The Inverse Problems Seminar

February 24

Speaker: Olalekan Babaniyi
Title: hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems

Abstract: Inverse problems governed by differential equation models show up in many applications in science and engineering. I present an Inverse Problem PYthon library (hIPPYlib) for solving large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs). hIPPYlib implements state-of-the-art scalable algorithms that exploit the structure of the problem, notably the Hessian of the log posterior. The key property of the algorithms implemented in hIPPYlib is that the solution is computed at a cost, measured in forward PDE solves, that is independent of the parameter dimension. I will show applications of some of the algorithms to inverse problems in the ice sheet flow modeling, and medical imaging fields.

October 1

Speaker: Olalekan Babaniyi
Title: Uncertainty quantification in the biomechanical imaging field

Abstract: Biomechanical imaging (aka elastography) is a technique used to estimate the mechanical properties of tissue from measurements of its deformation. These mechanical properties can be used to noninvasively diagnose and monitor the treatment of various diseases. To compute the mechanical properties, one needs to solve an inverse problem. Standard methods to solve these inverse problems fail to account for the uncertainties in the mechanical properties stemming from measurement and modeling errors. In this talk, I present a inversion method that can be used to estimate both the mechanical properties and its uncertainties. I evaluate the performance of the method with noisy simulated data.

October 29

Speaker: Pradeep Bajracharya and Md Shakil Zaman
Title: Deterministic and Probabilistic Estimation of Cardiac Electrophysiological Model Parameters via Bayesian Active Learning

Abstract: The deterministic and probabilistic estimation of patient-specific tissue properties in the form of model parameters is important for personalized physiological models. The expensive computation associated with these model simulations, however, makes direct Markov Chain Monte Carlo (MCMC) sampling inefficient for both deterministic optimization and probabilistic estimation of model parameters. Approximated models resulting from replacing the simulation model with a computationally efficient surrogate, on the other hand, have seen limited accuracy. In this work, we present a Bayesian active learning method to directly approximate (1) the mode as deterministic optimization and (2) the posterior probability density function (pdf) as probabilistic estimation of cardiac model parameters with small number of samples where we intelligently select training points to query the expensive simulation process. We integrate a generative model into Bayesian active learning to allow high-dimensional model parameter estimation in low-dimensional space. Furthermore we introduce new acquisition functions to focus the selection of training points in this Bayesian active learning. We evaluated the presented methods in estimating tissue excitability in a 3D cardiac electrophysiological model in a range of synthetic and real-data experiments. Our method demonstrated its improved accuracy in approximating the mode and the posterior pdf compared to Bayesian active learning using regular acquisition functions, and substantially reduced computational cost in comparison to existing standard or accelerated MCMC sampling.

November 12

Speaker: Xiajun Jiang and Maryam Toloubidokhti
Title: Physics-Informed Personalization in Electrocardiographic Imaging via Machine Learning

Abstract: Incorporating patient-specific physics-informed knowledge as model parameters is important in electrocardiographic imaging (ECGI) models. Traditional approaches to ECGI are heavily physics-informed under the framework of state-space modeling (SSM), while the strong reliance on prior knowledge for model assumption precludes it from being identified from data only. Modern deep learning methods have no such reliance but are not good at disentangling the latent dynamics and its observation at data space and lack in interpretability. Our works present physics-informed personalized methods that 1) consider subject-specific geometry in the inverse mapping, 2) disentangle the emission model and latent dynamics, and 3) incorporate and correct the forward modeling physics in inverse estimation. We model both observation and unknown variables over geometric space and learn the inverse mapping in the latent space as a function of geometry. We then disentangle the latent dynamics to its emission to the data domain and supervise the model by physics laws governing the system. Furthermore, we incorporate the forward model in the form of a generative model in the inverse estimation to uncover and optimize the forward model parameters and deliver more accurate inverse solutions. Our methods demonstrate improved performance on patient-specific data and substantially increase its interpretability compared to existing deep learning methods.

November 19

Joint inverse problems and SMS coloquium seminar. See here for more information.

February 19

Speaker: Olalekan Babaniyi
Title: The modified error in constitutive equation formulation for the shear wave elastography inverse problem

Abstract: Shear wave elastography is a technique used to estimate the mechanical properties of tissue from measurements of its deformation. These mechanical properties can be used to noninvasively diagnose and help with the treatment of various diseases. To compute the mechanical properties, one needs to solve an inverse problem governed by differential equations. I present a modified error in constitutive equations formulation to solve this inverse problem. Some advantages of this formulation include the fact that it is robust in dealing with noisy or missing measurements, and it does not require boundary conditions in the forward model. I explain some of these mathematical properties and show some computational experiments with simulated and measured data.

March 5

Speaker: Quinn Kolt
Title: Direct methods for shear modulus inversion in the time-harmonic viscoelastic scalar wave model

Abstract: Shear-wave elasticity imaging is a modern approach for the noninvasive identification of diseased soft tissue. This technique is performed by first measuring the propagation of artificially generated shear waves through the tissue and then reconstructing the corresponding shear modulus from the measured displacement. We seek to identify accurate and efficient methods for this reconstruction. As a continuation of the previous talk, I develop five direct methods for shear modulus inversion in the scalar wave problem. I derive the variational formulations for each of these methods and then compare them with synthetic examples.

