Suboptimal Convergence Rate Research Articles

Stochastic First-Order Methods for Average-Reward Markov Decision Processes

We study average-reward Markov decision processes (AMDPs) and develop novel first-order methods with strong theoretical guarantees for both policy optimization and policy evaluation. Compared with intensive research efforts in finite sample analysis of policy gradient methods for discounted MDPs, existing studies on policy gradient methods for AMDPs mostly focus on regret bounds under restrictive assumptions, and they often lack guarantees on the overall sample complexities. Toward this end, we develop an average-reward stochastic policy mirror descent method for solving AMDPs with and without regularizers and provide convergence guarantees in terms of the long-term average reward. For policy evaluation, existing on-policy methods suffer from suboptimal convergence rates as well as failure in handling insufficiently random policies due to the lack of exploration in the action space. To remedy these issues, we develop a variance-reduced temporal difference (VRTD) method with linear function approximation for randomized policies along with optimal convergence guarantees, and design an exploratory VRTD method that resolves the exploration issue and provides comparable convergence guarantees. By combining the policy evaluation and policy optimization parts, we establish sample complexity results for solving AMDPs under both generative and Markovian noise models. It is worth noting that when linear function approximation is utilized, our algorithm only needs to update in the low-dimensional parameter space, and thus can handle MDPs with large state and action spaces. Funding: T. Li and G. Lan were supported in part by the National Science Foundation [Grant CCF-1909298] and the Office of Navel Research [Grant N00014-20-1-2089].

High-dimensional IV cointegration estimation and inference

A semiparametric triangular systems approach shows how multicointegrating linkages occur naturally in an I(1) cointegrated regression model when the long run error variance matrix in the system is singular. Under such singularity, cointegrated I(1) systems embody a multicointegrated structure that makes them useful in many empirical settings. Earlier work shows that such systems may be analyzed and estimated without appealing to the associated I(2) system but with suboptimal convergence rates and potential asymptotic bias. The present paper develops a robust approach to estimation and inference of such systems using high dimensional IV methods that have appealing asymptotic properties like those known to apply in the optimal estimation of cointegrated systems (Phillips, 1991). The approach uses an extended version of high-dimensional trend IV (Phillips, 2006, 2014) estimation with deterministic orthonormal instruments. The methods and derivations involve new results on high-dimensional IV techniques and matrix normalization in the limit theory that are of independent interest. Wald tests of general linear restrictions are constructed using a fixed-b long run variance estimator that leads to robust pivotal HAR inference in both cointegrated and multicointegrated cases. Simulations show good properties of the estimation and inferential procedures in finite samples. An empirical illustration to housing stocks, starts and completions is provided.

Rational reparameterization of unstructured quadrilateral meshes for isogeometric analysis with optimal convergence

Unstructured quadrilateral meshes are widely used in representing surfaces of arbitrary topology. However, in connection with isogeometric analysis, unstructured meshes suffer from sub-optimal convergence rates in extraordinary regions, especially for solving higher-order PDEs (partial differential equations). In this work, a rational reparameterization method is proposed to achieve optimal convergence rates for isogeometric analysis using Catmull-Clark surfaces. The scheme lifts the convergence of analysis solutions by tuning the weights of control vertices of the input geometry in extraordinary regions, which are often neglected in constructing analysis-suitable parameterizations. Preferred tuning parameters with optimal convergence rates in L2-norm are determined statistically based on a test suite composed of typical testing solutions with a range of parameters. Numerical experiments confirm the validity of the proposed method for PDEs up to the fourth-order. Comparisons with the original Catmull-Clark scheme and a state-of-the-art tuned subdivision scheme also show the superiority of the proposed method in terms of both optimal convergence and reduced absolute errors of resulting analysis solutions.

