PDEs with Compressed Solutions

Sparsity plays a central role in recent developments in signal processing, linear algebra, statistics, optimization, and other fields. In these developments, sparsity is promoted through the addition of an $L^1$ norm (or related quantity) as a constraint or penalty in a variational principle. We apply this approach to partial differential equations that come from a variational quantity, either by minimization (to obtain an elliptic PDE) or by gradient flow (to obtain a parabolic PDE). Also, we show that some PDEs can be rewritten in an $L^1$ form, such as the divisible sandpile problem and signum-Gordon. Addition of an $L^1$ term in the variational principle leads to a modified PDE where a subgradient term appears. It is known that modified PDEs of this form will often have solutions with compact support, which corresponds to the discrete solution being sparse. We show that this is advantageous numerically through the use of efficient algorithms for solving $L^1$ based problems.Comment: 21 pages, 15 figure

Dynamics and termination cost of spatially coupled mean-field models

This work is motivated by recent progress in information theory and signal processing where the so-called `spatially coupled' design of systems leads to considerably better performance. We address relevant open questions about spatially coupled systems through the study of a simple Ising model. In particular, we consider a chain of Curie-Weiss models that are coupled by interactions up to a certain range. Indeed, it is well known that the pure (uncoupled) Curie-Weiss model undergoes a first order phase transition driven by the magnetic field, and furthermore, in the spinodal region such systems are unable to reach equilibrium in sub-exponential time if initialized in the metastable state. By contrast, the spatially coupled system is, instead, able to reach the equilibrium even when initialized to the metastable state. The equilibrium phase propagates along the chain in the form of a travelling wave. Here we study the speed of the wave-front and the so-called `termination cost'--- \textit{i.e.}, the conditions necessary for the propagation to occur. We reach several interesting conclusions about optimization of the speed and the cost.Comment: 12 pages, 11 figure

Estimating localized sources of diffusion fields using spatiotemporal sensor measurements

We consider diffusion fields induced by a finite number of spatially localized sources and address the problem of estimating these sources using spatiotemporal samples of the field obtained with a sensor network. Within this framework, we consider two different time evolutions: the case where the sources are instantaneous, as well as, the case where the sources decay exponentially in time after activation. We first derive novel exact inversion formulas, for both source distributions, through the use of Green's second theorem and a family of sensing functions to compute generalized field samples. These generalized samples can then be inverted using variations of existing algebraic methods such as Prony's method. Next, we develop a novel and robust reconstruction method for diffusion fields by properly extending these formulas to operate on the spatiotemporal samples of the field. Finally, we present numerical results using both synthetic and real data to verify the algorithms proposed herein

Model based learning for accelerated, limited-view 3D photoacoustic tomography

Recent advances in deep learning for tomographic reconstructions have shown great potential to create accurate and high quality images with a considerable speed-up. In this work we present a deep neural network that is specifically designed to provide high resolution 3D images from restricted photoacoustic measurements. The network is designed to represent an iterative scheme and incorporates gradient information of the data fit to compensate for limited view artefacts. Due to the high complexity of the photoacoustic forward operator, we separate training and computation of the gradient information. A suitable prior for the desired image structures is learned as part of the training. The resulting network is trained and tested on a set of segmented vessels from lung CT scans and then applied to in-vivo photoacoustic measurement data

Accelerated High-Resolution Photoacoustic Tomography via Compressed Sensing

Current 3D photoacoustic tomography (PAT) systems offer either high image quality or high frame rates but are not able to deliver high spatial and temporal resolution simultaneously, which limits their ability to image dynamic processes in living tissue. A particular example is the planar Fabry-Perot (FP) scanner, which yields high-resolution images but takes several minutes to sequentially map the photoacoustic field on the sensor plane, point-by-point. However, as the spatio-temporal complexity of many absorbing tissue structures is rather low, the data recorded in such a conventional, regularly sampled fashion is often highly redundant. We demonstrate that combining variational image reconstruction methods using spatial sparsity constraints with the development of novel PAT acquisition systems capable of sub-sampling the acoustic wave field can dramatically increase the acquisition speed while maintaining a good spatial resolution: First, we describe and model two general spatial sub-sampling schemes. Then, we discuss how to implement them using the FP scanner and demonstrate the potential of these novel compressed sensing PAT devices through simulated data from a realistic numerical phantom and through measured data from a dynamic experimental phantom as well as from in-vivo experiments. Our results show that images with good spatial resolution and contrast can be obtained from highly sub-sampled PAT data if variational image reconstruction methods that describe the tissues structures with suitable sparsity-constraints are used. In particular, we examine the use of total variation regularization enhanced by Bregman iterations. These novel reconstruction strategies offer new opportunities to dramatically increase the acquisition speed of PAT scanners that employ point-by-point sequential scanning as well as reducing the channel count of parallelized schemes that use detector arrays.Comment: submitted to "Physics in Medicine and Biology

Differential growth of wrinkled biofilms

Biofilms are antibiotic-resistant bacterial aggregates that grow on moist surfaces and can trigger hospital-acquired infections. They provide a classical example in biology where the dynamics of cellular communities may be observed and studied. Gene expression regulates cell division and differentiation, which affect the biofilm architecture. Mechanical and chemical processes shape the resulting structure. We gain insight into the interplay between cellular and mechanical processes during biofilm development on air-agar interfaces by means of a hybrid model. Cellular behavior is governed by stochastic rules informed by a cascade of concentration fields for nutrients, waste and autoinducers. Cellular differentiation and death alter the structure and the mechanical properties of the biofilm, which is deformed according to Foppl-Von Karman equations informed by cellular processes and the interaction with the substratum. Stiffness gradients due to growth and swelling produce wrinkle branching. We are able to reproduce wrinkled structures often formed by biofilms on air-agar interfaces, as well as spatial distributions of differentiated cells commonly observed with B. subtilis.Comment: 19 pages, 13 figure

Computation and Learning in High Dimensions (hybrid meeting)

The most challenging problems in science often involve the learning and accurate computation of high dimensional functions. High-dimensionality is a typical feature for a multitude of problems in various areas of science. The so-called curse of dimensionality typically negates the use of traditional numerical techniques for the solution of high-dimensional problems. Instead, novel theoretical and computational approaches need to be developed to make them tractable and to capture fine resolutions and relevant features. Paradoxically, increasing computational power may even serve to heighten this demand, since the wealth of new computational data itself becomes a major obstruction. Extracting essential information from complex problem-inherent structures and developing rigorous models to quantify the quality of information in a high-dimensional setting pose challenging tasks from both theoretical and numerical perspective. This has led to the emergence of several new computational methodologies, accounting for the fact that by now well understood methods drawing on spatial localization and mesh-refinement are in their original form no longer viable. Common to these approaches is the nonlinearity of the solution method. For certain problem classes, these methods have drastically advanced the frontiers of computability. The most visible of these new methods is deep learning. Although the use of deep neural networks has been extremely successful in certain application areas, their mathematical understanding is far from complete. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computational methods and to promote the exchange of ideas emerging in various disciplines about how to treat multiscale and high-dimensional problems