Michihiro Nagase

Behind and beyond the Matlab ODE suite

Abstract The paper explains the concepts of order and absolute stability of numerical methods for solving systems of first-order ordinary differential equations (ODE) of the form describes the phenomenon of problem stiffness , and reviews explicit Runge-Kutta methods, and explicit and implicit linear multistep methods. It surveys the five numerical methods contained in the Matlab ODE suite (three for nonstiff problems and two for stiff problems) to solve the above system, lists the available options, and uses the odedemo command to demonstrate the methods. One stiff ode code in Matlab can solve more general equations of the form M ( t ) y ′ = f ( t , y ) provided the Mass option is on.

The Lp-boundedness of pseudo-differential operators with non-regular symbols

(1977). The Lp-boundedness of pseudo-differential operators with non-regular symbols. Communications in Partial Differential Equations: Vol. 2, No. 10, pp. 1045-1061.

Weyl quantized Hamiltonians of relativistic spinless particles in magnetic fields

Let pm(x, ξ) be the classical Hamiltonian of a relativistic particle in a magnetic field: pm(x, ξ) = [¦cξ − ea(x)¦2 + m2c4]12. We study some properties of the Weyl quantized Hamiltonian pmw(X, D). Under the assumption allowing constant magnetic fields, we prove that for m ⩾ 0, pmw(X, D) is essentially self-adjoint on l(Rd). Moreover, we give some results on the convergence of pmw(X, D) as m → 0, and on the lower bounds of the Hamiltonians.

Microlocal filtering with multiwavelets

Abstract Hyperfunctions in R n are intuitively considered as sums of boundary values of holomorphic functions defined in infinitesimal wedges in C n. Orthonormal multiwavelets, which are a generalization of orthonormal single wavelets, generate a multiresolution analysis by means of several scaling functions. Microlocal analysis is briefly reviewed and a multiwavelet system adapted to microlocal filtering is proposed. A rough estimate of the microlocal content of functions or signals is obtained from their multiwavelet expansions. A fast algorithm for multiwavelet microlocal filtering is presented and several numerical examples are considered.

Smooth tight frame wavelets and image microanalyis in the fourier domain

Abstract General results on microlocal analysis and tight frames in R 2 are summarized. To perform microlocal analysis of tempered distributions, orthogonal multiwavelets, whose Fourier transforms consist of characteristic functions of squares or sectors of annuli, are constructed in the Fourier domain and are shown to satisfy a multiresolution analysis with several choices of scaling functions. To have good localization in both the x and Fourier domains, redundant smooth tight wavelet frames, with frame bounds equal to one, called Parseval wavelet frames, are obtained in the Fourier domain by properly tapering the above characteristic functions. These nonorthogonal frame wavelets can be generated by two-scale equations from a multiresolution analysis. A natural formulation of the problem is by means of pseudodifferential operators. Singularities, which are added to smooth images, can be localized in position and direction by means of the frame coefficients of the filtered images computed in the Fourier domain. Using Plancherels theorem, the frame expansion of the filtered images is obtained in the x domain. Subtracting this expansion from the scarred images restores the original images.

Comparing Multiresolution SVD with Other Methods for Image Compression

Digital image compression with multiresolution singular value decomposition is compared with discrete cosine transform, discrete 9/7 biorthogonal wavelet transform, Karhunen–Loeve transform, and a hybrid wavelet-svd transform. Compression uses SPIHT and run-length with Huffmann coding. The performances of these methods differ little from each other. Generally, the 9/7 biorthogonal wavelet transform is superior for most images that were tested for given compression rates. But for certain block transforms and certain images other methods are slightly superior. To appear in Proc. Fourth ISAAC Congress (York University), H. Begehr, R. P. Gilbert, M. Muldoon, and M. W. Wong, eds., Kluwer.

A construction of multiwavelets

A class of r-regular multiwavelets, depending on the smoothness of the multiwavelet functions, is introduced with the appropriate notation and definitions. Oscillation properties of orthonormal systems are obtained in Lemma 1 and Corollary 1 without assuming any vanishing moments for the scaling functions, and in Theorem 1 the existence of r-regular multiwavelets in L2(Rn) is established. In Theorem 2, a particular r-regular multiresolution analysis for multiwavelets is obtained from an r-regular multiresolution analysis for uniwavelets. In Theorem 3, an r-regular multiresolution analysis of split-type multiwavelets, which are perhaps the simplest multiwavelets, is easily obtained by using an r-regular multiresolution analysis for uniwavelets and a (2n − 1)-fold regular multiresolution analysis for uniwavelets. For some split-type multiwavelets, the support or width of the wavelets is shorter than the support or width of the scaling functions without loss of regularity nor of vanishing moments. Examples of split-type multiwavelets in L2(R) are constructed and illustrated by means of figures. Symmetry and antisymmetry are preserved in the case of infinite support.

On the Cauchy problem for parabolic pseudo-differential equations

In the recent paper [8] S. Kaplan has obtained an analogue of Gardings inequality for parabolic differential operators and applied it to a Hubert space treatment of the Cauchy problem. D. Ellis [3] has extended those results to higher order parabolic differential operators (see also [4]). On the other hand in [13] the author has studied a Hubert space treatment of the Cauchy problem for parabolic pseudo-differential equations and generalized the results of S. Kaplan [8]. In the present paper we shall study the Cauchy problem for higher order parabolic pseudo-differential equations of the form

The nonrelativistic limit for the Weyl quantized Hamiltonians and pseudo-differential operators

We consider the quantum Hamiltonian H(c) associated, via the Weyl correspondence, with the classical Hamiltonian c j/| ξ — a(x) | + m c + V(x), where c is the velocity of light. We show that H(c) — mc converges to the Schr dinger operator (l/2m) £j=i (d/idXj — a^x)) 4V(x) in the strong resolvent sense. 1991 Mathematics Subject Classification: 35S99, 47G05. §

Pseudo-Differential Operators, Microlocal Analysis and Image Restoration

Pseudo-differential operators with symbols supported on sectors of dyadic annuli in the Fourier domain are used to perform microlocal analysis of tempered distributions. Microlocal analysis is recalled. The above symbols are made of smooth wavelet frames which are constructed in the Fourier domain by means of modulated smooth tapered functions. The method is used to localize a line of singularities in an image.