Hayden Schaeffer

A Low Patch-Rank Interpretation of Texture

We propose a novel cartoon-texture separation model using a sparse low-rank decomposition. Our texture model connects the separate ideas of robust principal component analysis (PCA) [E. J. Candes, X. Li, Y. Ma, and J. Wright, J. ACM, 58 (2011), 11], nonlocal methods [A. Buades, B. Coll, and J.-M. Morel, Multiscale Model. Simul., 4 (2005), pp. 490--530], [A. Buades, B. Coll, and J.-M. Morel, Numer. Math., 105 (2006), pp. 1--34], [G. Gilboa and S. Osher, Multiscale Model. Simul., 6 (2007), pp. 595--630], [G. Gilboa and S. Osher, Multiscale Model. Simul., 7 (2008), pp. 1005--1028], and cartoon-texture decompositions in an interesting way, taking advantage of each of these methodologies. We define our texture norm using the nuclear norm applied to patches in the image, interpreting the texture patches to be low-rank. In particular, this norm is easier to implement than many of the weak function space norms in the literature and is computationally faster than nonlocal methods since there is no explicit weight ...

Sparse dynamics for partial differential equations

We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms.

On the Compressive Spectral Method

The authors of [Proc. Natl. Acad. Sci. USA, 110 (2013), pp. 6634--6639] proposed sparse Fourier domain approximation of solutions to multiscale PDE problems by soft thresholding. We show here that the method enjoys a number of desirable numerical and analytic properties, including convergence for linear PDEs and a modified equation resulting from the sparse approximation. We also extend the method to solve elliptic equations and introduce sparse approximation of differential operators in the Fourier domain. The effectiveness of the method is demonstrated on homogenization examples, where its complexity is dependent only on the sparsity of the problem and constant in many cases.

Learning partial differential equations via data discovery and sparse optimization

We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed methods robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.

An L^1 Penalty Method for General Obstacle Problems

We construct an efficient numerical scheme for solving obstacle problems in divergence form. The numerical method is based on a reformulation of the obstacle in terms of an

Active Contours with Free Endpoints

L^1

Sparse model selection via integral terms

-like penalty on the variational problem. The reformulation is an exact regularizer in the sense that for a large (but finite) penalty parameter, we recover the exact solution. Our formulation is applied to classical elliptic obstacle problems as well as some related free boundary problems, for example, the two-phase membrane problem and the Hele--Shaw model. One advantage of the proposed method is that the free boundary inherent in the obstacle problem arises naturally in our energy minimization without any need for problem specific or complicated discretization. In addition, our scheme also works for nonlinear variational inequalities arising from convex minimization problems.

Space-Time Regularization for Video Decompression

Image segmentation methods with length regularized edge sets are known to have segments whose endpoints either terminate perpendicularly to the boundary of the domain, terminate at a triple junction where three segments connect, or terminate at a free endpoint where the segment does not connect to any other edges. However, level set based segmentation methods are only able to capture edge structures which contain the first two types of segments. In this work, we propose an extension to the level set based image segmentation method in order to detect free endpoint structures. By generalizing the curve representation used in Chan and Vese (Trans. Image Proces. 10(2):266–277, 2001; Int. J. Comput. Vis. 50(3):271–293, 2002) to also include free endpoint structures, we are able to segment a larger class of edge types. Since our model is formulated using the level set framework, the curve evolution inherits useful properties such as the ability to change its topology by splitting and merging. The numerical method is provided as well as experimental results on both synthetic and real images.

Sparse + low-energy decomposition for viscous conservation laws

Model selection and parameter estimation are important for the effective integration of experimental data, scientific theory, and precise simulations. In this work, we develop a learning approach for the selection and identification of a dynamical system directly from noisy data. The learning is performed by extracting a small subset of important features from an overdetermined set of possible features using a nonconvex sparse regression model. The sparse regression model is constructed to fit the noisy data to the trajectory of the dynamical system while using the smallest number of active terms. Computational experiments detail the models stability, robustness to noise, and recovery accuracy. Examples include nonlinear equations, population dynamics, chaotic systems, and fast-slow systems.

An Accelerated Method for Nonlinear Elliptic PDE

We consider the problem of reconstructing frames from a video which has been compressed using the video compressive sensing (VCS) method. In VCS data, each frame comes from first subsampling the original video data in space and then averaging the subsampled sequence in time. This results in a large linear system of equations whose inversion is ill-posed. We introduce a convex regularizer to invert the system, where the spatial component is regularized by the total variation seminorm, and the temporal component is regularized by enforcing sparsity on the difference between the spatial gradients of each frame. Since the regularizers are