Richard Tsai

University of Texas at Austin | UT · Department of Mathematics

In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves) via discretization of the suitably extended versions of these equations in a thin tubular neighborhood around the surfaces (or curves), with some corresponding boundary conditions. In particular, we consider...

We propose a deep learning approach for wave propagation in media with multiscale wave speed, using a second-order linear wave equation model. We use neural networks to enhance the accuracy of a given inaccurate coarse solver, which under-resolves a class of multiscale wave media and wave fields of interest. Our approach involves generating trainin...

The low dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low dimensional manifolds embedded in a high dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as...

We present a simple algorithm to approximate the viscosity solution of Hamilton-Jacobi~(HJ) equations by means of an artificial deep neural network. The algorithm uses a stochastic gradient descent-based algorithm to minimize the least square principle defined by a monotone, consistent numerical scheme. We analyze the least square principle's criti...

We propose a new framework for the sampling, compression, and analysis of distributions of point sets and other geometric objects embedded in Euclidean spaces. Our approach involves constructing a tensor called the RaySense sketch, which captures nearest neighbors from the underlying geometry of points along a set of rays. We explore various operat...

Applying parallel-in-time algorithms to multiscale Hamiltonian systems to obtain stable long time simulations is very challenging. In this paper, we present novel data-driven methods aimed at improving the standard parareal algorithm developed by Lion, Maday, and Turinici in 2001, for multiscale Hamiltonian systems. The first method involves constr...

We consider a surveillance-evasion game in an environment with obstacles. In such an environment, a mobile pursuer seeks to maintain the visibility with a mobile evader, who tries to get occluded from the pursuer in the shortest time possible. In this two-player zero-sum game setting, we study the discontinuities of the value of the game near the b...

We present a family of high-order trapezoidal rule-based quadratures for a class of singular integrals, where the integrand has a point singularity. The singular part of the integrand is expanded in a Taylor series involving terms of increasing smoothness. The quadratures are based on the trapezoidal rule, with the quadrature weights for Cartesian...

In this paper, we study linear regression applied to data structured on a manifold. We assume that the data manifold is smooth and is embedded in a Euclidean space, and our objective is to reveal the impact of the data manifold's extrinsic geometry on the regression. Specifically, we analyze the impact of the manifold's curvatures (or higher order...

The low-dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low-dimensional manifolds embedded in a high-dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as...

We propose a deep learning approach for wave propagation in media with multiscale wave speed, using a second-order linear wave equation model. We use neural networks to enhance the accuracy of a given inaccurate coarse solver, which under-resolves a class of multiscale wave media and wave fields of interest. Our approach involves generating trainin...

In this paper, we solve the linearized Poisson-Boltzmann equation, used to model the electric potential of macromolecules in a solvent. We derive a corrected trapezoidal rule with improved accuracy for a boundary integral formulation of the linearized Poisson-Boltzmann equation. More specifically, in contrast to the typical boundary integral formul...

We present convergence theory for corrected quadrature rules on uniform Cartesian grids for functions with a point singularity. We begin by deriving an error estimate for the punctured trapezoidal rule, and then derive error expansions. We define the corrected trapezoidal rules, based on the punctured trapezoidal rule, where the weights for the nod...

We present a family of high order trapezoidal rule-based quadratures for a class of singular integrals, where the integrand has a point singularity. The singular part of the integrand is expanded in a Taylor series involving terms of increasing smoothness. The quadratures are based on the trapezoidal rule, with the quadrature weights for Cartesian...

We present new higher-order quadratures for a family of boundary integral operators re-derived using the approach introduced in Kublik et al. (2013) [7]. In this formulation, a boundary integral over a smooth, closed hypersurface is transformed into an equivalent volume integral defined in a sufficiently thin tubular neighborhood of the surface. Th...

We present new higher-order quadratures for a family of boundary integral operators re-derived using the approach introduced in [Kublik, Tanushev, and Tsai - J. Comp. Phys. 247: 279-311, 2013]. In this formulation, a boundary integral over a smooth, closed hypersurface is transformed into an equivalent volume integral defined in a sufficiently thin...