March 19

Speaker: Niels Otani
Title: Estimating electrical wave activity in the heart from its mechanical deformations

Abstract: Electrical waves in the heart, called action potentials, are responsible for coordinating the heart’s mechanical pumping function.  When abnormalities occur in the spatial and temporal patterns of these waves, irregular and sometimes life-threatening cardiac rhythms result.  Thus, we believe that the ability to visualize these waves in both research and clinical settings is important in managing these arrhythmias.  In this talk, I will describe our efforts to develop methods to “see” these waves, using mechanical deformations as input data.  As this is an ill-posed inverse problem, I will focus on a Tikhonov regularization process, which operates in a Krylov-like subspace obtained from Lanczos bidiagonalization.  Other approaches we have tried will also be discussed briefly.

April 2

Speaker: Michael Richards
Title: The Elastic Inverse Problem: In Search of Boundary Conditions

Abstract: Elasticity Imaging, or elastography, is a technique that utilizes image sequences of deforming soft tissues, typically acquired with common clinical modalities such as ultrasound or MRI, to measure and quantify their deformations or displacement fields. With careful considerations of the tissues constitutive behavior, the measured deformations can then be used to solve the mechanical inverse problem. A common constitutive assumption for many applications is that the tissue can be approximated as linear elastic, incompressible and deforming quasi-statically. If clinical ultrasound is used to measure the deformation, an additional assumption of plane strain or plane stress may also made to accommodate the 2D image data. With the constitutive assumptions, a finite element approximation is often used to iteratively solve the weak form of the forward elastic problem, with continuous updates to the mechanical properties (i.e. the shear modulus field) until the measured data sufficiently matches the model predicted deformations. With these finite element techniques, however, solving the forward elastic problem requires knowledge of forces or displacements on the boundary of the entire problem domain. In the past, these boundary conditions were either assumed or measured displacement data were used. This talk will introduce and examine the problems and errors introduced by assuming boundary conditions or using measured data. It will also discuss one novel strategy for searching for boundary conditions, in addition to the mechanical properties, and compare modulus reconstructions performed with assumptions to those where the boundary conditions were recovered. Example reconstructions will be given for simulated data applicable to breast cancer and vascular pathologies. A potential future application of the technique to musculoskeletal disorders will also be discussed.

April 23

Speaker: Raluca Felea
Title: Microlocal analysis in inverse problems

Abstract: We will discuss some inverse problems related to Synthetic aperture radar imaging and seismology. The forward operator $F$, which maps the image to the data, is a Fourier Integral Operator with singularities. We will describe $F$ and the normal operator $F^{*}F$ which is used to find the image. In these problems artifacts appear. We will analyze the artifacts and find their strength.

April 30

Speaker: Tony Wong
Title: Characterizing future coastal risks through probabilistic inversion

Abstract: Future sea-level rise poses severe risks for many coastal communities. The development of strategies to manage these risks hinges on a sound characterization of the often deep uncertainties. For example, recent studies suggest that large fractions of the Antarctic ice sheet may rapidly disintegrate in response to rising global temperatures, leading to potentially several meters of sea-level rise during the next few centuries. Whether such an Antarctic ice sheet disintegration will be triggered and if so, the resulting contributions to local sea-level rise, are deeply uncertain. In this talk, we will explore probabilistic methods for inferring likely values for the parameters governing these Antarctic dynamics. We will sample a wide range of future scenarios in order to characterize the implications of these uncertainties for managing risks in coastal areas, and assess what are the key drivers of coastal hazards.

February 5

Speaker: Olalekan Babaniyi
Title: An inverse problem for the basal sliding coefficient in a nonlinear Stokes ice sheet model

Abstract: Simulating the flow of ice sheets is an important problem for predicting the mass flux of ice into the oceans and sea level rise. This problem is modeled by the nonlinear Stokes equations, which contain several uncertain rheological parameters. The most uncertain of these parameters is the basal sliding coefficient, and it shows up as a parameter at the bottom boundary of the ice. In this talk, I will describe an inexact Newton conjugate gradient algorithm to infer the basal sliding coefficient from velocity data measured at the surface of the ice. I will show some simulation results where the algorithm is used to reconstruct the basal sliding coefficient of the Antarctic ice sheet.

Paper: see paper

March 4

Speaker: Nathan Cahill
Title: Deformable Registration of Biomedical Images

Abstract: Image registration is a crucial component of many biomedical image analysis systems. In clinical settings, images from different modalities, such as CT, MR, Ultrasound, and PET can provide vital complementary information about a disease, but they are difficult to directly compare unless they can be fused into a common frame of reference. Images from the same modality might also be collected longitudinally, and a radiologist or oncologist might need to align the images in order to assess how fast a tumor is growing or shrinking in response to treatment. In research settings such as ultrasound elastography, accurate estimates of local displacement enable solving inverse problems to approximate mechanical properties of the underlying tissue. In this talk, we will discuss how to approach image registration from a modeling perspective: by making assumptions about the underlying images and how they are related, an appropriate optimization problem can be posed and then solved numerically. We will then provide a variety of different examples in various practical contexts.

April 1

Speaker: Matthew Roberts
Title: Approximating the Generalized Singular Value Expansion

Abstract: The singular value expansion (SVE) of a compact operator is invaluable for analyzing Tikhonov regularization.  For semi-norm regularization, where a regularization operator such as the gradient is also involved, the generalized singular value expansion (GSVE) can be used in place of the SVE.  In this talk, we discuss the approximation of the GSVE of a pair of linear operators and present sufficient conditions for the convergence of a sequence of discretizations.