L1 Approximation Research Articles

Optimal subspaces for mean square approximation of classes of differentiable functions on a segment

In this paper, a set of optimal subspaces is specified for L2 approximation of three classes of functions in the Sobolev spaces $$W_{2}^{{(r)}}$$ defined on a segment and subject to certain boundary conditions. A subspace X of a dimension not exceeding n is called optimal for a function class A if the best approximation of A by X is equal to the Kolmogorov n-width of A. These boundary conditions correspond to subspaces of periodically extended functions with symmetry properties. All approximating subspaces are generated by equidistant shifts of a single function. The conditions of optimality are given in terms of Fourier coefficients of a generating function. In particular, we indicate optimal spline spaces of all degrees d $$ \geqslant $$ r – 1 with equidistant knots of several different types.

An interpolatory view of polynomial least squares approximation

We derive a formula for weighted polynomial least squares approximation which expresses the approximant as a convex combination of interpolants. There is a similar formula for L2 approximation and the same principle applies to multivariate approximation.

Numerical analysis for rotating electro-osmotic flow of fractional Maxwell fluids

In this paper, a rotating electro-osmotic flow of a fractional Maxwell fluid in a parallel plate microchannel with high zeta potentials is examined. The Navier’s slip law at walls is considered. The electric double layer potential distribution is derived by using the nonlinear Poisson–Boltzmann equation. Based on the L1 approximation of the Caputo derivative, a Crank–Nicolson numerical scheme is developed for obtaining the numerical solutions of the rotating electro-osmotic flow velocity profiles. With a purpose to verify the correctness of our numerical results, a comparison has been made with the analytical solutions of the Newtonian fluid given by the previous work and the excellent agreement between the solutions is clear. Finally, the influences of the fractional parameters of α and β, the slip length d and the wall zeta potential ζ on the velocity distribution are also discussed in detail.

Tight bounds on ℓ1 approximation and learning of self-bounding functions

We study the complexity of learning and approximation of self-bounding functions over the uniform distribution on the Boolean hypercube {0,1}n. Informally, a function f:{0,1}n→R is self-bounding if for every x∈{0,1}n, f(x) upper bounds the sum of all the n marginal decreases in the value of the function at x. Self-bounding functions include such well-known classes of functions as submodular and fractionally-subadditive (XOS) functions. They were introduced by Boucheron et al. (2010) in the context of concentration of measure inequalities. Our main result is a nearly tight ℓ1-approximation of self-bounding functions by low-degree juntas. Specifically, all self-bounding functions can be ϵ-approximated in ℓ1 by a polynomial of degree O˜(1/ϵ) over 2O˜(1/ϵ) variables. We show that both the degree and junta-size are optimal up to logarithmic terms. Previous techniques considered stronger ℓ2 approximation and proved nearly tight bounds of Θ(1/ϵ2) on the degree and 2Θ(1/ϵ2) on the number of variables. Our bounds rely on the analysis of noise stability of self-bounding functions together with a stronger connection between noise stability and ℓ1 approximation by low-degree polynomials. This technique can also be used to get tighter bounds on ℓ1 approximation by low-degree polynomials and a faster learning algorithm for halfspaces.These results lead to improved and in several cases almost tight bounds for PAC and agnostic learning of self-bounding functions relative to the uniform distribution. In particular, assuming hardness of learning juntas, we show that PAC and agnostic learning of self-bounding functions have complexity of nΘ˜(1/ϵ).

L2 Estimates and Vanishing Theorems for Holomorphic Vector Bundles Equipped with Singular Hermitian Metrics

We investigate singular Hermitian metrics on vector bundles, especially strictly Griffiths positive ones. $L^2$ esitimates and vanishing theorems usually require an assumption that vector bundles are Nakano positive. However there is no general definition of the Nakano positivity in singular settings. In this paper, we show various $L^2$ estimates and vanishing theromes by assuming that the vector bundle is strictly Griffiths positive and the base manifold is projective.