We consider surveillance-evasion differential games, where a pursuer must try to constantly maintain visibility of a moving evader. The pursuer loses as soon as the evader becomes occluded. Optimal controls for game can be formulated as a Hamilton-Jacobi-Isaac equation. We use an upwind scheme to compute the feedback value function, corresponding t...

We present a new formulation for the computation of solutions of a class of Hamilton Jacobi Bellman (HJB) equations on closed smooth surfaces of co-dimension one. For the class of equations considered in this paper, the viscosity solution of the HJB equation is equivalent to the value function of a corresponding optimal control problem. In this wor...

This paper describes a novel method for fast colonic polyp detection in colonoscopy images. Firstly, polyp detection is formulated as a similarity-based anomaly detection method, which formally involves non-dominated sorting based on multiple objectives. The chosen objectives rely on the main physical and visible differences, observed in colonoscop...

A new parallel-in-time iterative method is proposed for solving the homogeneous second-order wave equation. The new method involves a coarse scale propagator, allowing for larger time steps, and a fine scale propagator which fully resolves the medium using finer spatial grid and uses shorter time steps. The fine scale propagator is run in parallel...

We propose a new framework for the sampling, compression, and analysis of distributions of point sets and other geometric objects embedded in Euclidean spaces. A set of randomly selected rays are projected onto their closest points in the data set, forming the ray signature. From the signature, statistical information about the data set, as well as...

This paper studies a two-player game with a quantitative surveillance requirement on an adversarial target moving in a discrete state space and a secondary objective to maximize short-term visibility of the environment. We impose the surveillance requirement as a temporal logic constraint.We then use a greedy approach to determine vantage points th...

This paper studies a two-player game with a quantitative surveillance requirement on an adversarial target moving in a discrete state space and a secondary objective to maximize short-term visibility of the environment. We impose the surveillance requirement as a temporal logic constraint. We then use a greedy approach to determine vantage points t...

Colon cancer prevention, diagnosis, and prognosis are directly related to the identification of colonic polyps, in colonoscopy video sequences. In addition, diagnosing colon cancer in the early stages improves significantly the chance of surviving and effective treatment. Due to the large number of images that come from colonoscopy, the identificat...

Colorectal cancer (CRC) is one of the most common cancers worldwide and after a certain age (≥50) regular colonoscopy examination for CRC screening is highly recommended. One of the most prominent precursors of CRC are abnormal growths known as polyps. If a polyp is detected during colonoscopy examination the endoscopist needs to decide whether the...

Two general challenges faced by data analysis are the existence of noise and the extraction of meaningful information from collected data. In this study, we used a multiscale framework to reduce the effects caused by noise and to extract explainable geometric properties to characterize finite metric spaces. We conducted lab experiments that integra...

We develop a coupled grid based particle and implicit boundary integral method for simulation of three-dimensional interfacial flows with the presence of insoluble surfactant. The grid based particle method (GBPM, Leung and Zhao [20]) tracks the interface by the projection of the neighboring Eulerian grid points and does not require stitching of pa...

A new parallel-in-time iterative method is proposed for solving the homogeneous second-order wave equation. The new method involves a coarse scale propagator, allowing for larger time steps, and a fine scale propagator which fully resolves the medium using finer spatial grid and shorter time steps. The fine scale propagator is run in parallel for s...

We present a new formulation for the computation of solutions of a class of Hamilton Jacobi Bellman (HJB) equations on closed smooth surfaces of co-dimension one. For the class of equations considered in this paper, the viscosity solution of the HJB equation is equivalent to the value function of a corresponding optimal control problem. In this wor...

Background and study aims Detection of polyps during colonoscopy is essential for screening colorectal cancer and computer-aided-diagnosis (CAD) could be helpful for this objective. The goal of this study was to assess the efficacy of CAD in detection of polyps in video colonoscopy by using three methods we have proposed and applied for diagnosis o...

We study the problem of visibility-based exploration, reconstruction and surveillance in the context of supervised learning. Using a level set representation of data and information, we train a convolutional neural network to determine vantage points that maximize visibility. We show that this method drastically reduces the on-line computational co...

