We present recent finite element numerical results on a model convection-diffusion problem in the singular perturbed case when the convection term dominates the problem. We compare the standard Galerkin discretization using the linear element with a saddle point least square discretization that uses quadratic test functions, trying to control and explain the non-physical oscillations of the discrete solutions. We also relate the up-winding Petrov-Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. Some results can be extended to the multidimensional case in order to come up with efficient approximations for more general singular perturbed problems, including convection dominated models.
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We study a class of generalized linear programs (GLP) in a large-scale setting, which includes simple, possibly nonsmooth convex regularizer and simple convex set constraints. By reformulating (GLP) as an equivalent convex-concave min-max problem, we show that the linear structure in the problem can be used to design an efficient, scalable first-order algorithm, to which we give the name \emph{Coordinate Linear Variance Reduction} (\textsc{clvr}; pronounced "clever"). \textsc{clvr} yields improved complexity results for (GLP) that depend on the max row norm of the linear constraint matrix in (GLP) rather than the spectral norm. When the regularization terms and constraints are separable, \textsc{clvr} admits an efficient lazy update strategy that makes its complexity bounds scale with the number of nonzero elements of the linear constraint matrix in (GLP) rather than the matrix dimensions. On the other hand, for the special case of linear programs, by exploiting sharpness, we propose a restart scheme for \textsc{clvr} to obtain empirical linear convergence. Then we show that Distributionally Robust Optimization (DRO) problems with ambiguity sets based on both $f$-divergence and Wasserstein metrics can be reformulated as (GLPs) by introducing sparsely connected auxiliary variables. We complement our theoretical guarantees with numerical experiments that verify our algorithm's practical effectiveness, in terms of wall-clock time and number of data passes.
Given a Hilbert space $\mathcal H$ and a finite measure space $\Omega$, the approximation of a vector-valued function $f: \Omega \to \mathcal H$ by a $k$-dimensional subspace $\mathcal U \subset \mathcal H$ plays an important role in dimension reduction techniques, such as reduced basis methods for solving parameter-dependent partial differential equations. For functions in the Lebesgue--Bochner space $L^2(\Omega;\mathcal H)$, the best possible subspace approximation error $d_k^{(2)}$ is characterized by the singular values of $f$. However, for practical reasons, $\mathcal U$ is often restricted to be spanned by point samples of $f$. We show that this restriction only has a mild impact on the attainable error; there always exist $k$ samples such that the resulting error is not larger than $\sqrt{k+1} \cdot d_k^{(2)}$. Our work extends existing results by Binev at al. (SIAM J. Math. Anal., 43(3):1457--1472, 2011) on approximation in supremum norm and by Deshpande et al. (Theory Comput., 2:225--247, 2006) on column subset selection for matrices.
Generalized Zero-Shot Learning (GZSL) has emerged as a pivotal research domain in computer vision, owing to its capability to recognize objects that have not been seen during training. Despite the significant progress achieved by generative techniques in converting traditional GZSL to fully supervised learning, they tend to generate a large number of synthetic features that are often redundant, thereby increasing training time and decreasing accuracy. To address this issue, this paper proposes a novel approach for synthetic feature selection using reinforcement learning. In particular, we propose a transformer-based selector that is trained through proximal policy optimization (PPO) to select synthetic features based on the validation classification accuracy of the seen classes, which serves as a reward. The proposed method is model-agnostic and data-agnostic, making it applicable to both images and videos and versatile for diverse applications. Our experimental results demonstrate the superiority of our approach over existing feature-generating methods, yielding improved overall performance on multiple benchmarks.
In this paper, we propose a mesh-free numerical method for solving elliptic PDEs on unknown manifolds, identified with randomly sampled point cloud data. The PDE solver is formulated as a spectral method where the test function space is the span of the leading eigenfunctions of the Laplacian operator, which are approximated from the point cloud data. While the framework is flexible for any test functional space, we will consider the eigensolutions of a weighted Laplacian obtained from a symmetric Radial Basis Function (RBF) method induced by a weak approximation of a weighted Laplacian on an appropriate Hilbert space. Especially, we consider a test function space that encodes the geometry of the data yet does not require us to identify and use the sampling density of the point cloud. To attain a more accurate approximation of the expansion coefficients, we adopt a second-order tangent space estimation method to improve the RBF interpolation accuracy in estimating the tangential derivatives. This spectral framework allows us to efficiently solve the PDE many times subjected to different parameters, which reduces the computational cost in the related inverse problem applications. In a well-posed elliptic PDE setting with randomly sampled point cloud data, we provide a theoretical analysis to demonstrate the convergent of the proposed solver as the sample size increases. We also report some numerical studies that show the convergence of the spectral solver on simple manifolds and unknown, rough surfaces. Our numerical results suggest that the proposed method is more accurate than a graph Laplacian-based solver on smooth manifolds. On rough manifolds, these two approaches are comparable. Due to the flexibility of the framework, we empirically found improved accuracies in both smoothed and unsmoothed Stanford bunny domains by blending the graph Laplacian eigensolutions and RBF interpolator.
