In this work we present a data-driven method for the discovery of parametric partial differential equations (PDEs), thus allowing one to disambiguate between the underlying evolution equations and their parametric dependencies. Group sparsity is used to ensure parsimonious representations of observed dynamics in the form of a parametric PDE, while also allowing the coefficients to have arbitrary time series, or spatial dependence. This work builds on previous methods for the identification of constant coefficient PDEs, expanding the field to include a new class of equations which until now have eluded machine learning based identification methods. We show that group sequentially thresholded ridge regression outperforms group LASSO in identifying the fewest terms in the PDE along with their parametric dependency. The method is demonstrated on four canonical models with and without the introduction of noise. Key word. data-driven method, sparse regression, parametric models AMS subject classifications. 37M02, 65P02, 49M02 * Submitted 6/2/2018 Funding: S. Rudy, S. L. Brunton and J. N. Kutz acknowledge support from the Defense Advanced Research Projects Agency (DARPA HR0011-16-C-0016). J. N. Kutz acknowledges support from the Air Force Office of Scientific Research (AFOSR) grant FA9550-15-1-0385. S. L. Brunton acknowledges support from the Air Force Office of Scientific Research (AFOSR FA9550-18-1-0200).

Abstract. We present an accelerated algorithm for the solution of static Hamilton-JacobiBellman equations related to optimal control problems. Our scheme is based on a classic policy iteration procedure, which is known to have superlinear convergence in many relevant cases provided the initial guess is sufficiently close to the solution. This limitation often degenerates into a behavior similar to a value iteration method, with an increased computation time. The new scheme circumvents this problem by combining the advantages of both algorithms with an efficient coupling. The method starts with a coarse-mesh value iteration phase and then switches to a fine-mesh policy iteration procedure when a certain error threshold is reached. A delicate point is to determine this threshold in order to avoid cumbersome computations with the value iteration and, at the same time, to be reasonably sure that the policy iteration method will finally converge to the optimal solution. We analyze the methods and efficient coupling in a number of examples in dimension two, three and four illustrating their properties.

Abstract. We propose a new technique for obtaining reduced order models for nonlinear dynamical systems. Specifically, we advocate the use of the recently developed Dynamic Mode Decomposition (DMD), an equation-free method, to approximate the nonlinear term. DMD is a spatio-temporal matrix decomposition of a data matrix that correlates spatial features while simultaneously associating the activity with periodic temporal behavior. With this decomposition, one can obtain a fully reduced dimensional surrogate model and avoid the evaluation of the nonlinear term in the online stage. This allows for an impressive speed up of the computational cost, and, at the same time, accurate approximations of the problem. We present a suite of numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches.Key words. nonlinear dynamical systems, proper orthogonal decomposition, dynamic mode decomposition, data-driven modeling, reduced-order modeling, dimensionality reduction AMS subject classifications. 65L02, 65M02, 37M05, 62H251. Introduction. Reduced-order models (ROMs) are of growing importance in scientific computing as they provide a principled approach to approximating highdimensional PDEs with low-dimensional models. Indeed, the dimensionality reduction provided by ROMs help reduce the computational complexity and time needed to solve large-scale, engineering systems [24,3], enabling simulation based scientific studies not possible even a decade ago. One of the primary challenges in producing the low-rank dynamical system is efficiently projecting the nonlinearity of the governing PDEs (inner products) [2,6] on to the proper orthogonal decomposition (POD) [18,10,30] basis. This fact was recognized early on in the ROM community, and methods such as gappy POD [8,31,32] where proposed to more efficiently enable the task. More recently, the empirical interpolation method (EIM) [2], and the simplified discrete empirical interpolation method (DEIM) [6] for the proper orthogonal decomposition (POD) [18,10,30], have provided a computationally efficient method for discretely (sparsely) sampling and evaluating the nonlinearity. These broadly used and highly-successful methods ensure that the computational complexity of ROMs scale favorably with the rank of the approximation, even for complex nonlinearities. As an alternative to the EIM/DEIM architecture, we propose using the recently developed Dynamic Mode Decomposition (DMD) for producing low-rank approximations of the PDE nonlinearities. DMD provides a decomposition of data into spatio-temporal modes that correlates the data across spatial features (like POD), but also associates the correlated data to unique temporal Fourier modes, allowing for a computationally efficient regression of the nonlinear terms to a least-square fit linear dynamics approximation. We demonstrate that the POD-DMD method produces a viable ROM architecture, scaling favorable in computational efficiency relative to

The classical Dynamic Programming (DP) approach to optimal control problems is based on the characterization of the value function as the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. The DP scheme for the numerical approximation of viscosity solutions of Bellman equations is typically based on a time discretization which is projected on a fixed state-space grid. The time discretization can be done by a one-step scheme for the dynamics and the projection on the grid typically uses a local interpolation. Clearly the use of a grid is a limitation with respect to possible applications in high-dimensional problems due to the curse of dimensionality.Here, we present a new approach for finite horizon optimal control problems where the value function is computed using a DP algorithm with a tree structure algorithm (TSA) constructed by the time discrete dynamics. In this way there is no need to build a fixed space triangulation and to project on it. The tree will guarantee a perfect matching with the discrete dynamics and drop off the cost of the space interpolation allowing for the solution of very high-dimensional problems. Numerical tests will show the effectiveness of the proposed method.

Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. In this work, a ROM is developed and tested for the treatment of nonlinear PDEs whose solutions bifurcate as input parameter values change. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does no respect not take advantage of the often large differences in the PDE solutions corresponding to different subregions. Instead, in the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE. 1 For finite-dimensional spaces, we use the nomenclature "points" and "vectors" interchangeably.2 In §3, we use the Navier-Stokes equations as a concrete setting to illustrate our methodology. 1 arXiv:1807.08851v1 [math.NA] 23 Jul 20183 In general, the forms N and F themselves are also discretized, e.g., because quadrature rules are used to approximate integrals appearing in their definition. However, here, we ignore such approximations, again to keep the exposition simple.4 Equation (1.2) represents a Galerkin type setting in which the trial function u N and test function v belong to the same space V N . We could easily generalize our discussion to the Petrov-Galerkin case for which these function would belong to different approximating spaces.