On the design of stable, consistent, and conservative high-order methods for multi-material hydrodynamics

Obtaining stable and high-order numerical solutions for multi-material hydrodynamics is an open challenge. Although slope limiters are widely used to maintain monotonicity near discontinuities, typical limiting procedures violate closure laws at the discrete level when applied to multi-material hydrodynamics equations. Due to this, the high-order expansions of quantities related by the closure laws are no longer consistent. The commonly observed symptom of this consistency-violation is that the numerical method fails to maintain constant pressure and velocity across material interfaces. This leads to sub-optimal convergence rates for smooth multi-material problems as well. Specialized limiting procedures that satisfy consistency while maintaining conservation need to be developed for such equations. A novel procedure that re-instates consistency into slope-limited high-order discretizations applied to the multi-material hydrodynamics equations is presented here. Using simple examples, it is demonstrated that the presented method satisfies closure laws at the discrete level, while maintaining conservative properties of the high-order method. Furthermore, this procedure involves a projection step which relies on the compact basis of the underlying spatial discretization, i.e. for discontinuous schemes (viz. DG and FV) the projection is local, and does not involve global matrix solves. Comparisons with conventional approaches emphasizes the necessity of the consistent closure-law preserving limiting approach, in order to maintain design order of accuracy for smooth multi-material problems.

An hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem

The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics, and has wide applications. In this paper, we study an hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem. Based on the work of Gudi et al. [J Sci Comput, 37 (2008)], using piecewise polynomials of degree p≥3, we derive the a priori error estimates of the approximate eigenfunction in the broken H1 norm and L2 norm which are optimal in h and suboptimal in p. When p=2, the approximate eigenfunctions converge but with only suboptimal convergence rate. When p≥2 and the eigenfunction u∈Hs(Ω)(s≥p+1), we prove that the convergence rate of approximate eigenvalues reaches 2p−2 in h and p−(2s−7) in p. We also discuss the a posterior error estimates of the approximate eigenvalues and implement the adaptive calculation. Numerical experiments show that the methods are easy to implement and can efficiently compute biharmonic eigenvalues.

A reconstructed discontinuous approximation on unfitted meshes to [formula omitted] and [formula omitted] interface problems

We propose a high-order discontinuous Galerkin finite element method for solving the H(div)- and H(curl)-elliptic interface problems on unfitted meshes. The vector-valued approximation space is constructed by the patch reconstruction with at most d degrees of freedom per element. The C2-smooth interface is allowed to intersect elements in a quite general setup. The patch reconstruction provides the stability near the interface naturally without any additional stabilization strategy. The method is based on the symmetric interior penalty method and the optimal and suboptimal convergence rates under the energy norm and L2 norm are derived. Numerical examples in both two and three dimensions are presented to demonstrate the accuracy of the method.

Convergence rates for the Allen–Cahn equation with boundary contact energy: the non-perturbative regime

We extend the recent rigorous convergence result of Abels and Moser (SIAM J Math Anal 54(1):114–172, 2022. https://doi.org/10.1137/21M1424925) concerning convergence rates for solutions of the Allen–Cahn equation with a nonlinear Robin boundary condition towards evolution by mean curvature flow with constant contact angle. More precisely, in the present work we manage to remove the perturbative assumption on the contact angle being close to 90^circ . We establish under usual double-well type assumptions on the potential and for a certain class of boundary energy densities the sub-optimal convergence rate of order {varepsilon ^{frac{1}{2}}} for general contact angles alpha in (0,pi ). For a very specific form of the boundary energy density, we even obtain from our methods a sharp convergence rate of order varepsilon ; again for general contact angles alpha in (0,pi ). Our proof deviates from the popular strategy based on rigorous asymptotic expansions and stability estimates for the linearized Allen–Cahn operator. Instead, we follow the recent approach by Fischer et al. (SIAM J Math Anal 52(6):6222–6233, 2020. https://doi.org/10.1137/20M1322182), thus relying on a relative entropy technique. We develop a careful adaptation of their approach in order to encode the constant contact angle condition. In fact, we perform this task at the level of the notion of gradient flow calibrations. This concept was recently introduced in the context of weak-strong uniqueness for multiphase mean curvature flow by Fischer et al. (arXiv:2003.05478v2).

