A Fast Algorithm for Parabolic PDE-based Inverse Problems Based on Laplace Transforms and Flexible Krylov Solvers

We consider the problem of estimating parameters in large-scale weakly nonlinear inverse problems for which the underlying governing equations is a linear, time-dependent, parabolic partial differential equation. A major challenge in solving these inverse problems using Newton-type methods is the computational cost associated with solving the forward problem and with repeated construction of the Jacobian, which represents the sensitivity of the measurements to the unknown parameters. Forming the Jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time-dependent parabolic partial differential equations corresponding to multiple sources and receivers. We propose an efficient method based on a Laplace transform-based exponential time integrator combined with a flexible Krylov subspace approach to solve the resulting shifted systems of equations efficiently. Our proposed solver speeds up the computation of the forward and adjoint problems, thus yielding significant speedup in total inversion time. We consider an application from Transient Hydraulic Tomography (THT), which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests. The algorithms discussed are applied to a synthetic example taken from THT to demonstrate the resulting computational gains of this proposed method

Multipreconditioned GMRES for Shifted Systems

An implementation of GMRES with multiple preconditioners (MPGMRES) is proposed for solving shifted linear systems with shift-and-invert preconditioners. With this type of preconditioner, the Krylov subspace can be built without requiring the matrix-vector product with the shifted matrix. Furthermore, the multipreconditioned search space is shown to grow only linearly with the number of preconditioners. This allows for a more efficient implementation of the algorithm. The proposed implementation is tested on shifted systems that arise in computational hydrology and the evaluation of different matrix functions. The numerical results indicate the effectiveness of the proposed approach.U.S. National Science Foundation under grant DMS–1418882 and and by the Department of Energy grant DE–SC00165

Preconditioners for Krylov subspace methods: An overview

When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind

Fast algorithms for biophysically-constrained inverse problems in medical imaging

We present algorithms and software for parameter estimation for forward and inverse tumor growth problems and diffeomorphic image registration. Our methods target the following scenarios: automatic image registration of healthy images to tumor bearing medical images and parameter estimation/calibration of tumor models. This thesis focuses on robust and scalable algorithms for these problems. Although the proposed framework applies to many problems in oncology, we focus on primary brain tumors and in particular low and high-grade gliomas. For the tumor model, the main quantity of interest is the extent of tumor infiltration into the brain, beyond what is visible in imaging. The inverse tumor problem assumes that we have patient images at two (or more) well-separated times so that we can observe the tumor growth. Also, the inverse problem requires that the two images are segmented. But in a clinical setting such information is usually not available. In a typical case, we just have multimodal magnetic resonance images with no segmentation. We address this lack of information by solving a coupled inverse registration and tumor problem. The role of image registration is to find a plausible mapping between the patient's tumor-bearing image and a normal brain (atlas), with known segmentation. Solving this coupled inverse problem has a prohibitive computational cost, especially in 3D. To address this challenge we have developed novel schemes, scaled up to 200K cores. Our main contributions is the design and implementation of fast solvers for these problems. We also study the performance for the tumor parameter estimation and registration solvers and their algorithmic scalability. In particular, we introduce the following novel algorithms: An adjoint formulation for tumor-growth problems with/without mass-effect; The first parallel 3D Newton-Krylov method for large diffeomorphic image registration; A novel parallel semi-Lagrangian algorithm for solving advection equations in image registration and its parallel implementation on shared and distributed memory architectures; and Accelerated FFT (AccFFT), an open-source parallel FFT library for CPU and GPUs scaled up to 131,000 cores with optimized kernels for computing spectral operators. The scientific outcomes of this thesis, has appeared in the proceedings of three ACM/IEEE SCxy conferences (two best student paper finalist, and one ACM SRC gold medal), two journal papers, two papers in review, four papers in preparation (coupling, mass effect, segmentation, and multi-species tumor model), and seven conference presentations.Computational Science, Engineering, and Mathematic