Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough ($L^\infty$) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution $H$) minimizing the $L^2$ norm of the source terms; its (pre-)computation involves minimizing $\mathcal{O}(H^{-d})$ quadratic (cell) problems on (super-)localized sub-domains of size $\mathcal{O}(H \ln (1/ H))$. The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for $d\leq 3$, and polyharmonic for $d\geq 4$, for the operator -\diiv(a\nabla \cdot) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method ($\mathcal{O}(H)$ in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincar\'{e} inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.Comment: ESAIM: Mathematical Modelling and Numerical Analysis. Special issue (2013

Analysis of Adjoint Error Correction for Superconvergent Functional Estimates

Earlier work introduced the notion of adjoint error correction for obtaining superconvergent estimates of functional outputs from approximate PDE solutions. This idea is based on a posteriori error analysis suggesting that the leading order error term in the functional estimate can be removed by using an adjoint PDE solution to reveal the sensitivity of the functional to the residual error in the original PDE solution. The present work provides a priori error analysis that correctly predicts the behaviour of the remaining leading order error term. Furthermore, the discussion is extended from the case of homogeneous boundary conditions and bulk functionals, to encompass the possibilities of inhomogeneous boundary conditions and boundary functionals. Numerical illustrations are provided for both linear and nonlinear problems.\ud \ud This research was supported by EPSRC under grant GR/K91149, and by NASA/Ames Cooperative Agreement No. NCC 2-5431

Univariate spline quasi-interpolants and applications to numerical analysis

We describe some new univariate spline quasi-interpolants on uniform partitions of bounded intervals. Then we give some applications to numerical analysis: integration, differentiation and approximation of zeros

An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS

We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C1-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching

Local interpolation schemes for landmark-based image registration: a comparison

In this paper we focus, from a mathematical point of view, on properties and performances of some local interpolation schemes for landmark-based image registration. Precisely, we consider modified Shepard's interpolants, Wendland's functions, and Lobachevsky splines. They are quite unlike each other, but all of them are compactly supported and enjoy interesting theoretical and computational properties. In particular, we point out some unusual forms of the considered functions. Finally, detailed numerical comparisons are given, considering also Gaussians and thin plate splines, which are really globally supported but widely used in applications

B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness

Let $\xi = \{x^j\}_{j=1}^n$ be a grid of $n$ points in the $d$-cube {\II}^d:=[0,1]^d, and $\Phi = \{\phi_j\}_{j =1}^n$ a family of $n$ functions on {\II}^d. We define the linear sampling algorithm $L_n(\Phi,\xi,\cdot)$ for an approximate recovery of a continuous function $f$ on {\II}^d from the sampled values $f(x^1), ..., f(x^n)$, by $$L_n(\Phi,\xi,f)\ := \ \sum_{j=1}^n f(x^j)\phi_j$$. For the Besov class $B^\alpha_{p,\theta}$ of mixed smoothness $\alpha$ (defined as the unit ball of the Besov space \MB), to study optimality of $L_n(\Phi,\xi,\cdot)$ in L_q({\II}^d) we use the quantity $$r_n(B^\alpha_{p,\theta})_q \ := \ \inf_{H,\xi} \ \sup_{f \in B^\alpha_{p,\theta}} \, \|f - L_n(\Phi,xi,f)\|_q$$, where the infimum is taken over all grids $\xi = \{x^j\}_{j=1}^n$ and all families $\Phi = \{\phi_j\}_{j=1}^n$ in L_q({\II}^d). We explicitly constructed linear sampling algorithms $L_n(\Phi,\xi,\cdot)$ on the grid \xi = \ G^d(m):= \{(2^{-k_1}s_1,...,2^{-k_d}s_d) \in \II^d : \ k_1 + ... + k_d \le m\}, with $\Phi$ a family of linear combinations of mixed B-splines which are mixed tensor products of either integer or half integer translated dilations of the centered B-spline of order $r$. The grid $G^d(m)$ is of the size $2^m m^{d-1}$ and sparse in comparing with the generating dyadic coordinate cube grid of the size $2^{dm}$. For various $0<p,q,\theta \le \infty$ and $1/p < \alpha < r$, we proved upper bounds for the worst case error $\sup_{f \in B^\alpha_{p,\theta}} \, \|f - L_n(\Phi,\xi,f)\|_q$ which coincide with the asymptotic order of $r_n(B^\alpha_{p,\theta})_q$ in some cases. A key role in constructing these linear sampling algorithms, plays a quasi-interpolant representation of functions $f \in B^\alpha_{p,\theta}$ by mixed B-spline series