Optimal quadrature algorithms inH p spaces

Sikorski, K.

doi:10.1007/bf01407871

Cited by 13 publications

(5 citation statements)

References 9 publications

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“…The same estimates are reported in [16], [17], together with the claim that the trapezoidal formula is near optimal in H ∞ (D π/2 , ω). However, the estimates are derived rather indirectly by noting the fact that the study on the optimal quadrature in the space…”

Section: Error Estimatessupporting

confidence: 56%

Optimality of the double exponential formula - functional analysis approach -

Sugihara

1997

Numerische Mathematik

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In the light of the functional analysis theory we establish the optimality of the double exponential formula. The argument consists of the following three ingredients: (1) introduction of a number of spaces of functions analytic in a strip region about the real axis, each space being characterized by the decay rate of their elements (functions) in the neighborhood of the infinity; (2) proof of the (near-) optimality of the trapezoidal formula in each space introduced in (1) by showing the (near-) equality between an upper estimate for the error norm of the trapezoidal formula and a lower estimate for the minimum error norm of quadratures; (3) nonexistence theorem for the spaces, the characterizing decay rate of which is more rapid than the double exponential.

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Section: Error Estimatessupporting

confidence: 56%

Optimality of the double exponential formula - functional analysis approach -

Sugihara

1997

Numerische Mathematik

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“…Taken together, and while understanding some limitations, we still believe that modeling the crosslinked alginate polymers as rigid rods provides a computationally accessible strategy to approximate properties of the polymer system and derive hydrogel mesh properties. The model also determined cross-linking regions and correspondingly mesh size based on the egg-box model, where the α-l-guluronate (G) residues in the alginate polymer chain adopt a 2 1 helical conformation and the pair of G residues ("binding cavity") surrounds the calcium ions 60 . Previous work has shown that cross-link density significantly increases for alginate hydrogels with more calcium 16,[61][62] or more polymer content 28 .…”

Section: Discussionmentioning

confidence: 99%

Computational-Based Design of Hydrogels with Predictable Mesh Properties

Campbell

Wysoczynski

Hadley

et al. 2019

ACS Biomater. Sci. Eng.

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Hydrogel systems are an appealing class of therapeutic delivery vehicles, though it can be challenging to design hydrogels that maintain desired spatiotemporal presentation of therapeutic cargo. In this work, we propose a different approach in which computational tools are developed that creates a theoretical representation of the hydrogel polymer network to design hydrogels with predefined mesh properties critical for controlling therapeutic delivery. We postulated and confirmed that the computational model could incorporate properties of alginate polymers, including polymer content, monomer composition and polymer chain radius, to accurately predict cross-link density and mesh size for a wide range of alginate hydrogels. Additionally, the simulations provided a robust strategy to determine the mesh size distribution and identified properties to control the mesh size of alginate hydrogels. Furthermore, the model was validated for additional hydrogel systems and provided a high degree of correlation (R 2 > 0.95) to the mesh sizes determined for both fibrin and polyethylene glycol (PEG) hydrogels. Finally, a full factorial and Box-Behnken design of experiments (DOE) approach utilized in combination with the computational model predicted that the mesh size of hydrogels could be varied from approximately 5 nm to 5 μm through controlling properties of the polymer network. Overall, this computational model of the hydrogel polymer network provides a rapid and accessible strategy to predict hydrogel mesh properties and ultimately design hydrogel systems with desired mesh properties for potential therapeutic applications.

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“…We use Lemma 3 to conclude that (10) implies (9). Moreover, given (10), we have equivalence of (8) and (11).…”

Section: Finite Complexitymentioning

confidence: 79%

“…Some progress has recently been made in the worst case setting for the approximation and integration problems over R and R d ; see, respectively, Woźniakowski (2000, 2001). See also Sikorski (1982), Traub et al (1983), Curbera (1998Curbera ( , 2000, and Mathé (1998). In the present paper we study complexity of approximating functions f: R + Q R and their integrals over R + =[0, .)…”

Section: Introductionmentioning

confidence: 99%

Average Case Complexity of Weighted Approximation and Integration over R+

Plaskota

Ritter

Wasilkowski

2002

Journal of Complexity

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We study weighted approximation and integration of Gaussian stochastic processes X defined over R + whose rth derivatives satisfy a Hölder condition with exponent b in the quadratic mean. We assume that the algorithms use samples of X at a finite number of points. We study the average case (information) complexity, i.e., the minimal number of samples that are sufficient to approximate/integrate X with the expected error not exceeding e. We provide sufficient conditions in terms of the weight and the parameters r and b for the weighted approximation and weighted integration problems to have finite complexity. For approximation, these conditions are necessary as well. We also provide sufficient conditions for these complexities to be proportional to the complexities of the corresponding problems defined over [0,1], i.e., proportional to e −1/a where a=r+b for the approximation and a=r+b+1/2 for the integration. © 2002 Elsevier Science (USA)

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Optimal quadrature algorithms inH p spaces

Cited by 13 publications

References 9 publications

Optimality of the double exponential formula - functional analysis approach -

Optimality of the double exponential formula - functional analysis approach -

Computational-Based Design of Hydrogels with Predictable Mesh Properties

Average Case Complexity of Weighted Approximation and Integration over R+

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