Piecewise Regularity Research Articles

Piecewise regularity results for linear elliptic systems with piecewise regular coefficients

Piecewise regularity results for linear elliptic systems with piecewise regular coefficients

The Application of Piecewise Regularization Reconstruction to the Calibration of Strain Beams.

Standard beams are mainly used for the calibration of strain sensors using their load reconstruction models. However, as an ill-posed inverse problem, the solution to these models often fails to converge, especially when dealing with dynamic loads of different frequencies. To overcome this problem, a piecewise Tikhonov regularization method (PTR) is proposed to reconstruct dynamic loads. The transfer function matrix is built both using the denoised excitations and the corresponding responses. After singular value decomposition (SVD), the singular values are divided into submatrices of different sizes by utilizing a piecewise function. The regularization parameters are solved by optimizing the piecewise submatrices. The experimental result shows that the MREs of the PTR method are 6.20% at 70 Hz and 5.86% at 80 Hz. The traditional Tikhonov regularization method based on GCV exhibits MREs of 28.44% and 29.61% at frequencies of 70 Hz and 80 Hz, respectively, whereas the L-curve-based approach demonstrates MREs of 29.98% and 18.42% at the same frequencies. Furthermore, the PREs of the PTR method are 3.54% at 70 Hz and 3.73% at 80 Hz. The traditional Tikhonov regularization method based on GCV exhibits PREs of 27.01% and 26.88% at frequencies of 70 Hz and 80 Hz, respectively, whereas the L-curve-based approach demonstrates PREs of 29.50% and 15.56% at the same frequencies. All in all, the method proposed in this paper can be extensively applied to load reconstruction across different frequencies.

A regularization–correction approach for adapting subdivision schemes to the presence of discontinuities

Linear approximation methods suffer from Gibbs oscillations when approximating functions with jumps. Essentially non oscillatory subcell-resolution (ENO-SR) is a local technique avoiding oscillations and with a full order of accuracy, but a loss of regularity of the approximant appears. The goal of this paper is to introduce a new approach having both properties of full accuracy and regularity. In order to obtain it, we propose a three-stage algorithm: first, the data is smoothed by subtracting an appropriate non-smooth data sequence; then a chosen high order linear approximation operator is applied to the smoothed data and finally, an approximation with the proper jump or corner (jump in the first order derivative) discontinuity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. This new procedure can be applied as subdivision scheme to design curves and surfaces both in point-value and in cell-average contexts. Using the proposed algorithm, we are able to construct approximations with high precision, with high piecewise regularity, and without smearing nor oscillations in the presence of discontinuities. These are desired properties in real applications as computer aided design or car design, among others.

Sharp lower error bounds for strong approximation of SDEs with discontinuous drift coefficient by coupling of noise

In the past decade, an intensive study of strong approximation of stochastic differential equations (SDEs) with a drift coefficient that has discontinuities in space has begun. In the majority of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and that the diffusion coefficient is globally Lipschitz continuous and nondegenerate at the discontinuities of the drift coefficient. Under this type of assumptions the best Lp-error rate obtained so far for approximation of scalar SDEs at the final time is 3/4 in terms of the number of evaluations of the driving Brownian motion. In the present article, we prove for the first time in the literature sharp lower error bounds for such SDEs. We show that for a huge class of additive noise driven SDEs of this type the Lp-error rate 3/4 can not be improved. For the proof of this result we employ a novel technique by studying equations with coupled noise: we reduce the analysis of the Lp-error of an arbitrary approximation based on evaluation of the driving Brownian motion at finitely many times to the analysis of the Lp-distance of two solutions of the same equation that are driven by Brownian motions that are coupled at the given time-points and independent, conditioned on their values at these points. To obtain lower bounds for the latter quantity, we prove a new quantitative version of positive association for bivariate normal random variables (Y,Z) by providing explicit lower bounds for the covariance Cov(f(Y),g(Z)) in case of piecewise Lipschitz continuous functions f and g. In addition it turns out that our proof technique also leads to lower error bounds for estimating occupation time functionals ∫01f(Wt)dt of a Brownian motion W, which substantially extends known results for the case of f being an indicator function.

