Richard Tsai

Olivier Bokanowski, Carlos Esteve-Yagüe, Richard Tsai

We establish a well-posedness and error-estimation framework that solves Hamilton-Jacobi equations by minimizing the least-squares residual of monotone finite-difference discretizations. This approach also applies naturally to second-order elliptic and parabolic problems. We prove that, under suitable monotonicity conditions, every critical point of the residual loss functional is the unique global minimizer and coincides with the solution of the discrete scheme. We derive \emph{a~posteriori} error estimates that bound the approximation error by the magnitude of the residual with explicit, computable constants, and extend the full analysis to time-dependent problems with implicit discretization of the time derivatives. A spectral analysis of the linearized system shows that the condition number scales as $O(Δx^{-1})$ for proper schemes, and as $O(\exp(Δx^{-1}))$ under a uniform ellipticity condition. These results quantify the increasing difficulty of solving the optimization problem on finer meshes, and motivates a progressive multi-level warm-start strategy using Artificial Neural Networks. Combined with the convergence theorem of Barles and Souganidis for monotone and consistent schemes, our results guarantee that the solutions obtained converge to the unique viscosity solution as the mesh is refined. Numerical experiments demonstrate the scalability of the approach to high-dimensional Eikonal equations, level-set problems, and Hamilton--Jacobi--Isaacs equations with genuine second-order diffusion arising from stochastic differential games.

Juncai He, Richard Tsai, Rachel Ward

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 input. However, one often needs to consider evaluating the optimized network at points outside the training distribution. This paper considers the case in which the training data is distributed in a linear subspace of $\mathbb R^d$. We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace. We study the potential regularization effects associated with the network's depth and noise in the codimension of the data manifold. We also present additional side effects in training due to the presence of noise.

Yimin Zhong, Kui Ren, Richard Tsai

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 surrounding the closed surface that separate the molecule and the solvent. The needed implicit surfaces is constructed from the given atomic description of the molecules, by a sequence of standard level set algorithms. A fast multipole method is applied to accelerate the solution of the linear system. A few numerical studies involving some standard test cases are presented and compared to other existing results.

Jay Chu, Richard Tsai

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 formulate extensions of the variational integrals, derive the Euler-Lagrange equations of the extended problem, including the boundary conditions} that can be easily discretized on uniform Cartesian grids or adaptive meshes. In our approach, the surfaces are defined implicitly by the distance functions or by the closest point mapping. As such extensions are not unique, we investigate how a class of simple extensions can influence the resulting PDEs. In particular, we reduce the surface PDEs to model problems defined on a periodic strip and the corresponding boundary conditions, and use classical Fourier and Laplace transform methods to study the well-posedness of the resulting problems. For elliptic and parabolic problems, our boundary closure mostly yields stable algorithms to solve nonlinear surface PDEs. For hyperbolic problems, the proposed boundary closure is unstable in general, but the instability can be easily controlled by either adding a higher order regularization term or by periodically but infrequently "reinitializing" the computed solutions. Some numerical examples for each representative surface PDEs are presented.

Catherine Kublik, Richard Tsai

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 function. The novelty lies in the choice of kernels, specifically its number of vanishing moments, which enables accurate computations of integrals over a class of closed, continuous, piecewise smooth, curves or surfaces; e.g. curves in two dimensions that contain finite number of corners. We prove that for smooth interfaces, if the kernel has enough vanishing moments (related to the dimension of the embedding space), the analytical integral formulation coincides exactly with the integral one wishes to calculate. For curves with corners and cusps, the formulation is not exact but we provide an analytical result relating the severity of the corner or cusp with the width of the tubular neighborhood. We show numerical examples demonstrating the capability of the approach, especially for integrating over piecewise smooth interfaces and for computing integrals where the integrand is only Lipschitz continuous or has an integrable singularity.

Gil Ariel, Seong Jun Kim, Richard Tsai

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. Convergence is obtained using an alignment algorithm for fast phase-like variables. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances. We propose an alignment algorithm for almost-periodic solution, in which case convergence of the parareal iterations is proved. The applicability of the method is demonstrated in numerical examples.

Chieh Chen, Catherine Kublik, Richard Tsai

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 propagating the interface using a normal velocity obtained from the solution of the PDE at each time step. Our method is based on a simple formulation for implicit interfaces, which rewrites boundary integrals as volume integrals over the entire space. The resulting algorithm thus inherits the benefits of both level set methods and boundary integral methods to simulate the nonlocal front propagation problem with possible topological changes. We present numerical results in both two and three dimensions to demonstrate the effectiveness of the algorithm.

Gil Ariel, Hieu Nguyen, Richard Tsai

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 examples. The weights are optimized using information from past iterations, providing a systematic framework for using the parareal iterations as an approach to multiscale coupling. The advantage of the method is demonstrated using numerical examples, including some well-studied nonlinear Hamiltonian systems.

Liangchen Liu, Juncai He, Richard Tsai

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 nonlinearity in the parameterization when the curvatures are locally zero) on the uniqueness of the regression solution. Our findings suggest that the corresponding linear regression does not have a unique solution when the embedded submanifold is flat in some dimensions. Otherwise, the manifold's curvature (or higher order nonlinearity in the embedding) may contribute significantly, particularly in the solution associated with the normal directions of the manifold. Our findings thus reveal the role of data manifold geometry in ensuring the stability of regression models for out-of-distribution inferences.

Lindsay Martin, Richard Tsai

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 adapt a weighted version of the parareal method for stability, and the optimal weights are studied via a model problem. Numerical examples are given to demonstrate the method.

Lukas Taus, Vrushabh Zinage, Takashi Tanaka et al.