Iteration Dependent Waveform Relaxation for Hybrid Field Nonlinear Circuit Problems

This article presents a novel waveform relaxation scheme to solve electromagnetically large structures loaded with lumped linear and nonlinear elements. The scheme partitions the problem into a linear electromagnetic structure and a possibly nonlinear lumped circuit, which are coupled using Thévenin interfaces across the steps of an iterative waveform relaxation scheme. The main novel contribution is an adaptive selection of the decoupling resistances used as port references to define incident and reflected scattering signals, whose time-domain samples are refined through iterations. The decoupling resistances are updated through iterations, with the main objective of improving convergence speed and ultimately runtime. The resulting scheme is self-adapting to terminations exploiting high dynamic range in their impedance profiles and is able to provide a suboptimal convergence rate. Three-dimensional shielding structures loaded with nonlinear elements are employed as numerical examples to demonstrate the proposed method.

Unlocking the secrets of locking: Finite element analysis in planar linear elasticity

Finite element methods have been the subject of active research for the last six decades. However, gaps remain in the understanding of even the simplest applications, including linear elasticity — the area where finite elements were first developed as a serious technique for the approximation of partial differential equations by engineers interested in the stress analysis of structures. For nearly incompressible materials, such as rubber, the standard, conforming finite element method sometimes exhibits suboptimal convergence rates for the energy and/or stresses. This type of behavior, termed “locking” or “non-robustness”, is still not completely understood despite receiving a great deal of investigation. This paper reviews the concept of locking in conforming finite element approximations to planar linear elasticity and seeks to quantify the effects of locking for the h- and p-version finite element method with a particular emphasis on quantifying the effect of mesh topology and geometry. As a by-product, we show that the inf–sup constant, which is responsible for locking, is independent of the mesh size h and polynomial degree p when p≥4. Numerical examples are provided throughout to illustrate the theoretical results.

Tangent stiffness for nonlinear heat transfer and its application to additive manufacturing

Prediction-correction method is commonly employed in the solution of nonlinear heat transfer equation with temperature dependent thermophysical parameters arising in various engineering problems such as additive manufacturing of metal components. However, the stiffness of such conventional method is secant rather than tangent and thus it only results in suboptimal rate of convergence in Newton's iteration. In this paper, tangent stiffness for nonlinear heat transfer is proposed through the linearization of the weak form and an additional term presents which disappears in the conventional prediction-correction method. Numerical results show that the solution with the proposed tangent stiffness converges much faster than that with secant stiffness and approximately only half of the iteration steps are needed. Especially, in the case that the thermal conductivity varies dramatically with the temperature, the conventional method diverges whereas the tangent stiffness still shows quite good convergence. Application of the proposed method to the modeling of heat transfer in additive manufacturing (AM) process is presented and the obtained temperature results agree quite well with the existing experimental studies. It is worth mentioning that almost half of CPU time is saved due to the application of the proposed tangent stiffness. This saving is of great significance for modeling the AM process of complicated components, which is usually very time-consuming.

Optimally convergent mixed finite element methods for the stochastic Stokes equations

Abstract We propose some new mixed finite element methods for the time-dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known (Langa, J. A., Real, J. & Simon, J. (2003) Existence and regularity of the pressure for the stochastic Navier--Stokes equations. Appl. Math. Optim., 48, 195--210) that the pressure solution has low regularity, which manifests in suboptimal convergence rates for well-known inf-sup stable mixed finite element methods in numerical simulations; see Feng X., & Qiu, H. (Analysis of fully discrete mixed finite element methods for time-dependent stochastic Stokes equations with multiplicative noise. arXiv:1905.03289v2 [math.NA]). We show that eliminating this gradient part from the noise in the numerical scheme leads to optimally convergent mixed finite element methods and that this conceptual idea may be used to retool numerical methods that are well known in the deterministic setting, including pressure stabilization methods, so that their optimal convergence properties can still be maintained in the stochastic setting. Computational experiments are also provided to validate the theoretical results and to illustrate the conceptual usefulness of the proposed numerical approach.

