Computation of differential operators in wavelet coordinates

In [Found. Comput. Math., 2 (2002), pp. 203-245], Cohen, Dahmen, and DeVore proposed an adaptive wavelet algorithm for solving general operator equations. Assuming that the operator defines a boundedly invertible mapping between a Hilbert space and its dual, and that a Riesz basis of wavelet type for this Hilbert space is available, the operator equation is transformed into an equivalent well-posed infinite matrix-vector system. This system is solved by an iterative method, where each application of the infinite stiffness matrix is replaced by an adaptive approximation. It was shown that if the errors of the best linear combinations from the wavelet basis with terms are for some , which is determined by the Besov regularity of the solution and the order of the wavelet basis, then approximations yielded by the adaptive method with terms also have errors of . Moreover, their computation takes only operations, provided , with being a measure of how well the infinite stiffness matrix with respect to the wavelet basis can be approximated by computable sparse matrices. Under appropriate conditions on the wavelet basis, for both differential and singular integral operators and for the relevant range of , in [SIAM J. Math. Anal., 35(5) (2004), pp. 1110-1132] we showed that , assuming that each entry of the stiffness matrix is exactly available at unit cost.

Generally these entries have to be approximated using numerical quadrature. In this paper, restricting ourselves to differential operators, we develop a numerical integration scheme that computes these entries giving an additional error that is consistent with the approximation error, whereas in each column the average computational cost per entry is . As a consequence, we can conclude that the adaptive wavelet algorithm has optimal computational complexity.

     

Analysis of a General Family of Regularized Navier–Stokes and MHD Models

We consider a general family of regularized Navier–Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n≥2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier–Stokes equations, the Navier–Stokes-α model, the Leray-α model, the modified Leray-α model, the simplified Bardina model, the Navier–Stokes–Voight model, the Navier–Stokes-α-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. This family of models has become particularly important in the development of mathematical and computational models of turbulence. We give a unified analysis of the entire three-parameter family of models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the α→0 limit in α models. Next, we show existence of a global attractor for the general model, and then give estimates for the dimension of the global attractor and the number of degrees of freedom in terms of a generalized Grashof number. We then establish some results on determining operators for the two distinct subfamilies of dissipative and non-dissipative models. We finish by deriving some new length-scale estimates in terms of the Reynolds number, which allows for recasting the Grashof number-based results into analogous statements involving the Reynolds number. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, and determining operators for the well-known specific members of this family of regularized Navier–Stokes and MHD models, the framework we develop also makes possible a number of new results for all models in the general family, including some new results for several of the well-studied models. Analyzing the more abstract generalized model allows for a simpler analysis that helps bring out the core common structure of the various regularized Navier–Stokes and magnetohydrodynamics models, and also helps clarify the common features of many of the existing and new results. To make the paper reasonably self-contained, we include supporting material on spaces involving time, Sobolev spaces, and Grönwall-type inequalities.     

Plant Coumarins. 3. (+)-PTeryxin from <Emphasis Type="Italic">Peucedanum terebinthaceum</Emphasis>

The potentially medically valuable pyranocoumarin (+)-pteryxin was isolated for the first time from Peucedanum terebinthaceum Fischer et Turcz. The structure of (+)-pteryxin was rigorously proved using mass spectrometry, NMR, IR, and UV spectroscopy and comparison of the spectral characteristics of this compound and its basic hydrolysis products (+)-cis- and (−)-trans-khellactone and angelic and acetic acids. Translated from Khimiya Prirodnykh Soedinenii, No. 5, pp. 468–470, September-October, 2008.     

Rough Solutions of the Einstein Constraints on Closed Manifolds without Near-CMC Conditions

We consider the conformal decomposition of Einstein’s constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of non-CMC weak solutions using a combination of priori estimates for the individual Hamiltonian and momentum constraints, barrier constructions and fixed-point techniques for the Hamiltonian constraint, Riesz-Schauder theory for the momentum constraint, together with a topological fixed-point argument for the coupled system. Although we present general existence results for non-CMC weak solutions when the rescaled background metric is in any of the three Yamabe classes, an important new feature of the results we present for the positive Yamabe class is the absence of the near-CMC assumption, if the freely specifiable part of the data given by the traceless-transverse part of the rescaled extrinsic curvature and the matter fields are sufficiently small, and if the energy density of matter is not identically zero. In this case, the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, so that it can be arbitrarily far from constant, giving what is apparently the first existence results for non-CMC solutions without the near-CMC assumption. Using a coupled topological fixed-point argument that avoids near-CMC conditions, we establish existence of coupled non-CMC weak solutions with (positive) conformal factor ϕ ∈ W ^s,p , where p ∈ (1,∞) and s(p) ∈ (1 + 3/p,∞). In the CMC case, the regularity can be reduced to p ∈ (1,∞) and s(p) ∈ (3/p, ∞) ∩ [1,∞). In the case of s = 2, we reproduce the CMC existence results of Choquet-Bruhat [10], and in the case p = 2, we reproduce the CMC existence results of Maxwell [33], but with a proof that goes through the same analysis framework that we use to obtain the non-CMC results. The non-CMC results on closed manifolds here extend the 1996 non-CMC result of Isenberg and Moncrief in three ways: (1) the near-CMC assumption is removed in the case of the positive Yamabe class; (2) regularity is extended down to the maximum allowed by the background metric and the matter; and (3) the result holds for all three Yamabe classes. This last extension was also accomplished recently by Allen, Clausen and Isenberg, although their result is restricted to the near-CMC case and to smoother background metrics and data. Supported in part by NSF Awards 0715146, 0411723, and 0511766, and DOE Awards DE-FG02-05ER25707 and DE-FG02-04ER25620. Supported in part by NSF Awards 0715146 and 0411723.     

An optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems

In this paper, we modify the adaptive wavelet algorithm from Gantumur et al. [An optimal adaptive wavelet method without coarsening of the iterands, Technical Report 1325, Department of Mathematics, Utrecht University, March 2005, Math. Comp., to appear] so that it applies directly, i.e., without forming the normal equation, not only to self-adjoint elliptic operators but also to operators of the form L=A+B$L=A+B$, where A is self-adjoint elliptic and B is compact, assuming that the resulting operator equation is well posed. We show that the algorithm has optimal computational complexity.     

Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics

We establish new existence results for the Einstein constraint equations for mean extrinsic curvature arbitrarily far from constant. The results hold for rescaled background metric in the positive Yamabe class, with freely specifiable parts of the data sufficiently small, and with matter energy density not identically zero. Two technical advances make these results possible: A new topological fixed-point argument without smallness conditions on spatial derivatives of the mean extrinsic curvature, and a new global supersolution construction for the Hamiltonian constraint that is similarly free of such conditions. The results are presented for strong solutions on closed manifolds, but also hold for weak solutions and for compact manifolds with boundary. These results are apparently the first that do not require smallness conditions on spatial derivatives of the mean extrinsic curvature.     

An optimal adaptive wavelet method without coarsening of the iterands

In this paper, an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from [Math. Comp, 70 (2001), pp. 27-75] by Cohen, Dahmen and DeVore, in the sense that there is no recurrent coarsening of the iterands. Despite this, it will be shown that the method has optimal computational complexity. Numerical results for a simple model problem indicate that the new method is more efficient than an existing alternative adaptive wavelet method.

     

Olefins and Vinyl Polar Monomers: Bridging the Gap for Next Generation Materials

The inherent differences in reactivity between activated and non‐activated alkenes prevents copolymerization using established polymer synthesis techniques. Research over the past 20 years has greatly advanced the copolymerization of polar vinyl monomers and olefins. This Review highlights the challenges associated with conventional polymerization systems and evaluates the most relevant methods which have been developed to “bridge the gap” between polar vinyl monomers and olefins. We discuss advancements in heteroatom tolerant coordination–insertion polymerizations, methods of controlling radical polymerizations to incorporate olefinic monomers, as well as combined approaches employing sequential polymerizations. Finally, we discuss state‐of‐the‐art stimuli‐responsive systems capable of facile switching between catalytic pathways and provide an outlook towards applications in which tailored copolymers are ideally suited.     

GRP78‐Targeted HPMA Copolymer‐Photosensitizer Conjugate for Hyperthermia‐Induced Enhanced Uptake and Cytotoxicity in MCF‐7 Breast Cancer Cells

Photodynamic therapy (PDT) is a promising cancer treatment approach. However, the photosensitizers (PS) used for PDT are often limited by their poor solubility and selectivity for tumors. The goal of this study is to improve water solubility and delivery of the photosensitizer 2‐[1‐hexyloxyethyl]‐2‐divinyl pyropheophorbide‐a (HPPH) to breast cancer cells. An N‐(2‐hydroxypropyl)methacrylamide (HPMA) copolymer–HPPH photosensitizer conjugate is synthesized with heat shock receptor glucose‐regulated protein 78 (GRP78), targeting to GRP78 receptors of MCF‐7 cells, which are upregulated under mild hyperthermia. It is found that the uptake of the GRP78 targeted pep‐HPMA‐HPPH copolymer conjugate in MCF‐7 cells is improved through heat induction. Under mild hyperthermia the targeted copolymers are more effective compared to free HPPH. These results show potential for the utility of mild hyperthermia and copolymer delivery vehicles to enhance the efficacy of photodynamic therapy.     

Convergence Rates of Adaptive Methods,Besov Spaces,and Multilevel Approximation

This paper concerns characterizations of approximation classes associated with adaptive finite element methods with isotropic h-refinements. It is known from the seminal work of Binev, Dahmen, DeVore and Petrushev that such classes are related to Besov spaces. The range of parameters for which the inverse embedding results hold is rather limited, and recently, Gaspoz and Morin have shown, among other things, that this limitation disappears if we replace Besov spaces by suitable approximation spaces associated with finite element approximation from uniformly refined triangulations. We call the latter spaces multievel approximation spaces and argue that these spaces are placed naturally halfway between adaptive approximation classes and Besov spaces, in the sense that it is more natural to relate multilevel approximation spaces with either Besov spaces or adaptive approximation classes, than to go directly from adaptive approximation classes to Besov spaces. In particular, we prove embeddings of multilevel approximation spaces into adaptive approximation classes, complementing the inverse embedding theorems of Gaspoz and Morin. Furthermore, in the present paper, we initiate a theoretical study of adaptive approximation classes that are defined using a modified notion of error, the so-called total error, which is the energy error plus an oscillation term. Such approximation classes have recently been shown to arise naturally in the analysis of adaptive algorithms. We first develop a sufficiently general approximation theory framework to handle such modifications, and then apply the abstract theory to second-order elliptic problems discretized by Lagrange finite elements, resulting in characterizations of modified approximation classes in terms of memberships of the problem solution and data into certain approximation spaces, which are in turn related to Besov spaces. Finally, it should be noted that throughout the paper we paid equal attention to both conforming and non-conforming triangulations.