In this paper, we present efficient quantum algorithms that are exponentially faster than classical algorithms for solving the quantum optimal control problem. This problem involves finding the control variable that maximizes a physical quantity at time $T$, where the system is governed by a time-dependent Schr\"odinger equation. This type of control problem also has an intricate relation with machine learning. Our algorithms are based on a time-dependent Hamiltonian simulation method and a fast gradient-estimation algorithm. We also provide a comprehensive error analysis to quantify the total error from various steps, such as the finite-dimensional representation of the control function, the discretization of the Schr\"odinger equation, the numerical quadrature, and optimization. Our quantum algorithms require fault-tolerant quantum computers.
Inferring the parameters of ordinary differential equations (ODEs) from noisy observations is an important problem in many scientific fields. Currently, most parameter estimation methods that bypass numerical integration tend to rely on basis functions or Gaussian processes to approximate the ODE solution and its derivatives. Due to the sensitivity of the ODE solution to its derivatives, these methods can be hindered by estimation error, especially when only sparse time-course observations are available. We present a Bayesian collocation framework that operates on the integrated form of the ODEs and also avoids the expensive use of numerical solvers. Our methodology has the capability to handle general nonlinear ODE systems. We demonstrate the accuracy of the proposed method through a simulation study, where the estimated parameters and recovered system trajectories are compared with other recent methods. A real data example is also provided.
Privacy protection methods, such as differentially private mechanisms, introduce noise into resulting statistics which often results in complex and intractable sampling distributions. In this paper, we propose to use the simulation-based "repro sample" approach to produce statistically valid confidence intervals and hypothesis tests based on privatized statistics. We show that this methodology is applicable to a wide variety of private inference problems, appropriately accounts for biases introduced by privacy mechanisms (such as by clamping), and improves over other state-of-the-art inference methods such as the parametric bootstrap in terms of the coverage and type I error of the private inference. We also develop significant improvements and extensions for the repro sample methodology for general models (not necessarily related to privacy), including 1) modifying the procedure to ensure guaranteed coverage and type I errors, even accounting for Monte Carlo error, and 2) proposing efficient numerical algorithms to implement the confidence intervals and $p$-values.
We study the computational scalability of a Gaussian process (GP) framework for solving general nonlinear partial differential equations (PDEs). This framework transforms solving PDEs to solving quadratic optimization problem with nonlinear constraints. Its complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel of the GP and its partial derivatives at collocation points. We present a sparse Cholesky factorization algorithm for such kernel matrices based on the near-sparsity of the Cholesky factor under a new ordering of Diracs and derivative measurements. We rigorously identify the sparsity pattern and quantify the exponentially convergent accuracy of the corresponding Vecchia approximation of the GP, which is optimal in the Kullback-Leibler divergence. This enables us to compute $\epsilon$-approximate inverse Cholesky factors of the kernel matrices with complexity $O(N\log^d(N/\epsilon))$ in space and $O(N\log^{2d}(N/\epsilon))$ in time. With the sparse factors, gradient-based optimization methods become scalable. Furthermore, we can use the oftentimes more efficient Gauss-Newton method, for which we apply the conjugate gradient algorithm with the sparse factor of a reduced kernel matrix as a preconditioner to solve the linear system. We numerically illustrate our algorithm's near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Amp\`ere equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs.
This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones and seeks the numerical solution in the range of this projection. As the projection is not injective, a stabilisation based upon the complementary projection is added in order to restore well-posedness. Within the framework of elliptic problems, the discrete problem may be viewed as a reformulation of a discrete obstacle problem, incorporating the inequality constraints through Lipschitz projections. The derivation of the proposed method is exemplified for linear and nonlinear reaction-diffusion problems. Near-best approximation results in suitable norms are established. In particular, we prove that, in the linear case, the numerical solution is the best approximation in the energy norm among all nodally bound-preserving finite element functions. A series of numerical experiments for such problems showcase the good behaviour of the proposed bound-preserving finite element method.
The stochastic approximation (SA) algorithm is a widely used probabilistic method for finding a zero or a fixed point of a vector-valued funtion, when only noisy measurements of the function are available. In the literature to date, one makes a distinction between ``synchronous'' updating, whereby every component of the current guess is updated at each time, and ``asynchronous'' updating, whereby only one component is updated. In this paper, we study an intermediate situation that we call ``batch asynchronous stochastic approximation'' (BASA), in which, at each time instant, \textit{some but not all} components of the current estimated solution are updated. BASA allows the user to trade off memory requirements against time complexity. We develop a general methodology for proving that such algorithms converge to the fixed point of the map under study. These convergence proofs make use of weaker hypotheses than existing results. Specifically, existing convergence proofs require that the measurement noise is a zero-mean i.i.d\ sequence or a martingale difference sequence. In the present paper, we permit biased measurements, that is, measurement noises that have nonzero conditional mean. Also, all convergence results to date assume that the stochastic step sizes satisfy a probabilistic analog of the well-known Robbins-Monro conditions. We replace this assumption by a purely deterministic condition on the irreducibility of the underlying Markov processes. As specific applications to Reinforcement Learning, we analyze the temporal difference algorithm $TD(\lambda)$ for value iteration, and the $Q$-learning algorithm for finding the optimal action-value function. In both cases, we establish the convergence of these algorithms, under milder conditions than in the existing literature.
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