In this article, we propose numerical scheme for solving a multi-term time-fractional nonlocal parabolic partial differential equation (PDE). The scheme comprises $L2$-$1_{\sigma}$ scheme on a graded mesh in time and Galerkin finite element method (FEM) in space. We present the discrete fractional Gr$\ddot{{o}}$nwall inequality for $L2$-$1_{\sigma}$ scheme in case of multi-term time-fractional derivative, which is a multi-term analogue of~\cite[Lemma 4.1]{[r16]}. We derive \textit{a priori} bound and error estimate for the fully-discrete solution. The theoretical results are confirmed via numerical experiments. We should note that, though the way of proving the discrete fractional Gr$\ddot{{o}}$nwall inequality is similar to~\cite{[r5]}, the calculation parts are more complicated in this article.
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Trajectory optimization is a powerful tool for robot motion planning and control. State-of-the-art general-purpose nonlinear programming solvers are versatile, handle constraints effectively and provide a high numerical robustness, but they are slow because they do not fully exploit the optimal control problem structure at hand. Existing structure-exploiting solvers are fast, but they often lack techniques to deal with nonlinearity or rely on penalty methods to enforce (equality or inequality) path constraints. This work presents Fatrop: a trajectory optimization solver that is fast and benefits from the salient features of general-purpose nonlinear optimization solvers. The speed-up is mainly achieved through the integration of a specialized linear solver, based on a Riccati recursion that is generalized to also support stagewise equality constraints. To demonstrate the algorithm's potential, it is benchmarked on a set of robot problems that are challenging from a numerical perspective, including problems with a minimum-time objective and no-collision constraints. The solver is shown to solve problems for trajectory generation of a quadrotor, a robot manipulator and a truck-trailer problem in a few tens of milliseconds. The algorithm's C++-code implementation accompanies this work as open source software, released under the GNU Lesser General Public License (LGPL). This software framework may encourage and enable the robotics community to use trajectory optimization in more challenging applications.
We extend and analyze the deep neural network multigrid solver (DNN-MG) for the Navier-Stokes equations in three dimensions. The idea of the method is to augment of finite element simulations on coarse grids with fine scale information obtained using deep neural networks. This network operates locally on small patches of grid elements. The local approach proves to be highly efficient, since the network can be kept (relatively) small and since it can be applied in parallel on all grid patches. However, the main advantage of the local approach is the inherent good generalizability of the method. Since the network is only ever trained on small sub-areas, it never ``sees'' the global problem and thus does not learn a false bias. We describe the method with a focus on the interplay between finite element method and deep neural networks. Further, we demonstrate with numerical examples the excellent efficiency of the hybrid approach, which allows us to achieve very high accuracies on coarse grids and thus reduce the computation time by orders of magnitude.
We propose a novel scale-invariant version of the mean and variance multi-level Monte Carlo estimate. The computation cost across grid levels is optimised using a normalized error based on t-statistics. By doing so, the algorithm achieves convergence independent of the physical scale at which the estimate is computed. The effectiveness of this algorithm is demonstrated through testing on a linear elastic example, where material uncertainty incorporating both heterogeneity and anisotropy is considered in the constitutive law.
Constraint programming is known for being an efficient approach for solving combinatorial problems. Important design choices in a solver are the branching heuristics, which are designed to lead the search to the best solutions in a minimum amount of time. However, developing these heuristics is a time-consuming process that requires problem-specific expertise. This observation has motivated many efforts to use machine learning to automatically learn efficient heuristics without expert intervention. To the best of our knowledge, it is still an open research question. Although several generic variable-selection heuristics are available in the literature, the options for a generic value-selection heuristic are more scarce. In this paper, we propose to tackle this issue by introducing a generic learning procedure that can be used to obtain a value-selection heuristic inside a constraint programming solver. This has been achieved thanks to the combination of a deep Q-learning algorithm, a tailored reward signal, and a heterogeneous graph neural network architecture. Experiments on graph coloring, maximum independent set, and maximum cut problems show that our framework is able to find better solutions close to optimality without requiring a large amounts of backtracks while being generic.
The proximal Galerkin finite element method is a high-order, nonlinear numerical method that preserves the geometric and algebraic structure of bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free-boundary problems, enforce discrete maximum principles, and develop scalable, mesh-independent algorithms for optimal design. The paper begins with a derivation of the latent variable proximal point (LVPP) method: an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive (Bayesian) barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout this work, we arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization. The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis.