An indirect collocation method for variable-order fractional wave equations on uniform or graded meshes and its optimal error estimates

We develop an indirect collocation method for a variable-order fractional wave equation. Fractional differential equations are well known to exhibit initial weak singularity, which makes it unrealistic to carry out error estimates of their numerical approximations based on the smoothness assumptions of the true solutions. In this paper, we analyze the convergence behaviour of the method without artificially assuming the (often untrue) full regularity of the true solution of the problem, but only based on the behaviour of the coefficients and variable order of the problem. More precisely, we prove the following results: (i) If the variable order has an integer initial value, the method discretized on a uniform partition has an optimal-order convergence rate in the norm. (ii) Otherwise, the method discretized on a uniform mesh has only a sub-optimal order convergence rate. The method discretized on a graded mesh with the mesh grading parameter determined by the initial value of the variable order has an optimal-order convergence rate in the norm. Numerical experiments are performed to substantiate the theoretical results.

A discontinuous least squares finite element method for time-harmonic Maxwell equations

Abstract We propose and analyze a discontinuous least squares finite element method for solving the indefinite time-harmonic Maxwell equations. The scheme is based on the $L^2$ norm least squares functional with weak imposition of continuity across interior faces. We minimize the functional over the piecewise polynomial spaces to seek numerical solutions. The method is shown to be stable without any constraint on the mesh size. We prove the optimal convergence rate under the energy norm and sub-optimal convergence rate under the $L^2$ norm. Numerical results in two and three dimensions are presented to verify the error estimates.

Nitsche’s method for linear Kirchhoff–Love shells: Formulation, error analysis, and verification

Stable and accurate modeling of thin shells requires proper enforcement of all types of boundary conditions. Unfortunately, for Kirchhoff–Love shells, strong enforcement of Dirichlet boundary conditions is difficult because both displacement and normal rotation boundary conditions must be applied. A popular alternative is to employ Nitsche’s method to weakly enforce all boundary conditions. However, while many Nitsche-based formulations have been proposed in the literature, they lack comprehensive error analyses and verification. In fact, existing formulations are variationally inconsistent and yield sub-optimal convergence rates when used with common boundary condition specifications. In this paper, we present a novel Nitsche-based formulation for the linear Kirchhoff–Love shell that is provably stable and optimally convergent for general sets of admissible boundary conditions. To arrive at our formulation, we first present a framework for constructing Nitsche’s method for any abstract variationally constrained minimization problem. We then apply this framework to the linear Kirchhoff–Love shell and, for the particular case of NURBS-based isogeometric analysis, we prove that the resulting formulation yields optimal convergence rates in both the shell energy norm and the standard L2-norm. To arrive at this formulation, we derive the Euler–Lagrange equations for general sets of admissible boundary conditions and show that the Euler–Lagrange boundary conditions typically presented in the literature are incorrect. We verify our formulation by manufacturing solutions for a new shell obstacle course that encompasses flat, parabolic, hyperbolic, and elliptic geometric configurations with a variety of common boundary condition specifications. These manufactured solutions allow us to robustly measure the error across the entire shell in contrast with current best practices where displacement and stress errors are only measured at specific locations. We use NURBS discretizations to represent the shell geometry and show optimal convergence rates in both the shell energy norm and the standard L2-norm with varying polynomial degrees for all of the problems in the obstacle course.

