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60
Data networks as cascades: Investigating the multifractal nature of Internet WAN traffic
, 1998
"... In apparent contrast to the welldocumented selfsimilar (i.e., monofractal) scaling behavior of measured LAN traffic, recent studies have suggested that measured TCP/IP and ATM WAN traffic exhibits more complex scaling behavior, consistent with multifractals. To bring multifractals into the realm o ..."
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Cited by 192 (12 self)
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In apparent contrast to the welldocumented selfsimilar (i.e., monofractal) scaling behavior of measured LAN traffic, recent studies have suggested that measured TCP/IP and ATM WAN traffic exhibits more complex scaling behavior, consistent with multifractals. To bring multifractals into the realm of networking, this paper provides a simple construction based on cascades (also known as multiplicative processes) that is motivated by the protocol hierarchy of IP data networks. The cascade framework allows for a plausible physical explanation of the observed multifractal scaling behavior of data traffic and suggests that the underlying multiplicative structure is a traffic invariant for WAN traffic that coexists with selfsimilarity. In particular, cascades allow us to refine the previously observed selfsimilar nature of data traffic to account for local irregularities in WAN traffic that are typically associated with networking mechanisms operating on small time scales, such as TCP flo...
Triple Product Wavelet Integrals for AllFrequency Relighting
, 2004
"... This paper focuses on efficient rendering based on precomputed light transport, with realistic materials and shadows under allfrequency direct lighting such as environment maps. The basic difficulty is representation and computation in the 6D space of light direction, view direction, and surface po ..."
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Cited by 92 (9 self)
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This paper focuses on efficient rendering based on precomputed light transport, with realistic materials and shadows under allfrequency direct lighting such as environment maps. The basic difficulty is representation and computation in the 6D space of light direction, view direction, and surface position. While imagebased and synthetic methods for realtime rendering have been proposed, they do not scale to high sampling rates with variation of both lighting and viewpoint. Current approaches are therefore limited to lower dimensionality (only lighting or viewpoint variation, not both) or lower sampling rates (low frequency lighting and materials) . We propose a new mathematical and computational analysis of precomputed light transport. We use factored forms, separately precomputing and representing visibility and material properties. Rendering then requires computing triple product integrals at each vertex, involving the lighting, visibility and BRDF. Our main contribution is a general analysis of these triple product integrals, which are likely to have broad applicability in computer graphics and numerical analysis. We first determine the computational complexity in a number of bases like point samples, spherical harmonics and wavelets. We then give efficient linear and sublineartime algorithms for Haar wavelets, incorporating nonlinear wavelet approximation of lighting and BRDFs. Practically, we demonstrate rendering of images under new lighting and viewing conditions in a few seconds, significantly faster than previous techniques.
Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 72 (12 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
The Curvelet Representation of Wave Propagators is Optimally Sparse
, 2004
"... This paper argues that curvelets provide a powerful tool for representing very general linear symmetric systems of hyperbolic differential equations. Curvelets are a recently developed multiscale system [10, 7] in which the elements are highly anisotropic at fine scales, with effective support shape ..."
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Cited by 60 (13 self)
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This paper argues that curvelets provide a powerful tool for representing very general linear symmetric systems of hyperbolic differential equations. Curvelets are a recently developed multiscale system [10, 7] in which the elements are highly anisotropic at fine scales, with effective support shaped according to the parabolic scaling principle width ≈ length 2 at fine scales. We prove that for a wide class of linear hyperbolic differential equations, the curvelet representation of the solution operator is both optimally sparse and well organized. • It is sparse in the sense that the matrix entries decay nearly exponentially fast (i.e. faster than any negative polynomial), • and wellorganized in the sense that the very few nonnegligible entries occur near a few shifted diagonals. Indeed, we show that the wave group maps each curvelet onto a sum of curveletlike waveforms whose locations and orientations are obtained by following the different Hamiltonian flows—hence the diagonal shifts in the curvelet representation. A physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time, with enough spatial localization so that they simultaneously behave like particles.
Multilinear Calderón Zygmund theory
 ADV. IN MATH. 40
, 1996
"... A systematic treatment of multilinear CalderónZygmundoperators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, the multilinear T1 theorem, anda variety of results regarding multilinear multiplier operators. ..."
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Cited by 46 (16 self)
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A systematic treatment of multilinear CalderónZygmundoperators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, the multilinear T1 theorem, anda variety of results regarding multilinear multiplier operators.
PSEUDODIFFERENTIAL OPERATORS WITH GENERALIZED SYMBOLS AND REGULARITY THEORY
, 2005
"... We study pseudodifferential operators with amplitudes aε(x, ξ) depending on a singular parameter ε → 0 with asymptotic properties measured by different scales. We prove, taking into account the asymptotic behavior for ε → 0, refined versions of estimates for classical pseudodifferential operators. W ..."
