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We develop a novel computational framework to approximate solution operators of evolution partial differential equations (PDEs). By employing a general nonlinear reduced-order model, such as a deep neural network, to approximate the solution of a given PDE, we realize that the evolution of the model parameters is a control problem in the parameter space. Based on this observation, we propose to approximate the solution operator of the PDE by learning the control vector field in the parameter space. From any initial value, this control field can steer the parameter to generate a trajectory such that the corresponding reduced-order model solves the PDE. This allows for substantially reduced computational cost to solve the evolution PDE with arbitrary initial conditions. We also develop comprehensive error analysis for the proposed method when solving a large class of semilinear parabolic PDEs. Numerical experiments on different high-dimensional evolution PDEs with various initial conditions demonstrate the promising results of the proposed method. [ABSTRACT FROM AUTHOR]

We study the optimal control formulation for stochastic nonlinear Schrödinger equation (SNLSE) on a finite graph. By viewing the SNLSE as a stochastic Wasserstein Hamiltonian flow on density manifold, we show the global existence of a unique strong solution for SNLSE with a linear drift control or a linear diffusion control on graph. Furthermore, we provide the gradient formula, the existence of the optimal control and a description on the optimal condition via the forward and backward stochastic differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method that combines the heterogeneous multiscale method (HMM) and the empirical mode decomposition (EMD) type of filtering techniques to compute the slow dynamics in a multiscale system. The main idea is that we apply the adaptive local iterative filtering (ALIF) algorithm, an EMD-like nonlinear signal analysis strategy which decomposes a signal into several intrinsic mode functions (IMFs), to extract essential information, and then use it in the HMM framework to compute the coarse scale behavior without fully resolving the fine scale solutions. Our numerical examples demonstrate that the new method has a number of advantages over the existing ones: (1) The effective rate of change for the slow dynamics is calculated as the result of the local solution decomposition by ALIF, and an effective ordinary differential equation is obtained on the fly by analyzing the trend of the dynamics. (2) ALIF can find hidden intermediate time scales, with possible noninteger exponents, and the method can treat multiple (> 2) time scale systems hierarchically. (3) The time-frequency analysis in ALIF allows us to identify some frequency related slow variables, such as the relative phase. This is especially useful for the systems where the slow variables are not explicitly known. [ABSTRACT FROM AUTHOR]

We study the Wasserstein Hamiltonian flow with a common noise on the density manifold of a finite graph. Under the framework of the stochastic variational principle, we first develop the formulation of stochastic Wasserstein Hamiltonian flow and show the local existence of a unique solution. We also establish a sufficient condition for the global existence of the solution. Consequently, we obtain the global well-posedness for the nonlinear Schr\"odinger equations with common noise on a graph. In addition, using Wong--Zakai approximation of common noise, we prove the existence of the minimizer for an optimal control problem with common noise. We show that its minimizer satisfies the stochastic Wasserstein Hamiltonian flow on a graph as well. [ABSTRACT FROM AUTHOR]

We propose a denoising algorithm for medical images based on a combination of the total variation minimization scheme and the wavelet scheme. We show that our scheme offers effective noise removal in real noisy medical images while maintaining sharpness of objects. More importantly, this scheme allows us to implement an effective automatic stopping time criterion. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method to solve the classic L²-optimal transport problem. Our algorithm is based on the use of multiple shooting, in combination with a continuation procedure, to solve the boundary value problem associated to the transport problem. Based on the viewpoint of Wasserstein Hamiltonian flow with initial and target densities, our algorithm reflects the Hamiltonian structure of the underlying problem and exploits it in the numerical discretization. Several numerical examples are presented to illustrate the performance of the method. [ABSTRACT FROM AUTHOR]

We propose a unified data-driven framework based on inverse optimal transport that can learn adaptive, nonlinear interaction cost function from noisy and incomplete empirical matching matrix and predict new matching in various matching contexts. We emphasize that the discrete optimal transport plays the role of a variational principle which gives rise to an optimization based framework for modeling the observed empirical matching data. Our formulation leads to a non-convex optimization problem which can be solved efficiently by an alternating optimization method. A key novel aspect of our formulation is the incorporation of marginal relaxation via regularized Wasserstein distance, significantly improving the robustness of the method in the face of noisy or missing empirical matching data. Our model falls into the category of prescriptive models, which not only predict potential future matching, but is also able to explain what leads to empirical matching and quantifies the impact of changes in matching factors. The proposed approach has wide applicability including predicting matching in online dating, labor market, college application and crowdsourcing. We back up our claims with numerical experiments on both synthetic data and real world data sets. [ABSTRACT FROM AUTHOR]

We propose a computational method to optimally position a sensor system. Each sensor has limited range and viewing angle, and it may fail with a certain failure rate. The goal is to find the optimal locations as well as the viewing directions of all the sensors and achieve the maximal surveillance of the known environment. We set up the problem using the level set framework. Both the environment and the viewing range of the sensors are represented by level set functions. Then we solve a system of ordinary differential equations (ODEs) to find the optimal viewing directions and locations of all sensors together. Furthermore, we use the intermittent diffusion, which converts the ODEs into stochastic differential equations, to find the global maximum of the total surveillance area. The numerical examples include various failure rates of sensors, different rates of importance of the surveillance region, and 3-dimensional setups. They show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We propose a new dynamical model, inspired by optimal transport theory and Fisher information, for finite player discrete strategy games. For a potential game, the model is a gradient flow, known as Fokker--Planck equations (FPEs), in a probability simplex set equipped with an optimal transport metric. Based on FPEs, we introduce the best-reply Markov processes, which model players' myopia, greed, and uncertainty when making decisions. The stationary measure of the dynamics provides each pure Nash equilibrium a probability by which it is ranked. We extend the proposed dynamical model to rank/select equilibria for nonpotential games as well. [ABSTRACT FROM AUTHOR]

This paper is concerned with an inverse random source problem for the one-dimensional stochastic Helmholtz equation, which is to reconstruct the statistical properties of the random source function from boundary measurements of the radiating random electric field. Although the emphasis of the paper is on the inverse problem, we adapt a computationally more efficient approach to study the solution of the direct problem in the context of the scattering model. Specifically, the direct model problem is equivalently formulated into a two-point spatially stochastic boundary value problem, for which the existence and uniqueness of the pathwise solution is proved. In particular, an explicit formula is deduced for the solution from an integral representation by solving the two-point boundary value problem. Based on this formula, a novel and efficient strategy, which is entirely done by using the fast Fourier transform, is proposed to reconstruct the mean and the variance of the random source function from measurements at one boundary point, where the measurements are assumed to be available for many realizations of the source term. Numerical examples are presented to demonstrate the validity and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]