A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony‐type equation with nonsmooth solutions

AbstractTo recover the full accuracy of discretized fractional derivatives, nonuniform mesh technique is a natural and simple approach to efficiently resolve the initial singularities that always appear in the solutions of time‐fractional linear and nonlinear differential equations. We first construct a nonuniform L2 approximation for the fractional Caputo's derivative of order 1 < α < 2 and present a global consistency analysis under some reasonable regularity assumptions. The temporal nonuniform L2 formula is then utilized to develop a linearized difference scheme for a time‐fractional Benjamin–Bona–Mahony‐type equation. The unconditional convergence of our scheme on both uniform and nonuniform (graded) time meshes are proven with respect to the discrete H1‐norm. Numerical examples are provided to justify the accuracy.

Optimizing Deep Learning RNN Topologies on Intel Architecture

Recurrent neural network (RNN) models have been found to be well suited for processing temporal data. In this work, we present an optimized implementation of vanilla RNN cell and its two popular variants: LSTM and GRU for Intel Xeon architecture. Typical implementations of these RNN cells employ one or two large matrix multiplication (GEMM) calls and then apply the element-wise operations (sigmoid/tanh) onto the GEMM results. While this approach is easy to implement by exploiting vendor-optimized GEMM library calls, the data reuse relies on how GEMMs are parallelized and is sub-optimal for GEMM sizes stemming from small minibatch. Also, the element-wise operations are exposed as a bandwidth-bound kernel after the GEMM which is typically a compute-bound kernel. To address this discrepancy, we implemented a parallel blocked matrix GEMM in order to (a) achieve load balance, (b) maximize weight matrix reuse, (c) fuse the element-wise operations after partial GEMM blocks are computed and while they are hot in cache. Additionally, we bring the time step loop in our cell to further increase the weight reuse and amortize the overhead to transform the weights into blocked layout. The results show that our implementation is generally faster than Intel MKL-DNN library implementations, e.g. for RNN, forward pass is up to ~3× faster whereas the backward/weight update pass is up to ~5× faster. Furthermore, we investigate high-performance implementations of sigmoid and tanh activation functions that achieve various levels of accuracy. These implementations rely on minimax polynomial approximations, rational polynomials, Taylor expansions and exponential approximation techniques. Our vectorized implementations can be flexibly integrated into deep learning computations with different accuracy requirements without compromising performance; in fact, these are able to outperform vectorized and reduced accuracy vendor-optimized (Intel SVML) libraries by 1.6–2.6× while speep up over GNU libm is close to two orders of magnitude. All our experiments are conducted on Intel’s latest CascadeLake architecture.

Global small data smooth solutions of 2-D null-form wave equations with non-compactly supported initial data

For the 2-D nonlinear wave equations □u=F(∂u,∂2u) with initial data (u(0,x),∂tu(0,x))=(εu0(x),εu1(x)), where x=(x1,x2), x0=t, ∂=(∂0,∂1,∂2), ε>0 is small enough, u0(x),u1(x)∈C0∞(R2), and the smooth nonlinearity F(∂u,∂2u)=O(|∂u|2+|∂2u|2), when F(∂u,∂2u) satisfies the null conditions, S. Alinhac in [2] shows that the smooth solution u exists globally. The proof relies on the compactness of the support of (u0(x),u1(x)). Recently, for a class of quasilinear wave equations □u=Nαβμν∂αβ2u∂μν2u or □u=Aα∂α(|∂tu|2−|∇u|2) with small and non-compactly supported initial data, where Nαβμν and Aα are constants (0≤α,β,μ,ν≤2), when the related null condition hold, the authors in [6] prove the global existence of solution u. In this paper, we will prove the global existence for the general 2-D null-form wave equations □u=F(∂u,∂2u) with non-compactly supported initial data. The new key ingredient is to establish a class of weighted L∞-L∞ estimates of solution w to the 2-D linear wave equation □w=f(t,x) instead of the usual L∞-L2 estimates used in [2], [6] and so on. From this, we also get a better time-decay rate for the “good derivatives” of solution u of nonlinear wave equations.

