint with a trade-off of additional computational cost. We show results on Cifar-10/100 datasets using ResNet and SqueezeNext neural networks.

cs.LG↗

arXiv2019

ANODEV2: A Coupled Neural ODE Evolution Framework

It has been observed that residual networks can be viewed as the explicit Euler discretization of an Ordinary Differential Equation (ODE). This observation motivated the introduction of so-called Neural ODEs, which allow more general discretization schemes with adaptive time stepping. Here, we propose ANODEV2, which is an extension of this approach that also allows evolution of the neural network parameters, in a coupled ODE-based formulation. The Neural ODE method introduced earlier is in fact a special case of this new more general framework. We present the formulation of ANODEV2, derive optimality conditions, and implement a coupled reaction-diffusion-advection version of this framework in PyTorch. We present empirical results using several different configurations of ANODEV2, testing them on multiple models on CIFAR-10. We report results showing that this coupled ODE-based framework is indeed trainable, and that it achieves higher accuracy, as compared to the baseline models as well as the recently-proposed Neural ODE approach.

cs.LG↗

arXiv2019

Simulation of glioblastoma growth using a 3D multispecies tumor model with mass effect

In this article, we present a multispecies reaction-advection-diffusion partial differential equation (PDE) coupled with linear elasticity for modeling tumor growth. The model aims to capture the phenomenological features of glioblastoma multiforme observed in magnetic resonance imaging (MRI) scans. These include enhancing and necrotic tumor structures, brain edema and the so called "mass effect", that is, the deformation of brain tissue due to the presence of the tumor. The multispecies model accounts for proliferating, invasive and necrotic tumor cells as well as a simple model for nutrition consumption and tumor-induced brain edema. The coupling of the model with linear elasticity equations with variable coefficients allows us to capture the mechanical deformations due to the tumor growth on surrounding tissues. We present the overall formulation along with a novel operator-splitting scheme with components that include linearly-implicit preconditioned elliptic solvers, and semi-Lagrangian method for advection. Also, we present results showing simulated MRI images which highlight the capability of our method to capture the overall structure of glioblastomas in MRIs.

physics.med-ph↗

arXiv2019

Identifying the Best Machine Learning Algorithms for Brain Tumor Segmentation, Progression Assessment, and Overall Survival Prediction in the BRATS Challenge

Gliomas are the most common primary brain malignancies, with different degrees of aggressiveness, variable prognosis and various heterogeneous histologic sub-regions, i.e., peritumoral edematous/invaded tissue, necrotic core, active and non-enhancing core. This intrinsic heterogeneity is also portrayed in their radio-phenotype, as their sub-regions are depicted by varying intensity profiles disseminated across multi-parametric magnetic resonance imaging (mpMRI) scans, reflecting varying biological properties. Their heterogeneous shape, extent, and location are some of the factors that make these tumors difficult to resect, and in some cases inoperable. The amount of resected tumor is a factor also considered in longitudinal scans, when evaluating the apparent tumor for potential diagnosis of progression. Furthermore, there is mounting evidence that accurate segmentation of the various tumor sub-regions can offer the basis for quantitative image analysis towards prediction of patient overall survival. This study assesses the state-of-the-art machine learning (ML) methods used for brain tumor image analysis in mpMRI scans, during the last seven instances of the International Brain Tumor Segmentation (BraTS) challenge, i.e., 2012-2018. Specifically, we focus on i) evaluating segmentations of the various glioma sub-regions in pre-operative mpMRI scans, ii) assessing potential tumor progression by virtue of longitudinal growth of tumor sub-regions, beyond use of the RECIST/RANO criteria, and iii) predicting the overall survival from pre-operative mpMRI scans of patients that underwent gross total resection. Finally, we investigate the challenge of identifying the best ML algorithms for each of these tasks, considering that apart from being diverse on each instance of the challenge, the multi-institutional mpMRI BraTS dataset has also been a continuously evolving/growing dataset.

cs.CV↗

arXiv2019

Machine learning acceleration of simulations of Stokesian suspensions

Particulate Stokesian flows describe the hydrodynamics of rigid or deformable particles in Stokes flows. Due to highly nonlinear fluid-structure interaction dynamics, moving interfaces, and multiple scales, numerical simulations of such flows are challenging and expensive. In this Letter, we propose a generic machine-learning-augmented reduced model for these flows. Our model replaces expensive parts of a numerical scheme with multilayer perceptrons. Given the physical parameters of the particle, our model generalizes to arbitrary geometries and boundary conditions without the need to retrain the regression function. It is 10 times faster than a state-of-the-art numerical scheme having the same number of degrees of freedom and can reproduce several features of the flow quite accurately. We illustrate the performance of our model on integral equation formulation of vesicle suspensions in two dimensions.

