175 – Quiz 6

Here is quiz 6.

175 – Self test and Extra credit problems

Here are the two parts of the test (part 1, part 2), in case you want a blank copy. As I mentioned, they do not include all topics; they mostly cover material related to chapter 7 of the book, and even then, they are not comprehensive (for example, the second part does not include parts or word problems), but I hope you found them helpful in identifying topics that may require further study or review.

Here are two extra credit problems. Please let me know if you need me to clarify either one.

502 – Equivalents of the axiom of choice

The goal of this note is to show the following result:

Theorem 1 The following statements are equivalent in

 

1. The axiom of choice: Every set can be well-ordered.
2. Every collection of nonempty set admits a choice function, i.e., if for all then there is such that for all
3. Zorn’s lemma: If is a partially ordered set with the property that every chain has an upper bound, then has maximal elements.
4. Any family of pairwise disjoint nonempty sets admits a selector, i.e., a set such that for all in the family.
5. Any set is a well-ordered union of finite sets of bounded size, i.e., for every set there is a natural an ordinal and a function such that for all and
6. Tychonoff’s theorem: The topological product of compact spaces is compact.
7. Every vector space (over any field) admits a basis.

598 – Upcoming talk: Grady Wright

Grady Wright, Wed. November 18, 2:40-3:30 pm, MG 120.

Scattered Node Finite Difference-Type Formulas Generated from Radial Basis Functions with Applications

In the finite difference (FD) method for solving partial differential equations (PDEs), derivatives at a node are approximated by a weighted sum of function values at some surrounding nodes. In the one dimensional case, the weights of the FD formulas are conveniently computed using polynomial interpolation. These one dimensional formulas can be combined to create FD formulas for partial derivatives in two and higher dimensions. This strategy, however, requires that the nodes of the FD “stencils” are situated on some kind of structured grid (or collection of structured grids), which severely limits the application of the FD method to PDEs in irregular geometries.  In this talk, we present a novel approach that resolves this problem by allowing the nodes of the FD stencils to be placed freely and by using radial basis function (RBF) interpolation for computing the corresponding weights in the scattered node FD-type formulas. We show how this RBF approach can exactly reproduce all classical FD formulas and how compact FD formulas can be generalized to scattered nodes and RBFs. This latter result is important in that it allows the number of nodes in the stencils to remain relatively low without sacrificing accuracy.  For the Poisson equation, these new compact scattered node schemes can also be made diagonally dominant, which ensures both a high degree of robustness and applicability of iterative methods. We conclude the talk with some numerical examples and future applications of the method for geophysical problems.

502 – Cantor-Bendixson derivatives

Given a topological space and a set let be the set of accumulation points of i.e., those points of such that any open neighborhood of meets in an infinite set.

Suppose that is closed. Then Define for closed compact by recursion: and for limit. Note that this is a decreasing sequence, so that if we set there must be an such that for all  

[The sets are the Cantor-Bendixson derivatives of In general, a derivative operation is a way of associating to sets some kind of ``boundary.'']

502 – The Löwenheim-Skølem theorem

In this note I sketch the proof of the Löwenheim-Skølem (or Löwenheim-Skølem-Tarski) theorem for first order theories. This basic result of model theory is really a consequence of a set theoretic combinatorial lemma, as the proof will demonstrate.

Let be a first order language, understood as a set of constant, function, and relation symbols. Let

so is unless is finite, in which case we take Talking about rather than simplifies the presentation slightly.

The Löwenheim-Skølem theorem is concerned with the possible infinite sizes of models of first order theories. Of course, a theory could only have finite models; the result does not say anything about if that is the case.

Theorem 1 If is a first order theory in a language and there is at least one infinite model of then there are models of of size for all

We will prove a more precise statement. Before stating it, note that it is possible to have a theory in some uncountable language such that has models of certain infinite sizes, but not all. Theorem 1 does not say anything about infinite models of of size What cardinals in this range are the possible sizes of models of is actually a rather difficult problem, and we will not address it.

175 – Quiz 5

Here is quiz 5.

598 – Upcoming talk: Leming Qu

Leming Qu, Wed. November 11, 2:40-3:30 pm, MG 120.

Wavelet Image Restoration and Regularization Parameters Selection

For the restoration of an image based on its noisy distorted observations, we propose wavelet domain restoration by a scale-dependent penalized regularization method (WaveRSL1). The data-adaptive choice of the regularization parameters is based on the Akaike Information Criterion (AIC) and the degrees of freedom (df) are estimated by the number of nonzero elements in the solution. Experiments on some commonly used testing images illustrate that the proposed method possesses good empirical properties.

175 – Midterm 2

Here is the second midterm.

598 – Upcoming talk: Jodi Mead

Jodi Mead, Wed. November 4, 2:40-3:30 pm, MG 120.

Non-smooth Solutions to Least Squares Problems

In an attempt to overcome the ill-posedness or ill-conditioning of inverse problems, regularization methods are implemented by introducing assumptions on the solution.  Common regularization methods include total variation, L-curve, Generalized Cross Validation (GCV), and the discrepancy principle. It is generally accepted that all of these approaches except total variation unnecessarily smooth solutions, mainly because the regularization operator is in . Alternatively, statistical approaches to ill-posed problems typically involve specifying a priori information about the parameters in the form of Bayesian inference. These approaches can be more accurate than typical regularization methods because the regularization term is weighted with a matrix rather than a constant. The drawback is that the matrix weight requires information that is typically not available or is expensive to calculate.

The method developed by the author and colleagues can be viewed as a regularization method that uses statistical information to find matrices to weight the regularization term.  We will demonstrate that unique and simple  solutions found by this method do not unnecessarily smooth solutions when the regularization term is accurately weighted with a diagonal matrix.