A note on the Schrödinger smoothing effect

AbstractThe Kato–Yajima smoothing estimate is a smoothing weighted L2 estimate with a singular power weight for the Schrödinger propagator. The weight has been generalized relatively recently to Morrey–Campanato weights. In this paper we make this generalization more sharp in terms of the so‐called Kerman–Sawyer weights. Our result is based on a more sharpened Fourier restriction estimate in a weighted L2 space. Obtained results are also extended to the fractional Schrödinger propagator.

High-order composite finite element exact sequences based on tetrahedral–hexahedral–prismatic–pyramidal partitions

The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and quadrilateral faces. This paper presents high order exact sequences of finite element approximations in H1(Ω), H(curl,Ω), H(div,Ω), and L2(Ω) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular faces of these tetrahedral elements are constrained to match the quadrilateral shape functions on the quadrilateral face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nédélec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy’s problems based on compatible pairs of approximations in {H(div,Ω),L2(Ω)} for such tetrahedral–hexahedral–prismatic–pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables is obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems.

On Estimates in L2(ℝ) of Mean ν-Widths of Classes of Functions Defined via the Generalized Modulus of Continuity of ωℳ

For the classes of functions $${W^r}\left( {{\omega _{\cal M}},\;{\rm{\Phi }}} \right)\;: = \left\{ {f\; \in \;L_2^r\left(\mathbb{R}\right)\;:\;{\omega _{\cal M}}\left( {{f^{\left( r \right)}},\;t} \right)\; \le \;{\rm{\Phi }}\left( t \right)\;\forall \;t\; \in \;\left( {0,\;\infty } \right)} \right\},$$

Superconvergence of the direct discontinuous Galerkin method for a time‐fractional initial‐boundary value problem

AbstractA time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ≔ Ω × [0, T], where Ω is the interval [0, l]. Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2. In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L2(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L2(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H1(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp.

L1 − L1 estimates for the strongly damped plate equation

In this paper, we derive L1−L1 long time estimates for the strongly damped plate equationutt+Δ2u+Δ2ut=0x∈Rn,t∈R+,u(0,x)=u0(x),ut(0,x)=u1(x). In particular, we prove that‖u(t,⋅)‖L1≤C(1+t)n4(‖u0‖L1+(1+t)12‖u1‖L1), for any t≥0, in space dimension n≥5.

Boundary layer effect at the edge of fibrous composites using homogenization theory

Many studies in the theory of composite materials are based on the homogenization approach, which consists of the substitution of the heterogeneous medium by a homogeneous one with certain effective properties. These effective properties are obtained from internal asymptotics which excludes boundary layer effects. However, at the boundary of the individual ply the periodic solution is disrupted. This loss of periodicity of the interior asymptotic solution needs to be accounted for. Here, the boundary layer solution is a correction in the micro solution which accounts for the applied boundary condition. This error gives rise to a sharp boundary layer which decays very quickly when it travels to the interior of the domain, leaving behind a tail which has to be added to the interior solutions to get the correct representation of the global displacement field. It is shown that this correction is significant, local and affects local stress and strain predictions, which are missed by meso or classical micro mechanical solutions. This important outcome of the study proves the theoretical estimates available in the literature (via error estimates in L2 and H1 - norms) emphasizing significant improvement in the local solution at the boundary. It is shown, through numerical examples, that the corrected solution can significantly alter damage initiation predictions at the boundary.