Weak Galerkin finite element with curved edges

This paper studies a weak Galerkin finite element method on a domain with curved edges. We consider the two dimensional Poisson’s equation of a fixed curved boundary. Usually, the straight polygon meshes deliver a sub-optimal rate of convergence for the curved domain as approximation order bounded by O(h2) even for high order numerical schemes. In this paper, we shall develop a high order (optimal rate of convergence) WGFEM based on the curved mesh. The implementation details are studied for efficient numerical integration tools. Numerical experiments are provided to show the proposed curved WGFEM produces an optimal rate for errors measured in L2-norm and energy-norm of convergence as O(hk+1) and O(hk) for curved two-dimensional domain.

Smoothed residual stopping for statistical inverse problems via truncated SVD estimation

This work examines under what circumstances adaptivity for truncated SVD estimation can be achieved by an early stopping rule based on the smoothed residuals $\|(AA^{\top })^{\alpha /2}(Y-A\widehat{\mu }^{(m)})\|^{2}$. Lower and upper bounds for the risk are derived, which show that moderate smoothing of the residuals can be used to adapt over classes of signals with varying smoothness, while oversmoothing yields suboptimal convergence rates. The range of smoothness classes for which adaptation is possible can be controlled via $\alpha $. The theoretical results are illustrated by Monte-Carlo simulations.

Optimal Convergence Analysis of a Second Order Scheme for a Thin Film Model Without Slope Selection

In Li et al. (J Sci Comput 76(3):1905–1937, 2018), a temporal second-order mixed finite element scheme has been proposed for the thin film epitaxial growth model without slope selection. Using the super-convergence theory in a regular rectangular mesh, the authors of Li et al. (2018) proved an optimal $$O(h^{q+1}+\tau ^2)$$ convergence. However, in a quasi-uniform triangulation mesh setting, only a sub-optimal convergence rate $$O(h^q+\tau ^2)$$ is proved, while numerical results indicated an optimal $$O(h^{q+1}+\tau ^2)$$ convergence when the exact solution has $$H^{q+1}$$ regularity in space. Here h and $$\tau $$ are the discretization sizes in space and time, respectively, and $$q\ge 1$$ is the degree of the polynomial in the spatial discretization. In this paper, we provide a theoretical proof of the optimal convergence rate. The main difficulty lies in how to treat a nonlinear term $$\frac{\nabla u}{1+|\nabla u|^2}$$ . We solve this by using a discrete Laplacian operator $$-\varDelta _h$$ and some uncommon techniques in the analysis. Numerical results are also presented to demonstrate the $$(q+1)$$ -order convergence of the spatial approximation.

High order VEM on curved domains

We deal with the virtual element method (VEM) for solving the Poisson equation on a domain \Omega with curved boundary. Given a polygonal approximation \Omega_h of the domain \Omega , the standard order m VEM [6], for m increasing, leads to a suboptimal convergence rate. We adapt the approach of [14] to VEM and we prove that an optimal convergence rate can be achieved by using a suitable correction depending on high order normal derivatives of the discrete solution at the boundary edges of \Omega_h , which, to retain computability, is evaluated after applying the projector \Pi^{\nabla} onto the space of polynomials. Numerical experiments confirm the theory.

The Virtual Element Method with curved edges

In this paper we initiate the investigation of Virtual Elements with curved faces. We consider the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain or with a curved interface. While an approximation of the domain with polygons leads, for degree of accuracy k≥2, to a sub-optimal rate of convergence, we show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence.

Complete expected improvement converges to an optimal budget allocation

AbstractThe ranking and selection problem is a well-known mathematical framework for the formal study of optimal information collection. Expected improvement (EI) is a leading algorithmic approach to this problem; the practical benefits of EI have repeatedly been demonstrated in the literature, especially in the widely studied setting of Gaussian sampling distributions. However, it was recently proved that some of the most well-known EI-type methods achieve suboptimal convergence rates. We investigate a recently proposed variant of EI (known as ‘complete EI’) and prove that, with some minor modifications, it can be made to converge to the rate-optimal static budget allocation without requiring any tuning.