In this paper, we present a new multiscale domain decomposition algorithm for computing solutions of static Eikonal equations. The new method is an iterative two-scale method that uses a parareal-like update scheme in combination with standard Eikonal solvers. The purpose of the two scales is to accelerate convergence and maintain accuracy. We adap...

In this paper, we present a new multiscale domain decomposition algorithm for computing solutions of static Eikonal equations. The new method is an iterative two-scale method that uses a parareal-like update scheme in combination with standard Eikonal solvers. The purpose of the two scales is to accelerate convergence and maintain accuracy. We adap...

Wireless capsule endoscope (WCE) enables the visualization of the interior of the gastrointestinal (GI) tract. In particular it is very important for the examination of regions in the small bowel that cannot be reached by conventional endoscopy techniques. However, when an abnormality is found in WCE images of the small bowel, it is unknown how far...

We present an algorithm for computing the nonlinear interface dynamics of the Mullins-Sekerka model for interfaces that are defined implicitly (e.g. by a level set function) using integral equations. The computation of the dynamics involves solving Laplace’s equation with Dirichlet boundary conditions on multiply connected and unbounded domains and...

We propose a new algorithm for solving Helmholtz equations in exterior domains with implicitly represented boundaries. The algorithm not only combines the advantages of implicit surface representation and the boundary integral method, but also provides a new way to compute a class of the so-called hypersingular integrals. The keys to the proposed a...

A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equations that arise in mathematical models for the electrostatics of molecules in solvent. The proposed method used an implicit boundary integral formulation to derived a linear system defined on Cartesian nodes in a narrowband surroundi...

A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equations that arise in mathematical models for the electrostatics of molecules in solvent. The proposed method used an implicit boundary integral formulation to derived a linear system defined on Cartesian nodes in a narrowband surroundi...

In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves). In particular, we consider PDEs that originate from variational principles defined on the surfaces; these include Laplace-Beltrami equations and surface wave equations. The approach is to systematically for...

In a previous work we have shown that the curve representing the dissimilarity measure between consecutive frames of a wireless capsule endoscopic video of the small bowel, obtained by means of an image registration method, can be regarded as a rough indicator of the speed of the capsule, and simultaneously, it is also a valuable auxiliary medical...

A weighted version of the parareal method for parallel-in-time computation of time dependent problems is presented. Linear stability analysis for a scalar weighing strategy shows that the new scheme may enjoy favorable stability properties with marginal reduction in accuracy at worse. More complicated matrix-valued weights are applied in numerical...

A weighted version of the parareal method for parallel-in-time computation of time dependent problems is presented. Linear stability analysis for a scalar weighing strategy shows that the new scheme may enjoy favorable stability properties with marginal reduction in accuracy at worse. More complicated matrix-valued weights are applied in numerical...

We study analytically and numerical the growth rate of a crystal surface growing by several screw dislocations. To describe several spiral steps we use the revised level set method for spirals by the authors (Journal of Scientific Computing 62, 831-874, 2015). We carefully compare our simulation results on the growth rates with predictions in a cla...

We study analytically and numerical the growth rate of a crystal surface growing by several screw dislocations. To describe several spiral steps we use the revised level set method for spirals by the authors (Journal of Scientific Computing 62, 831-874, 2015). We carefully compare our simulation results on the growth rates with predictions in a cla...

We introduce a new parallel in time (parareal) algorithm which couples multiscale integrators with fully resolved fine scale integration and computes highly oscillatory solutions for a class of ordinary differential equations in parallel. The algorithm computes a low-cost approximation of all slow variables in the system. Then, fast phase-like vari...

We provide a new approach for computing integrals over hypersurfaces in the level set framework. In particular, this new approach is able to compute high order approximations of line or surface integrals in the case where the curve or surface has singularities such as corners. The method is based on the discretization (via simple Riemann sums) of t...

We provide a new approach for computing integrals over hypersurfaces in the level set framework. The method is based on the discretization (via simple Riemann sums) of the classical formulation used in the level set framework, with the choice of specific kernels supported on a tubular neighborhood around the interface to approximate the Dirac delta...