physics.comp-ph↗

arXiv2019

Optimal design of deterministic lateral displacement device for viscosity contrast based cell sorting

We solve a design optimization problem for deterministic lateral displacement (DLD) device to sort same-size biological cells by their deformability, in particular to sort red blood cells (RBCs) by their viscosity contrast between the fluid in the interior and the exterior of the cells. A DLD device optimized for efficient cell sorting enables rapid medical diagnoses of several diseases such as malaria since infected cells are stiffer than their healthy counterparts. The device consists of pillar arrays in which pillar rows are tilted and hence are not orthogonal to the columns. This arrangement leads cells to have different final vertical displacements depending on their deformability, therefore, it vertically separates the cells. Pillar cross section, tilt angle of the pillar rows and center-to-center distances between pillars define a unique device. For a given pair of viscosity contrast values of the cells we seek optimal DLD designs by fixing the tilt angle and the center-to-center distances. So the only design parameter is the pillar cross section which we parameterize with uniform 5th order B-splines. We propose an objective function to try to capture efficient cell sorting. The objective function is evaluated by simulating the cell flows through a device using our 2D model based on a boundary integral method (Kabacaoglu et al. Journal of Computational Physics, 357:43-77, 2018). We solve the optimization problem using the covariance matrix adaptation evolution strategy (CMA-ES), which is a stochastic, derivative-free algorithm. We present several scenarios where solving the optimization problem finds designs that can separate cells with similar viscosity contrast values. To the best of our knowledge, this is the first study which poses designing a DLD device as a constrained optimization problem and shows that solving this problem systematically discovers optimal designs.

physics.comp-ph↗

arXiv2018

A Novel Domain Adaptation Framework for Medical Image Segmentation

We propose a segmentation framework that uses deep neural networks and introduce two innovations. First, we describe a biophysics-based domain adaptation method. Second, we propose an automatic method to segment white and gray matter, and cerebrospinal fluid, in addition to tumorous tissue. Regarding our first innovation, we use a domain adaptation framework that combines a novel multispecies biophysical tumor growth model with a generative adversarial model to create realistic looking synthetic multimodal MR images with known segmentation. Regarding our second innovation, we propose an automatic approach to enrich available segmentation data by computing the segmentation for healthy tissues. This segmentation, which is done using diffeomorphic image registration between the BraTS training data and a set of prelabeled atlases, provides more information for training and reduces the class imbalance problem. Our overall approach is not specific to any particular neural network and can be used in conjunction with existing solutions. We demonstrate the performance improvement using a 2D U-Net for the BraTS'18 segmentation challenge. Our biophysics based domain adaptation achieves better results, as compared to the existing state-of-the-art GAN model used to create synthetic data for training.

cs.CV↗

arXiv2018

Sorting same-size red blood cells in deep deterministic lateral displacement devices

We study deformability-based sorting of same-size RBCs via DLD using an in-house integral equation solver. Our goal is to quantitatively characterize the physical mechanisms that enable the cell separation. To this end, we systematically investigate the effects of the interior fluid viscosity and membrane elasticity of a cell on its behavior. In particular, we consider deep devices in which a cell can show rich dynamics such as taking a particular angular orientation depending on its mechanical property. We have found out that cells moving with a sufficiently high positive inclination angle with respect to the flow direction displace laterally while those with smaller angles travel with the flow streamlines. Thereby, deformability-based cell sorting is possible. The underlying mechanism here is cell migration due to the cell's positive inclination and the shear gradient. The higher the inclination is, the farther the cell can travel laterally. We also assess the efficiency of the technique for dense suspensions. It turns out that most of the cells in dense suspensions does not displace in the lateral direction no matter what their deformability is. As a result, separating cells using a DLD device becomes harder.

physics.comp-ph↗

arXiv2018

Reconstruction of a compactly supported sound profile in the presence of a random background medium

In this paper, we present algorithms for reconstructing an unknown compact scatterer embedded in a random noisy background medium, given measurements of the scattered field and information about the background medium and the sound profile. We present six different methods for the solution of this inverse problem using different amounts of scattered data and prior information about the random background medium and the scatterer. The different inversion algorithms are defined by a combination of stochastic programming methods and Bayesian formulation. Our basic results show that if we have data for just one instance of the random background medium the best strategy is to invert for both random medium and unknown scatterer with appropriate regularization. However, if we have data for multiple instances of the medium it may be worth solving a coupled set of multiple inverse problems. We present several numerical results for inverting for various scatterer geometries under different inversion scenarios. The main take-away of our study is that one should invert for both unknown scatterer and random medium, with appropriate, prior-information based regularization. Furthermore, if data from multiple realizations of the background medium is available, then it may be beneficial to combine results from multiple inversions.