The nonlocal Allen-Cahn equation with nonlocal diffusion operator is a generalization of the classical Allen-Cahn equation. It satisfies the energy dissipation law and maximum bound principle (MBP), and is important for simulating a series of physical and biological phenomena involving long-distance interactions in space. In this paper, we construct first- and second-order (in time) accurate, unconditionally energy stable and MBP-preserving schemes for the nonlocal Allen-Cahn type model based on the stabilized exponential scalar auxiliary variable (sESAV) approach. On the one hand, we have proved the MBP and unconditional energy stability carefully and rigorously in the fully discrete levels. On the other hand, we adopt an efficient FFT-based fast solver to compute the nearly full coefficient matrix generated from the spatial discretization, which improves the computational efficiency. Finally, typical numerical experiments are presented to demonstrate the performance of our proposed schemes.
The nonlocal Cahn-Hilliard (NCH) equation with nonlocal diffusion operator is more suitable for the simulation of microstructure phase transition than the local Cahn-Hilliard (LCH) equation. In this paper, based on the exponential semi-implicit scalar auxiliary variable (ESI-SAV) method, the highly effcient and accurate schemes in time with unconditional energy stability for solving the NCH equation are proposed. On the one hand, we have demostrated the unconditional energy stability for the NCH equation with its high-order semi-discrete schemes carefully and rigorously. On the other hand, in order to reduce the calculation and storage cost in numerical simulation, we use the fast solver based on FFT and FCG for spatial discretization. Some numerical simulations involving the Gaussian kernel are presented and show the stability, accuracy, efficiency and unconditional energy stability of the proposed schemes.
The selection of Gaussian kernel parameters plays an important role in the applications of support vector classification (SVC). A commonly used method is the k-fold cross validation with grid search (CV), which is extremely time-consuming because it needs to train a large number of SVC models. In this paper, a new approach is proposed to train SVC and optimize the selection of Gaussian kernel parameters. We first formulate the training and parameter selection of SVC as a minimax optimization problem named as MaxMin-L2-SVC-NCH, in which the minimization problem is an optimization problem of finding the closest points between two normal convex hulls (L2-SVC-NCH) while the maximization problem is an optimization problem of finding the optimal Gaussian kernel parameters. A lower time complexity can be expected in MaxMin-L2-SVC-NCH because CV is not needed. We then propose a projected gradient algorithm (PGA) for training L2-SVC-NCH. The famous sequential minimal optimization (SMO) algorithm is a special case of the PGA. Thus, the PGA can provide more flexibility than the SMO. Furthermore, the solution of the maximization problem is done by a gradient ascent algorithm with dynamic learning rate. The comparative experiments between MaxMin-L2-SVC-NCH and the previous best approaches on public datasets show that MaxMin-L2-SVC-NCH greatly reduces the number of models to be trained while maintaining competitive test accuracy. These findings indicate that MaxMin-L2-SVC-NCH is a better choice for SVC tasks.
In this paper, we study a general low-rank matrix recovery problem with linear measurements corrupted by some noise. The objective is to understand under what conditions on the restricted isometry property (RIP) of the problem local search methods can find the ground truth with a small error. By analyzing the landscape of the non-convex problem, we first propose a global guarantee on the maximum distance between an arbitrary local minimizer and the ground truth under the assumption that the RIP constant is smaller than $1/2$. We show that this distance shrinks to zero as the intensity of the noise reduces. Our new guarantee is sharp in terms of the RIP constant and is much stronger than the existing results. We then present a local guarantee for problems with an arbitrary RIP constant, which states that any local minimizer is either considerably close to the ground truth or far away from it. Next, we prove the strict saddle property, which guarantees the global convergence of the perturbed gradient descent method in polynomial time. The developed results demonstrate how the noise intensity and the RIP constant of the problem affect the landscape of the problem.
The aim of this paper is to develop estimation and inference methods for the drift parameters of multivariate L\'evy-driven continuous-time autoregressive processes of order $p\in\mathbb{N}$. Starting from a continuous-time observation of the process, we develop consistent and asymptotically normal maximum likelihood estimators. We then relax the unrealistic assumption of continuous-time observation by considering natural discretizations based on a combination of Riemann-sum, finite difference, and thresholding approximations. The resulting estimators are also proven to be consistent and asymptotically normal under a general set of conditions, allowing for both finite and infinite jump activity in the driving L\'evy process. When discretizing the estimators, allowing for irregularly spaced observations is of great practical importance. In this respect, CAR($p$) models are not just relevant for "true" continuous-time processes: a CAR($p$) specification provides a natural continuous-time interpolation for modeling irregularly spaced data - even if the observed process is inherently discrete. As a practically relevant application, we consider the setting where the multivariate observation is known to possess a graphical structure. We refer to such a process as GrCAR and discuss the corresponding drift estimators and their properties. The finite sample behavior of all theoretical asymptotic results is empirically assessed by extensive simulation experiments.
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