398 research outputs found
A multi-level algorithm for the solution of moment problems
We study numerical methods for the solution of general linear moment
problems, where the solution belongs to a family of nested subspaces of a
Hilbert space. Multi-level algorithms, based on the conjugate gradient method
and the Landweber--Richardson method are proposed that determine the "optimal"
reconstruction level a posteriori from quantities that arise during the
numerical calculations. As an important example we discuss the reconstruction
of band-limited signals from irregularly spaced noisy samples, when the actual
bandwidth of the signal is not available. Numerical examples show the
usefulness of the proposed algorithms
Use of Anisotropic Radial Basis Functions
νμλ
Όλ¬Έ(λ°μ¬) -- μμΈλνκ΅λνμ : μμ°κ³Όνλν ν΅κ³νκ³Ό, 2021.8. μ€ν¬μ.Spatial inhomogeneity along the one-dimensional curve makes two-dimensional data non-stationary. Curvelet transform, first proposed by Candes and Donoho (1999), is one of the most well-known multiscale methods to represent the directional singularity, but it has a limitation in that the data needs to be observed on equally-spaced sites. On the other hand, radial basis function interpolation is widely used to approximate the underlying function from the scattered data. However, the isotropy of the radial basis functions lowers the efficiency of the directional representation. This thesis proposes a new multiscale method that uses anisotropic radial basis functions to efficiently represent the direction from the noisy scattered data in two-dimensional Euclidean space. Basis functions are orthogonalized across the scales so that each scale can represent a global or local directional structure separately. It is shown that the proposed method is remarkable for representing directional scattered data through numerical experiments. Convergence property and practical issues in implementation are discussed as well.2μ°¨μ 곡κ°μμ κ΄μΈ‘λλ λΉμ μ μλ£λ κ·Έ 곡κ°μ λΉλμ§μ±μ΄ 1μ°¨μ 곑μ μ λ°λΌ λνλλ€. μ΄λ¬ν λ°©ν₯μ νΉμ΄μ±μ νννκΈ° μν λ€μ€μ²λ λ°©λ²λ‘ μΌλ‘λ Candes and Donoho (1999)κ° μ²μ μ μν 컀λΈλ λ³νμ΄ λ리 μλ €μ Έ μμ§λ§ μ΄λ μλ£κ° μΌμ ν κ°κ²©μΌλ‘ κ΄μΈ‘λμ΄μΌ νλ€λ μ μ½μ΄ μλ€. ννΈ μ°μ¬λ μλ£μ λ΄μ¬λ ν¨μλ₯Ό κ·Όμ¬νκΈ° μν΄μλ λ°©μ¬κΈ°μ ν¨μλ₯Ό μ΄μ©ν λ΄μ½λ²μ΄ νν μ΄μ©λμ§λ§ λ±λ°©μ±μ΄ μλ λ°©μ¬κΈ°μ ν¨μλ‘λ λ°©ν₯μ±μ ν¨μ¨μ μΌλ‘ ννν μ μλ€. λ³Έ νμλ
Όλ¬Έμμλ 2μ°¨μ μ ν΄λ¦¬λ 곡κ°μμ μ‘μκ³Ό ν¨κ» μ°μ¬λμ΄ κ΄μΈ‘λλ λ°©ν₯μ± μλ£μ ν¨μ¨μ μΈ ννμ μν΄ λΉλ±λ°©μ± λ°©μ¬κΈ°μ ν¨μλ₯Ό μ΄μ©ν μλ‘μ΄ λ€μ€μ²λ λ°©λ²λ‘ μ μ μνλ€. μ΄λ κ° μ€μΌμΌμμ μ λ°μ μΈ λ°©ν₯μ± κ΅¬μ‘°μ κ΅μμ μΈ λ°©ν₯μ± κ΅¬μ‘°λ₯Ό λΆλ¦¬νμ¬ νννκΈ° μν΄ κΈ°μ ν¨μμ μ€μΌμΌ κ° μ§κ΅νκ° μ΄λ£¨μ΄μ§λ€. μ μλ λ°©λ²μ΄ μ°μ¬λ λ°©ν₯μ± μλ£λ₯Ό νννλ λ° μμ΄ μ°μν¨μ 보μ΄κΈ° μν΄ λͺ¨μμ€νκ³Ό μ€μ μλ£μ λν μμΉμ€νμ ν κ²°κ³Όλ₯Ό μ μνμλ€. ννΈ μ μλ λ°©λ²μ μλ ΄μ±κ³Ό μ€μ ꡬν λ°©λ²μ κ΄ν μ¬μλ€λ λ€λ£¨μλ€.1 Introduction 1
2 Multiscale Analysis 4
2.1 Classical wavelet transform 5
2.1.1 Continuous wavelet transform 5
2.1.2 Multiresolution analysis 7
2.1.3 Discrete wavelet transform 10
2.1.4 Two-dimensional wavelet transform 13
2.2 Wavelets for equally-spaced directional data 14
2.2.1 Ridgelets 15
2.2.2 Curvelets 16
2.3 Wavelets for scattered data 19
2.3.1 Lifting scheme 21
2.3.2 Spherical wavelets 23
3 Radial Basis Function Approximation 26
3.1 Radial basis function interpolation 27
3.1.1 Radial basis functions and scattered data interpolation 27
3.1.2 Compactly supported radial basis functions 29
3.1.3 Error bounds 32
3.2 Multiscale representation with radial basis functions 35
3.2.1 Multiscale approximation 35
3.2.2 Error bounds 37
4 Multiscale Representation of Directional Scattered Data 41
4.1 Anisotropic radial basis function approximation 41
4.1.1 Representation of a single linear directional structure 42
4.1.2 Representation of complex directional structure 46
4.1.3 Multiscale representation of the directional structure 46
4.2 Directional wavelets for scattered data 47
4.2.1 Directional wavelets 48
4.2.2 Estimation of coefficients 49
4.2.3 Practical issues in implementation 50
5 Numerical Experiments 57
5.1 Simulation study 57
5.1.1 Scattered observation sites 60
5.1.2 Equally-spaced observation sites 69
5.2 Real data analysis 70
5.2.1 Temperature data in South Korea 70
6 Concluding Remarks 74
6.1 Summary of results 74
6.2 Future research 74
Abstract (in Korean) 82λ°
Wavelet based Adaptive RBF Method for Nearly Singular Poisson-Type Problems on Irregular Domains
We present a wavelet based adaptive scheme and investigate the efficiency of this scheme for solving nearly singular potential PDEs over irregularly shaped domains. For a problem defined over Ξ© β βd, the boundary of an irregularly shaped domain, Ξ, is defined as a boundary curve that is a product of a Heaviside function along the normal direction and a piecewise continuous tangential curve. The link between the original wavelet based adaptive method presented in Libre, Emdadi, Kansa, Shekarchi, and Rahimian (2008, 2009) or LEKSR method and the generalized one is given through the use of simple Heaviside masking procedure. In addition level dependent thresholding were introduced to improve the efficiency and convergence rate of the solution. We will show how the generalized wavelet based adaptive method can be applied for detecting nearly singularities in Poisson type PDEs over irregular domains. The numerical examples have illustrated that the proposed method is powerful to analyze the Poisson type PDEs with rapid changes in gradients and nearly singularities
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