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Dynamic mode decomposition (DMD) is an emerging methodology that has recently attracted computational scientists working on nonintrusive reduced order modeling. One of the major strengths that DMD possesses is having ground theoretical roots from the Koopman approximation theory. Indeed, DMD may be viewed as the data-driven realization of the famous Koopman operator. Nonetheless, the stable implementation of DMD incurs computing the singular value decomposition of the input data matrix. This, in turn, makes the process computationally demanding for high dimensional systems. In order to alleviate this burden, we develop a framework based on sketching methods, wherein a sketch of a matrix is simply another matrix which is significantly smaller, but still sufficiently approximates the original system. Such sketching or embedding is performed by applying random transformations, with certain properties, on the input matrix to yield a compressed version of the initial system. Hence, many of the expensive computations can be carried out on the smaller matrix, thereby accelerating the solution of the original problem. We conduct numerical experiments conducted using the spherical shallow water equations as a prototypical model in the context of geophysical flows. The performance of several sketching approaches is evaluated for capturing the range and co-range of the data matrix. The proposed sketching-based framework can accelerate various portions of the DMD algorithm, compared to classical methods that operate directly on the raw input data. This eventually leads to substantial computational gains that are vital for digital twinning of high dimensional systems.

Element Method. The Finite Volume Method guarantees local and global mass conservation. A property not satisfied by the Finite Volume Method. On the down side, the Finite Volume Method requires non trivial modifications to attain high order approximations unlike the Finite Volume Method. It has been contended that the Discontinuous Galerkin Method, locally conservative and high order, is a natural progression for Coastal Ocean Modeling. Consequently, as a primer we consider the vertical ocean-slice model with the inclusion of density effects. To solve these non steady Partial Differential Equations, we develop a pressure projection method for solution. We propose a Hybridized Discontinuous Galerkin solution for the required Poisson Problem in each time step. The purpose, is to reduce the computational cost of classical applications of the Discontinuous Galerkin method. The Hybridized Discontinuous Galerkin method is first presented as a general elliptic problem solver. It is shown that a high order implementation yields fast and accurate approximations on coarse meshes.

Stabilized Runge-Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized Runge-Kutta methods overcome the stringent stability condition of standard methods without sacrificing explicitness. However, when stiffness is only induced by a few components, as in the presence of spatially local mesh refinement, their efficiency deteriorates. To remove the crippling effect of a few severely stiff components on the entire system of differential equations, we derive a modified equation, whose stiffness solely depend on the remaining mildly stiff components. By applying stabilized Runge-Kutta methods to this modified equation, we then devise an explicit multirate Runge-Kutta-Chebyshev (mRKC) method whose stability conditions are independent of a few severely stiff components. Stability of the mRKC method is proved for a model problem, whereas its efficiency and usefulness are demonstrated through a series of numerical experiments.

In this paper, we propose a semigroup method for solving high-dimensional elliptic partial differential equations (PDEs) and the associated eigenvalue problems based on neural networks. For the PDE problems, we reformulate the original equations as variational problems with the help of semigroup operators and then solve the variational problems with neural network (NN) parameterization. The main advantages are that no mixed second-order derivative computation is needed during the stochastic gradient descent training and that the boundary conditions are taken into account automatically by the semigroup operator. Unlike popular methods like PINN \cite{raissi2019physics} and Deep Ritz \cite{weinan2018deep} where the Dirichlet boundary condition is enforced solely through penalty functions and thus changes the true solution, the proposed method is able to address the boundary conditions without penalty functions and it gives the correct true solution even when penalty functions are added, thanks to the semigroup operator. For eigenvalue problems, a primal-dual method is proposed, efficiently resolving the constraint with a simple scalar dual variable and resulting in a faster algorithm compared with the BSDE solver \cite{han2020solving} in certain problems such as the eigenvalue problem associated with the linear Schr\"odinger operator. Numerical results are provided to demonstrate the performance of the proposed methods.

