NUMERICAL METHODS AND APPLICATIONS B
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- Assessment methods
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- Full programme
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Basic course in numerical analysis.
Oral examination and project implemented in Matlab.
Students will acquire the basic knowledge in order to model and to solve ill-posed problems.
Discrete least squares: normal equations, QR factorization and SVD.
Classification, overfitting and stability, Tikhonov regularization and introduction to machine learning. Generalized cross-validation (GCV) and bilevel optimization methods.
Ill-posed problems: regularization, filtering methods, iterative regularizing methods, variational model, sparsity in the wavelet domain.
Convolution and Fast Fourier Tranform (FFT).
Applications: compression and denoising of signals and images, reconstruction of blurred signals and images, computerized tomography.
Discrete least-square problems: minimum norm solution, singular value decomposition (SVD), truncated SVD (TSVD), pseudo-inverse, Golub-Kahan algorithm to compute the SVD, Landweber iteration. (24 h)
Ill-posed problems and regularization: filtering by TSVD, Tikhonov method, and SVD filtering. Strategies for estimating the regularization parameter: discrete Picard condition, discrepancy principle, L-curve, and GCV. Regularization iterative methods (Landweber method and conjugate gradients for normal equations). Convolution and discrete Fourier transform by fast Fourier transform. Compression, segmentation, and deconvolution of signals and images. (30 h)
Introduction to machine learning shows that the main idea of machine learning can be traced back to the regularization of an ill-posed inverse problem depending on the choice of the data fidelity measure and the predicting model. (10 h)
The numerical methods and some applications will be implemented and tested using MatLab.
Front lectures with some computer lab supplementary lessons.
Meeting by appointment.
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Degree course in: MATHEMATICS