math.NA↗

arXiv2018

Coupling Brain-Tumor Biophysical Models and Diffeomorphic Image Registration

We present the SIBIA (Scalable Integrated Biophysics-based Image Analysis) framework for joint image registration and biophysical inversion and we apply it to analyse MR images of glioblastomas (primary brain tumors). In particular, we consider the following problem. Given the segmentation of a normal brain MRI and the segmentation of a cancer patient MRI, we wish to determine tumor growth parameters and a registration map so that if we "grow a tumor" (using our tumor model) in the normal segmented image and then register it to the patient segmented image, then the registration mismatch is as small as possible. We call this "the coupled problem" because it two-way couples the biophysical inversion and registration problems. In the image registration step we solve a large-deformation diffeomorphic registration problem parameterized by an Eulerian velocity field. In the biophysical inversion step we estimate parameters in a reaction-diffusion tumor growth model that is formulated as a partial differential equation. In this paper, our contributions are the introduction of a PDE-constrained optimization formulation of the coupled problem, the derivation of the optimality conditions, and the derivation of a Picard iterative scheme for the solution of the coupled problem. In addition, we perform several tests to experimentally assess the performance of our method on synthetic and clinical datasets. We demonstrate the convergence of the SIBIA optimization solver in different usage scenarios. We demonstrate that we can accurately solve the coupled problem in three dimensions ($256^3$ resolution) in a few minutes using 11 dual-x86 nodes. Also, we demonstrate that, with our coupled approach, we can successfully register normal MRI to tumor-bearing MRI while obtaining Dice coefficients that match those achieved when registering of normal-to-normal MRI.

physics.med-ph↗

arXiv2018

PDE-constrained optimization in medical image analysis

PDE-constrained optimization problems find many applications in medical image analysis, for example, neuroimaging, cardiovascular imaging, and oncological imaging. We review related literature and give examples on the formulation, discretization, and numerical solution of PDE-constrained optimization problems for medical imaging. We discuss three examples. The first one is image registration. The second one is data assimilation for brain tumor patients, and the third one data assimilation in cardiovascular imaging. The image registration problem is a classical task in medical image analysis and seeks to find pointwise correspondences between two or more images. The data assimilation problems use a PDE-constrained formulation to link a biophysical model to patient-specific data obtained from medical images. The associated optimality systems turn out to be sets of nonlinear, multicomponent PDEs that are challenging to solve in an efficient way. The ultimate goal of our work is the design of inversion methods that integrate complementary data, and rigorously follow mathematical and physical principles, in an attempt to support clinical decision making. This requires reliable, high-fidelity algorithms with a short time-to-solution. This task is complicated by model and data uncertainties, and by the fact that PDE-constrained optimization problems are ill-posed in nature, and in general yield high-dimensional, severely ill-conditioned systems after discretization. These features make regularization, effective preconditioners, and iterative solvers that, in many cases, have to be implemented on distributed-memory architectures to be practical, a prerequisite. We showcase state-of-the-art techniques in scientific computing to tackle these challenges.

math.OC↗

arXiv2018

A Semi-Lagrangian two-level preconditioned Newton-Krylov solver for constrained diffeomorphic image registration

We propose an efficient numerical algorithm for the solution of diffeomorphic image registration problems. We use a variational formulation constrained by a partial differential equation (PDE), where the constraints are a scalar transport equation. We use a pseudospectral discretization in space and second-order accurate semi-Lagrangian time stepping scheme for the transport equations. We solve for a stationary velocity field using a preconditioned, globalized, matrix-free Newton-Krylov scheme. We propose and test a two-level Hessian preconditioner. We consider two strategies for inverting the preconditioner on the coarse grid: a nested preconditioned conjugate gradient method (exact solve) and a nested Chebyshev iterative method (inexact solve) with a fixed number of iterations. We test the performance of our solver in different synthetic and real-world two-dimensional application scenarios. We study grid convergence and computational efficiency of our new scheme. We compare the performance of our solver against our initial implementation that uses the same spatial discretization but a standard, explicit, second-order Runge-Kutta scheme for the numerical time integration of the transport equations and a single-level preconditioner. Our improved scheme delivers significant speedups over our original implementation. As a highlight, we observe a 20$\times$ speedup for a two dimensional, real world multi-subject medical image registration problem.

math.OC↗