Upcoming Seminars

Title: Iterative regularization methods for large-scale linear inverse problems
Seminar: Numerical Analysis and Scientific Computing
Speaker: Silvia Gazzola of University of Bath
Contact: James Nagy, jnagy@emory.edu
Date: 2019-08-27 at 2:00PM
Venue: MSC W301
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Abstract:
Inverse problems are ubiquitous in many areas of Science and Engineering and, once discretized, they lead to ill-conditioned linear systems, often of huge dimensions: regularization consists in replacing the original system by a nearby problem with better numerical properties, in order to find a meaningful approximation of its solution. After briefly surveying some standard regularization methods, both iterative (such as many Krylov methods) and direct (such as Tikhonov method), this talk will introduce a recent class of methods that merge an iterative and a direct approach to regularization. In particular, strategies for choosing the regularization parameter and the regularization matrix will be emphasized, eventually leading to the computation of approximate solutions of Tikhonov problems involving a regularization term expressed in some p-norms.
Title: Modular linear differential equations
Seminar: Algebra
Speaker: Kiyokazu Nagatomo of Osaka University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-09-03 at 4:00PM
Venue: MSC W303
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Abstract:
The most naive definition of \textit{modular linear differential equations} (MLDEs) would be linear differential equations whose space of solutions are invariant under the weight $k$ slash action of $\Gamma_1=SL_2(\mathbb{Z})$, for some $k$. Then under an analytic condition for coefficients functions and the Wronskians of a~basis of the space of solutions of equations, we have (obvious) expressions of MLDEs as: \[ L(f) \,=\,\mathfrak{d}_k^n(f)+\sum_{i=2}^nP_{2i}\mathfrak{d}_k^{n-i}(f) \] where $P_{2i}$ is a modular form of of weight $2i$ on $SL_2(\mathbb{Z})$ and $\mathfrak{d}_k(f)$ is the \textit{Serre derivative}. (We could replace $\Gamma$ by a Fuchsian subgroup of $SL_2(\mathbb{R})$ and allow the modular forms $P_{2i}$ to be meromorphic.) However, the iterated Serre derivative $\mathfrak{d}_k^n(f)$ (called a ``higher Serre derivation'' because as an operator it preserves modulality) is very complicated since it involves the Eisenstein series $E_2$. MLDEs, of course, can be given in the form % \[ % \mathsf{L}(f) \,=\, D^n(f)+\sum_{i=1}^nQ_iD^i(f)\quad\text{where $D=\frac{1}{2\pi\sqrt{-1}}\frac{d}{d\tau}$.} % \] \[ \mathsf{L}(f) \,=\, D^n(f)+\sum_{i=1}^nQ_iD^i(f) \] where \[ D=\frac{1}{2\pi\sqrt{-1}}\frac{d}{d\tau}. \] Then it is not easy to know if the equation above is an MLDE except the fact that $Q_i$ are quasimodular forms. Very recently, Y.~Sakai and D.~Zagier (my collaborators) found formulas of $\mathsf{L}(f)$ by using the Rankin--Cohen products between $f$ and $g_i$. This is a modular form of weight $2i$, which is a linear function of the differential of~$Q_{j}$. Moreover, there are \textit{inversion formulas} which express $Q_i$ as a linear function of the derivatives of $g_{j}$. The most important fact is that the order $n$ and $n-1$ parts are equal to the so-called higher Serre derivative in the sense of Kaneko and Koike, where the group is $\Gamma_1$. (This holds for any Fuchsian group.) \\ Finally, the most important nature of my talk is that I will use a \textbf{blackboard} instead of \textbf{slides}ss.