An adaptive strong order 1 method for SDEs with discontinuous drift coefficient

In recent years, a number of results has been proven in the literature for strong approximation of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space. In many of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and the diffusion coefficient is Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. For scalar SDEs of that type the best Lp-error rate known so far for approximation of the solution at the final time point is 3/4 in terms of the number of evaluations of the driving Brownian motion and it is achieved by the transformed equidistant quasi-Milstein scheme, see [24]. Recently in [27] it has been shown that for such SDEs the Lp-error rate 3/4 can not be improved in general by no numerical method based on evaluations of the driving Brownian motion at fixed time points. In the present article we construct for the first time in the literature a method based on sequential evaluations of the driving Brownian motion, which achieves an Lp-error rate of at least 1 in terms of the average number of evaluations of the driving Brownian motion for such SDEs.

Interior jump and contact singularity for compressible flows with inflow jump datum

Initial jump datum at a point must generate a jump curve directing into the region by the hyperbolic character of the continuity equation and the pressure gradient (in the momentum equation) is not well-defined. For this we construct a velocity vector field directed by the pressure difference on the jump curve. In bounded domains the jump curve must meet some points of the boundary and brings us contact singularities. In a rectangle domain we split the velocity vector field into a non-smooth part plus a contact singular one plus a smoother one and similar for the density function. We show piecewise regularity for the Navier-Stokes equations of compressible viscous flows and derive the Rankine-Hugoniot conditions.

Interior discontinuity for a stationary compressible Stokes system with inflow datum

In this paper we show an interior discontinuity and its piecewise regularity for a stationary compressible Stokes problem with inflow jump datum. The discontinuity is preserved along the interior curve directed by the ambient vector field. The difficulty lies in controlling the gradient of the pressure across the interior curve starting at the jump point that the inflow datum has. We deal with this issue by constructing a vector function K corresponding to the interior jump and splitting the vector K from the velocity vector. Finally the remainder for the velocity vector and pressure function is shown to be in the space H2,q×H1,q for q≥2.

Event-Based 3D Motion Flow Estimation Using 4D Spatio Temporal Subspaces Properties.

State of the art scene flow estimation techniques are based on projections of the 3D motion on image using luminance—sampled at the frame rate of the cameras—as the principal source of information. We introduce in this paper a pure time based approach to estimate the flow from 3D point clouds primarily output by neuromorphic event-based stereo camera rigs, or by any existing 3D depth sensor even if it does not provide nor use luminance. This method formulates the scene flow problem by applying a local piecewise regularization of the scene flow. The formulation provides a unifying framework to estimate scene flow from synchronous and asynchronous 3D point clouds. It relies on the properties of 4D space time using a decomposition into its subspaces. This method naturally exploits the properties of the neuromorphic asynchronous event based vision sensors that allows continuous time 3D point clouds reconstruction. The approach can also handle the motion of deformable object. Experiments using different 3D sensors are presented.

Convergence of a Second-Order Linearized BDF–IPDG for Nonlinear Parabolic Equations with Discontinuous Coefficients

We study the anti-symmetric interior over-penalized discontinuous Galerkin finite element methods for solving nonlinear parabolic interface problems with second-order backward difference formula for the time discretization, where the diffusion coefficient depends on the unknown solution and is discontinuous across the interface. We present optimal-order error estimates for the finite element solution based on piecewise regularity of the solution.

Jump dynamics due to jump datum of compressible viscous Navier–Stokes flows in a bounded plane domain

In this paper, when the initial density has a jump across an interior curve in a bounded domain, we show unique existence, piecewise regularity and jump discontinuity dynamics for the density and the velocity vector governed by the Navier–Stokes equations of compressible viscous barotropic flows. A critical difficulty is in controlling the gradient of the pressure across the jump curve. This is resolved by constructing a vector function associated with the pressure jump value on the convecting curve and extending it to the whole domain.

A Finite Element Method for a Curlcurl-Graddiv Eigenvalue Interface Problem

In this paper we propose and study a finite element method for a curlcurl-graddiv eigenvalue interface problem. Its solution may be of piecewise non-$H^1$. We would like to approximate such a solution in an $H^1$-conforming finite element space. With the discretizations of both curl and div operators of the underlying eigenvalue problem in two finite element spaces, the proposed method is essentially a standard $H^1$-conforming element method, up to element bubbles which can be statically eliminated at element levels. We first analyze the proposed method for the related source interface problem by establishing the stability and the error bounds. We then analyze the underlying eigenvalue interface problem, and we obtain the error bounds $\mathcal{O}(h^{2r_0})$ for eigenvalues which correspond to eigenfunctions in $\prod_{j=1}^J (H^r(\Omega_j))^3\hookrightarrow (H^{r_0}(\Omega))^3$ space, where the piecewise regularity $r$ and the global regularity $r_0$ may belong to the most interesting interval $[0,1]$.