We revisit the problem of motion planning in the Gaussian belief space. Motivated by the fact that most existing sampling-based planners suffer from high computational costs due to the high-dimensional nature of the problem, we propose an approach that leverages a deep learning model to predict optimal path candidates directly from the problem description. Our proposed approach consists of three steps. First, we prepare a training dataset comprising a large number of input-output pairs: the input image encodes the problem to be solved (e.g., start states, goal states, and obstacle locations), whereas the output image encodes the solution (i.e., the ground truth of the shortest path). Any existing planner can be used to generate this training dataset. Next, we leverage the U-Net architecture to learn the dependencies between the input and output data. Finally, a trained U-Net model is applied to a new problem encoded as an input image. From the U-Net's output image, which is interpreted as a distribution of paths,an optimal path candidate is reconstructed. The proposed method significantly reduces computation time compared to the sampling-based baseline algorithm.

Lukas Taus, Richard Tsai, Jeffrey G. Andrews

In a wireless network, the spatial location of the transmitters has a large impact on the achievable rate at each user location. The optimal placement of -- for example -- cellular base stations is a difficult non-convex problem, and is usually addressed with simplified propagation models and simplified heuristics that may account for specifics such as the site topology, building locations, and user density. We propose a mathematically rigorous framework for optimal transmitter placement that explicitly integrates detailed site-specific maps, spatial material properties, and realistic signal attenuation. We introduce a novel aggregated network quality functional which captures the essential trade-off between maximizing network coverage and minimizing cost, and establish the problem's sub-modularity under certain practical conditions. To solve the resulting resource-constrained optimization problem for sparse, discrete transmitter configurations, we propose the Interference-Aware Submodular Placement Algorithm (IA-SPA) and prove theoretical performance guarantees on its gap from optimality. IA-SPA is general and can incorporate existing BS locations and prohibited areas (e.g. a lake), making it useful for either clean-slate or incremental deployments. We show the utility of our approach using a ray tracing-based simulation framework applied to 3D maps of San Francisco and Florence, where we compare to known base station deployments by AT&T, T-Mobile and Iliad. We demonstrate that our proposed placement strategy achieves significant increases in mean data rate (about 2x) and edge rate ($2-8$x) compared to existing tower deployments, using the same number of transmitters.

Luis Kaiser, Richard Tsai, Christian Klingenberg

In a variety of scientific and engineering domains, the need for high-fidelity and efficient solutions for high-frequency wave propagation holds great significance. Recent advances in wave modeling use sufficiently accurate fine solver outputs to train a neural network that enhances the accuracy of a fast but inaccurate coarse solver. In this paper we build upon the work of Nguyen and Tsai (2023) and present a novel unified system that integrates a numerical solver with a deep learning component into an end-to-end framework. In the proposed setting, we investigate refinements to the network architecture and data generation algorithm. A stable and fast solver further allows the use of Parareal, a parallel-in-time algorithm to correct high-frequency wave components. Our results show that the cohesive structure improves performance without sacrificing speed, and demonstrate the importance of temporal dynamics, as well as Parareal, for accurate wave propagation.

Bing-Ze Lu, Richard Tsai

In many applications, one needs to learn a dynamical system from its solutions sampled at a finite number of time points. The learning problem is often formulated as an optimization problem over a chosen function class. However, in the optimization procedure, it is necessary to employ a numerical scheme to integrate candidate dynamical systems and assess how their solutions fit the data. This paper reveals potentially serious effects of a chosen numerical scheme on the learning outcome. In particular, our analysis demonstrates that a damped oscillatory system may be incorrectly identified as having "anti-damping" and exhibiting a reversed oscillation direction, despite adequately fitting the given data points.

Suda Bharadwaj, Louis Ly, Bo Wu et al.

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 that optimize a notion of information gain, namely, the number of newly-seen states. By using a convolutional neural network trained on a class of environments, we can efficiently approximate the information gain at each potential vantage point.Subsequent vantage points are chosen such that moving to that location will not jeopardize the surveillance requirement, regardless of any future action chosen by the target. Our method combines guarantees of correctness from formal methods with the scalability of machine learning to provide an efficient approach for surveillance-constrained visibility optimization.

Catherine Kublik, Richard Tsai

We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, 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 to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of the Jacobian matrix of the closest point mapping. Additionally, we extend the formulation - initially derived to integrate over manifolds of codimension one - to include integration along curves in three dimensions. Some numerical examples using very simple discretizations are presented to demonstrate the efficacy of the formulation.

Isabel N. Figueiredo, Carlos Leal, Luís Pinto et al.

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 that remains unsolved is the WCE precise location in the human body during its operating time. This is mainly due to the complex physiological environment and the inherent capsule effects during its movement. When an abnormality is detected, in the WCE images, medical doctors do not know precisely where this abnormality is located relative to the intestine and therefore they can not proceed efficiently with the appropriate therapy. The primary objective of the present paper is to give a contribution to WCE localization, using image-based methods. The main focus of this work is on the description of a multiscale elastic image registration approach, its experimental application on WCE videos, and comparison with a multiscale affine registration. The proposed approach includes registrations that capture both rigid-like and non-rigid deformations, due respectively to the rigid-like WCE movement and the elastic deformation of the small intestine originated by the GI peristaltic movement. Under this approach a qualitative information about the WCE speed can be obtained, as well as the WCE location and orientation via projective geometry. The results of the experimental tests with real WCE video frames show the good performance of the proposed approach, when elastic deformations of the small intestine are involved in successive frames, and its superiority with respect to a multiscale affine image registration, which accounts for rigid-like deformations only and discards elastic deformations.