In this work, we determine the full expression for the global truncation error of hyperbolic partial differential equations (PDEs). In particular, we use theoretical analysis and symbolic algebra to find exact expressions for the coefficients of the generic global truncation error. Our analysis is valid for any hyperbolic PDE, be it linear or non-linear, and employing finite difference, finite volume, or finite element discretization in space, and advanced in time with a predictor-corrector, multistep, or a deferred correction method, belonging to the Method of Lines. Furthermore, we discuss the practical implications of this analysis. If we employ a stable numerical scheme and the orders of accuracy of the global solution error and the global truncation error agree, we make the following asymptotic observations: (a) the order of convergence at constant ratio of $\Delta t$ to $\Delta x$ is governed by the minimum of the orders of the spatial and temporal discretizations, and (b) convergence cannot even be guaranteed under only spatial or temporal refinement. An implication of (a) is that it is impractical to invest in a time-stepping method of order higher than the spatial discretization. In addition to (b), we demonstrate that under certain circumstances, the error can even monotonically increase with refinement only in space or only in time, and explain why this phenomenon occurs. To verify our theoretical findings, we conduct convergence studies of linear and non-linear advection equations using finite difference and finite volume spatial discretizations, and predictor-corrector and multistep time-stepping methods. Finally, we study the effect of slope limiters and monotonicity-preserving strategies on the order of accuracy.

This paper is concerned with the efficient spectral solutions for weakly singular nonlocal diffusion equations with Dirichlet-type volume constraints. This type of equation contains an integral operator which typically has a singularity at the midpoint of the integral domain, and the approximation of such the integral operator is one of the essential difficulties in solving the nonlocal equations. To overcome this problem, two-sided Jacobi spectral quadrature rules are proposed to develop a Jacobi spectral collocation method for the nonlocal diffusion equations. Rigorous convergence analysis of the proposed method is presented in $L^\infty$ norms, and we further prove that the Jacobi collocation solution converges to its corresponding local limit as nonlocal interactions vanish. Numerical examples are given to verify the theoretical results.

Linear time-varying differential-algebraic equations with symmetries are studied. The structures that we address are self-adjoint and skew-adjoint systems. Local and global canonical forms under congruence are presented and used to classify the geometric properties of the flow associated with the differential equation as symplectic or generalized orthogonal flow. As applications, the results are applied to the analysis of dissipative Hamiltonian systems arising from circuit simulation and incompressible flow.

We propose a novel numerical method for high dimensional Hamilton--Jacobi--Bellman (HJB) type elliptic partial differential equations (PDEs). The HJB PDEs, reformulated as optimal control problems, are tackled by the actor-critic framework inspired by reinforcement learning, based on neural network parametrization of the value and control functions. Within the actor-critic framework, we employ a policy gradient approach to improve the control, while for the value function, we derive a variance reduced least-squares temporal difference method using stochastic calculus. To numerically discretize the stochastic control problem, we employ an adaptive step size scheme to improve the accuracy near the domain boundary. Numerical examples up to $20$ spatial dimensions including the linear quadratic regulators, the stochastic Van der Pol oscillators, the diffusive Eikonal equations, and fully nonlinear elliptic PDEs derived from a regulator problem are presented to validate the effectiveness of our proposed method.

In this paper we discuss a reduced basis method for linear evolution PDEs, which is based on the application of the Laplace transform. The main advantage of this approach consists in the fact that, differently from time stepping methods, like Runge-Kutta integrators, the Laplace transform allows to compute the solution directly at a given instant, which can be done by approximating the contour integral associated to the inverse Laplace transform by a suitable quadrature formula. In terms of the reduced basis methodology, this determines a significant improvement in the reduction phase - like the one based on the classical proper orthogonal decomposition (POD) - since the number of vectors to which the decomposition applies is drastically reduced as it does not contain all intermediate solutions generated along an integration grid by a time stepping method. We show the effectiveness of the method by some illustrative parabolic PDEs arising from finance and also provide some evidence that the method we propose, when applied to a simple advection equation, does not suffer the problem of slow decay of singular values which instead affects methods based on time integration of the Cauchy problem arising from space discretization.

The joint bidiagonalization (JBD) method has been used to compute some extreme generalized singular values and vectors of a large regular matrix pair $\{A,L\}$, where we propose three approaches to compute approximate generalized singular values and vectors. We make a numerical analysis of the underlying JBD process and establish relationships between it and two mathematically equivalent Lanczos bidiagonalizations in finite precision. Based on the results of numerical analysis, we investigate the convergence of the approximate generalized singular values and vectors of $\{A,L\}$. The results show that, under some mild conditions, the semiorthogonality of Lanczos type vectors suffices to deliver approximate generalized singular values with the same accuracy as the full orthogonality does, meaning that it is only necessary to seek for efficient semiorthogonalization strategies for the JBD process. We also establish a sharp bound for the residual norm of an approximate generalized singular value and corresponding approximate right generalized singular vectors, which can reliably estimate the residual norm without explicitly computing the approximate right generalized singular vectors before the convergence occurs.