The $L^2$-Projection and Quasi-Optimality of Galerkin Methods for Parabolic Equations

We consider linear parabolic initial-boundary value problems and analyze Galerkin approximation in space. With the help of the inf-sup theory, we derive quasi-optimality results with respect to norms that arise from the standard weak formulation and from a formulation requiring only integrability in time. Moreover, we reveal that the $H^1$-stability of the $L^2$-projection is not only sufficient but also necessary for these results. As application, we consider conforming finite element approximation in space and derive a priori error bounds in terms of the local meshsize and piecewise regularity. The regularity is the minimal one indicated by approximation theory and matches regularity results for linear parabolic problems.

On higher regularity for the Westervelt equation with strong nonlinear damping

We show higher interior regularity for the Westervelt equation with strong nonlinear damping term of q-Laplace type. Secondly, we investigate an interface coupling problem for these models, which arise, e.g. in the context of medical applications of high intensity focused ultrasound in the treatment of kidney stones. We show that the solution to the coupled problem exhibits piecewise regularity in space, provided that the gradient of the acoustic pressure is essentially bounded in space and time on the whole domain. This result is of importance in numerical approximations of the present problem as well as in gradient based algorithms for finding the optimal shape of a focusing acoustic lens in lithotripsy.

Approximating Gradients with Continuous Piecewise Polynomial Functions

Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by continuous piecewise polynomial functions over a simplicial mesh. The main result is that the global best approximation error is equivalent to an appropriate sum in terms of the local best approximations errors on elements. Thus, requiring continuity does not downgrade local approximability and discontinuous piecewise polynomials essentially do not offer additional approximation power, even for a fixed mesh. This result implies error bounds in terms of piecewise regularity over the whole admissible smoothness range. Moreover, it allows for simple local error functionals in adaptive tree approximation of gradients.

Homotopy Based Algorithms for $\ell _{\scriptscriptstyle 0}$-Regularized Least-Squares

Sparse signal restoration is usually formulated as the minimization of a quadratic cost function $\|y-Ax\|_2^2$, where A is a dictionary and x is an unknown sparse vector. It is well-known that imposing an $\ell_0$ constraint leads to an NP-hard minimization problem. The convex relaxation approach has received considerable attention, where the $\ell_0$-norm is replaced by the $\ell_1$-norm. Among the many efficient $\ell_1$ solvers, the homotopy algorithm minimizes $\|y-Ax\|_2^2+\lambda\|x\|_1$ with respect to x for a continuum of $\lambda$'s. It is inspired by the piecewise regularity of the $\ell_1$-regularization path, also referred to as the homotopy path. In this paper, we address the minimization problem $\|y-Ax\|_2^2+\lambda\|x\|_0$ for a continuum of $\lambda$'s and propose two heuristic search algorithms for $\ell_0$-homotopy. Continuation Single Best Replacement is a forward-backward greedy strategy extending the Single Best Replacement algorithm, previously proposed for $\ell_0$-minimization at a given $\lambda$. The adaptive search of the $\lambda$-values is inspired by $\ell_1$-homotopy. $\ell_0$ Regularization Path Descent is a more complex algorithm exploiting the structural properties of the $\ell_0$-regularization path, which is piecewise constant with respect to $\lambda$. Both algorithms are empirically evaluated for difficult inverse problems involving ill-conditioned dictionaries. Finally, we show that they can be easily coupled with usual methods of model order selection.