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Cited by 13 (9 self)
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We study pseudodifferential operators with amplitudes aε(x, ξ) depending on a singular parameter ε → 0 with asymptotic properties measured by different scales. We prove, taking into account the asymptotic behavior for ε → 0, refined versions of estimates for classical pseudodifferential operators. We apply these estimates to nets of regularizations of exotic operators as well as operators with amplitudes of low regularity, providing a unified method for treating both classes. Further, we develop a full symbolic calculus for pseudodifferential operators acting on algebras of Colombeau generalized functions. As an application, we formulate a sufficient condition of hypoellipticity in this setting, which leads to regularity results for generalized pseudodifferential equations.
An approximate wavelet MLE of short and long memory parameters
 Studies in Nonlinear Dynamics and Econometrics
, 1999
"... Abstract. By design a wavelet's strength rests in its ability to localize a process simultaneously in timescale space. The wavelet's ability to localize a time series in timescale space directly leads to the computational e ciency of the wavelet representation of a N N matrix operator by allowing ..."
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Cited by 11 (3 self)
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Abstract. By design a wavelet's strength rests in its ability to localize a process simultaneously in timescale space. The wavelet's ability to localize a time series in timescale space directly leads to the computational e ciency of the wavelet representation of a N N matrix operator by allowing the N largest elements of the wavelet represented operator to represent the matrix operator [Devore, et al. (1992a) and (1992b)]. This property allows many dense matrices to have sparse representation when transformed by wavelets. In this paper we generalize the longmemory parameter estimator of McCoy and Walden (1996) to estimate simultaneously the short and longmemory parameters. Using the sparse wavelet representation of a matrix operator, we are able to approximate an ARFIMA models likelihood function with the series's wavelet coe cients and their variances. Maximization of this approximate likelihood function over the short and longmemory parameter space results in the approximate wavelet maximum likelihood estimates of the ARFIMA model. By simultaneously maximizing the likelihood function over both the short and longmemory parameters and using only the wavelet coe cient's variances, the approximate wavelet MLE provides a fast alternative to the frequencydomain MLE. Furthermore, the simulation studies found herein reveal the approximate wavelet MLE to be robust over the invertible parameter region of the ARFIMA model's moving average parameter, whereas the frequencydomain MLE dramatically deteriorates as the moving average parameter approaches the boundaries of invertibility.
Discrete decompositions for bilinear operators and almost diagonal conditions
 TRANS. AMER. MATH. SOC
, 1998
"... Using discrete decomposition techniques,bilinear operators are naturally associated with trilinear tensors. An intrinsic size condition on the entries of such tensors is introduced and is used to prove boundedness for the corresponding bilinear operators on several products of function spaces. This ..."
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Cited by 10 (6 self)
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Using discrete decomposition techniques,bilinear operators are naturally associated with trilinear tensors. An intrinsic size condition on the entries of such tensors is introduced and is used to prove boundedness for the corresponding bilinear operators on several products of function spaces. This condition should be considered as the direct analogue of an almost diagonal condition for linear operators of CalderónZygmund type. Applications include a reduced T 1 theorem for bilinear pseudodifferential operators and the extension of an L p multiplier result of Coifman and Meyer to the full range of H p spaces. The results of this article rely on decomposition techniques developed by Frazier and Jawerth and on the vector valued maximal estimate of Fefferman and Stein.
Condition Number Estimates for Combined Potential Boundary Integral Operators in Acoustic Scattering
"... We study the classical combined field integral equation formulations for timeharmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to BrakhageWerner/Leis/Panič, and the direct formulation associated with the names of Burton and Miller. We obtain lower a ..."
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Cited by 9 (5 self)
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We study the classical combined field integral equation formulations for timeharmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to BrakhageWerner/Leis/Panič, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single and doublelayer potential operators.
ON THE H 1 –L 1 BOUNDEDNESS OF OPERATORS
"... Abstract. We prove that if q is in (1, ∞), Y is a Banach space, and T is a linear operator defined on the space of finite linear combinations of (1,q)atoms in Rn with the property that sup{‖Ta‖Y: a is a (1,q)atom} < ∞, then T admits a (unique) continuous extension to a bounded linear operator from ..."
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Cited by 9 (0 self)
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Abstract. We prove that if q is in (1, ∞), Y is a Banach space, and T is a linear operator defined on the space of finite linear combinations of (1,q)atoms in Rn with the property that sup{‖Ta‖Y: a is a (1,q)atom} < ∞, then T admits a (unique) continuous extension to a bounded linear operator from H1 (Rn)toY. We show that the same is true if we replace (1,q)atoms by continuous (1, ∞)atoms. This is known to be false for (1, ∞)atoms. 1.