We propose a new algorithm for solving Helmholtz equations in exterior domains with implicitly represented boundaries. The algorithm not only combines the advantages of implicit surface representation and the boundary integral method, but also provides a new way to compute a class of the so-called hyper-singular integrals. The keys to the proposed...

We propose a new formulation using the closest point mapping for integrating over smooth curves and surfaces with boundaries that are described by their closest point mappings. Contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wish t...

We present an algorithm for computing the nonlinear interface dynamics of the Mullins-Sekerka model for interfaces that are defined implicitly (e.g. by a level set function) using integral equations . The computation of the dynamics involves solving Laplace's equation with Dirichlet boundary conditions on multiply connected and unbounded domains an...

In a previous work we have shown that the curve representing the dissimilarity measure between consecutive frames of a wireless capsule endoscopic video of the small bowel, obtained by means of an image registration method, can be regarded as a rough indicator of the speed of the capsule, and simultaneously, it is also a valuable auxiliary medical...

We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the system using an appropriate multiscale integrator, which is refined using parallel fine scale integrations. Converg...

Wireless Capsule Endoscope (WCE) is an innovative imaging device that permits physicians to examine all the areas of the Gastrointestinal (GI) tract. It is especially important for the small intestine, where traditional invasive endoscopies cannot reach. Although WCE represents an extremely important advance in medical imaging, a major drawback tha...

We introduce a new level set method to simulate motion of spirals in a crystal surface governed by an eikonal–curvature flow equation. Our formulation allows collision of several spirals and different strength (different modulus of Burgers vectors) of screw dislocation centers. We represent a set of spirals by a level set of a single auxiliary func...

Colorectal polyps are important precursors to colon cancer, a major health problem. Colon capsule endoscopy (CCE) is a safe and minimally invasive examination procedure, in which the images of the intestine are obtained via digital cameras on board of a small capsule ingested by a patient. The video sequence is then analyzed for the presence of pol...

We address the problem of exploring an unknown environment using a range sensor. In the presence of opaque objects of arbitrary shape and topology, exploration is driven by minimization of residual uncertainty, that is primarily due to lack of visibility. The resulting optimal control is approximated with a computationally efficient heuristic, that...

We consider the inverse problem of finding sparse initial data from the sparsely sampled solutions of the heat equation. The initial data are assumed to be a sum of an unknown but finite number of Dirac delta functions at unknown locations. Pointwise values of the heat solution at only a few locations are used in an ℓ 1 constrained optimization to...

A theory of iterated averaging is developed for a class of highly oscillatory ordinary differential equations (ODEs) with three well separated time scales. The solutions of these equations are assumed to be (almost) periodic in the fastest time scales. It is proved that the dynamics on the slowest time scale can be approximated by an effective ODE...

This article focuses on the mathematical problem of reconstructing dynamic permeability K(ω) of two-phase composites from data at different frequencies, utilizing the analytic structure of K(ω). To numerically simulate the data by solving the unsteady Stokes equations in the 3D domain reconstructed from μ-CT scans of the composites, the issue of se...

Fast sweeping methods have become a useful tool for computing the solutions of static Hamilton-Jacobi equations. By adapting the main idea behind these methods, we describe a new approach for computing steady state solutions to systems of conservation laws. By exploiting the flow of information along characteristics, these fast sweeping methods can...

We present an integrated algorithm for computing vantage points of maximal visibility in an a priori unknown fully 3D environment using point clouds and a multi-resolution Hausdorff metric based surface reconstruction procedure. This work aims at demonstrate the proposed algorithm's ability to explore and learn complicated 3D urban environments. We...

We propose a new heterogeneous multiscale method (HMM) for computing the effective behavior of a class of highly oscillatory ordinary differential equations (ODEs). Without the need for identifying hidden slow variables, the proposed method is constructed based on the following ideas: a nonstandard splitting of the vector field (the right hand side...

In the limit of infinite stiffness, the differential equations of motion of stiff mechanical systems become differential algebraic equations whose solutions stay in a constraint submanifold P of the phase space. Even though solutions of the stiff differential equations are typically oscillatory with large frequency, there exists a slow manifold P c...