Cycle graph analysis for 3D roof structure modelling: Concepts and performance

The paper presents a cycle graph analysis approach to the automatic reconstruction of 3D roof models from airborne laser scanner data. The nature of convergences of topological relations of plane adjacencies, allowing for the reconstruction of roof corner geometries with preserved topology, can be derived from cycles in roof topology graphs. The topology between roof adjacencies is defined in terms of ridge-lines and step-edges. In the proposed method, the input point cloud is first segmented and roof topology is derived while extracting roof planes from identified non-terrain segments. Orientation and placement regularities are applied on weakly defined edges using a piecewise regularization approach prior to the reconstruction, which assists in preserving symmetries in building geometry. Roof corners are geometrically modelled using the shortest closed cycles and the outermost cycle derived from roof topology graph in which external target graphs are no longer required. Based on test results, we show that the proposed approach can handle complexities with nearly 90% of the detected roof faces reconstructed correctly. The approach allows complex height jumps and various types of building roofs to be firmly reconstructed without prior knowledge of primitive building types.

Convergence of a Discontinuous Galerkin Multiscale Method

We present a discontinuous Galerkin multiscale method for second order elliptic problems and prove convergence. We consider a heterogeneous and highly varying diffusion coefficient in $L^\infty(\Omega,\mathbb{R}^{d\times d}_{sym})$ with uniform spectral bounds without any assumption on scale separation or periodicity. The multiscale method uses a corrected basis that is computed on patches/subdomains. The error, due to truncation of the corrected basis, decreases exponentially with the size of the patches. Hence, to achieve an algebraic convergence rate of the multiscale solution on a uniform mesh with mesh size $H$ to a reference solution, it is sufficient to choose the patch sizes $\mathcal{O}(H|\log H|)$. We also discuss a way to further localize the corrected basis to elementwise support. Improved convergence rate can be achieved depending on the piecewise regularity of the forcing function. Linear convergence in energy norm and quadratic convergence in the $L^2$-norm is obtained independently of the ...

A Delta-Regularization Finite Element Method for a Double Curl Problem with Divergence-Free Constraint

To deal with the divergence-free constraint in a double curl problem, ${\rm curl\,} \mu^{-1} {\rm curl\,} u=f$ and ${\rm div\,} \varepsilon u=0$ in $\Omega$, where $\mu$ and $\varepsilon$ represent the physical properties of the materials occupying $\Omega$, we develop a $\delta$-regularization method, ${\rm curl\,} \mu^{-1} {\rm curl\,} u_\delta +\delta \varepsilon u_\delta=f$, to completely ignore the divergence-free constraint ${\rm div\,} \varepsilon u=0$. We show that $u_\delta$ converges to $u$ in $H({\rm curl\,};\Omega)$ norm as $\delta\rightarrow 0$. The edge finite element method is then analyzed for solving $u_\delta$. With the finite element solution $u_{\delta,h}$, a quasioptimal error bound in the $H({\rm curl\,};\Omega)$ norm is obtained between $u$ and $u_{\delta,h}$, including a uniform (with respect to $\delta$) stability of $u_{\delta,h}$ in the $H({\rm curl\,};\Omega)$ norm. All the theoretical analysis is done in a general setting, where $\mu$ and $\varepsilon$ may be discontinuous, anisotropic, and inhomogeneous, and the solution may have a very low piecewise regularity on each material subdomain $\Omega_j$ with $u,{\rm curl\,} u\in (H^r(\Omega_j))^3$ for some $0<r<1$, where $r$ may not be greater than $1/2$. To establish the uniform stability and the error bound for $r\le 1/2$, we have respectively developed a new theory for the $\mathscr{K}_h$ ellipticity (related to mixed methods) and a new theory for the Fortin interpolation operator. Numerical results confirm the theory.

Mesh of Linear Arrays for Template Matching

This paper presents the architecture and the implementation of template matching on a 3-D piece-wise regular processor space that forms a two-dimensional array of linear systolic arrays. Template matching can be considered as a 2-D convolution of an image of sizeN × Nwith a kernel of sizer× r. Conventional high-speed implementations use 2-D systolic arrays of sizeO(r2) which compute inO(N2) time. The drawback of this solution is that the size of the processor array follows on the size of the convolution kernel. This does not permit the allocation of more processors in order to meet the real-time requirements. With the approach used in this paper, the size of the processor array may be extended up toO(sr2), 1 ≤s≤N, thereby accomplishing the calculations inO(N2/s) time. In the case whens=r, ther × rmesh of 1-D systolic arrays of sizeO(r) is yielded. The piecewise regularity of the 3-D processor array allows also easy physical realization.