An extension of Riesz transform

In this paper, we consider the following singular integral Tjf(x)=Kj∗f(x),Kj(x)=xj|x|n+1−β,where x∈Rn,0≤β<n,j=1,2,…,n. When β=0, it corresponds to the Riesz transform. Based on the L2 estimate of Tjf in Yu et al. (2019) and making use of the refined Calderon–Zygmund decomposition, we establish an estimate of Tjf in the Lq space for 1<q<2. For 2<q<∞ and q′=qq−1 (the dual number of q), by the duality method, we prove an estimate of Tjf in the (Lq′∩Lp′)∗ space which is the dual space of Lq′∩Lp′ with 1q′=1p′(1−βn). The obtained estimates hold uniformly on β>0 when β is appropriately small. As a result, the strong (q,q) estimate with 1<q<∞ and the weak (1,1) estimate of the Riesz transform can be recovered from the obtained estimates as β→0, respectively.

Directional regression for functional data

We propose a method of sufficient dimension reduction for functional data based on directional regression. Functional principal component analysis with truncation acting as the regularization mechanism is used in the development. We establish optimal convergence rate for the estimation of the dimension reduction directions, in both L2 estimation error and prediction risk. Somewhat surprisingly, the optimal rates for both error measures are achieved at the same truncation level. Effect of discrete sampling of functional predictor on a dense grid is also considered. Simulations and a real data application illustrate the favorable finite sample performance of the method.

Unconditional optimal error estimate of the projection/Lagrange-Galerkin finite element method for the Boussinesq equations

This paper provides an unconditional optimal convergence of a fractional-step method for solving the Boussinesq equations. In this method, the convection is treated by the Lagrange-Galerkin technique, whereas the diffusion and the incompressibility are treated by the projection method. There are lots of authors who worked on this method, and some authors gave the error estimate of this method. But, to our best knowledge, the error estimate for this method is under certain time-step restrictions. In this paper, we prove that the methods are stable almost unconditionally, i.e., when τ and h are smaller than a given constant. The basic idea of our analysis is splitting the error function into three terms, one term between the finite element solution and the projection, the other term between the projection and the time-discrete solution, the third term between the time-discrete solution and the exact solution, and giving the error estimates for each term respectively. Then, we obtain the optimal error estimates in L2 and H1-norm for the velocity and L2-norm for the pressure. In order to show the efficiency of our method, some numerical results are presented.

Finite element analysis for coupled time-fractional nonlinear diffusion system

This paper is concerned with the L1-Galerkin finite element analysis of the coupled time-fractional nonlinear diffusion system. Existence and uniqueness results are discussed at continuous as well as discrete levels. Two different methods for linearizing the nonlinear fully discrete problems are used. We provide a priori bounds and convergence estimates in L2(Ω) norm for fully-discrete problems. Numerical results based on the presented finite element methods are provided to validate the theoretical estimates.

An application of L1 estimates for oscillating integrals to parabolic like semi-linear structurally damped σ-evolution models

We study the following Cauchy problems for semi-linear structurally damped σ-evolution models:utt+(−Δ)σu+μ(−Δ)δut=f(u,ut),u(0,x)=u0(x),ut(0,x)=u1(x) with σ≥1, μ>0 and δ∈(0,σ2). Here the function f(u,ut) stands for the power nonlinearities |u|p and |ut|p with a given number p>1. We are interested in investigating L1 estimates for oscillating integrals in the presentation of the solutions to the corresponding linear models with vanishing right-hand sides by applying the theory of modified Bessel functions and Faà di Bruno's formula. By assuming additional Lm regularity on the initial data, we use (Lm∩Lq)−Lq and Lq−Lq estimates with q∈(1,∞) and m∈[1,q), to prove the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on Lq spaces.

Solvability for a drift-diffusion system with Robin boundary conditions

This paper focuses on a drift-diffusion system subjected to boundedly non dissipative Robin boundary conditions. A general existence result with large initial conditions is established by using suitable L1, L2 and trace estimates. Finally, two examples coming from the corrosion and the self-gravitation model are analyzed.