# 세미나 및 콜로퀴엄

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For a reliable simulation of systems subject to noise, it is necessary to characterize the noise properly and develop efficient algorithms.

In the first part of this talk, I will present a numerical technique to model and simulate multiple correlated random processes. The method finds the appropriate expansion for each correlated random process by generalizing the Karhunen-Loeve (K-L) expansion, in particular, by releasing the bi-orthogonal condition of the K-L expansion. I will address the convergence and computational efficiency, in addition to some explicit formulae and analytical results.

In the second part, I will present an adaptive reduced basis method that enables an efficient simulation of parameterized stochastic PDEs. The method is employed by using an adaptive ANOVA and probabilistic collocation method to automatically identify the important dimensions and appropriate resolution in each dimension. The effectiveness of the method is demonstrated in anisotropic high-dimensional stochastic PDEs.

High-dimensionality is one of the major challenges in stochastic simulation of realistic physical systems. The most appropriate numerical scheme needs to balance accuracy and computational complexity, and it also needs to address issues such as multiple scales, lack of regularity, and long-term integration.

In this talk, I will review state-of-the-art numerical techniques for high-dimensional systems including low-rank tensor approximation, sparse grid collocation, and ANOVA decomposition. The presented numerical methods are tested and compared in the joint response-excitation PDF equation that generalizes the existing PDF equations and enables us to do kinetic simulations with non-Gaussian colored noise. The alternative to the numerical approach, I will discuss dimension reduction techniques such as Mori-Zwanzig approach and moment closures that can obtain reduced order equations in lower dimensions. I will also present numerical results including stochastic Burgers equation and Lorenz-96 system.

Based on the quantum white noise theory, we introduce the new concept of quantum white noise derivatives of white noise operators. As applications we solve implementation problems for the canonical commutation relation and for a quantum extension of Girsanov transformation.

‣ Date & Time : 7/21, 10:00~11:00, 11:10~12:10

In the second lecture we continue the discussion of orthogonal polynomials, now dealing with multi-variable functions. By introducing creation, annihilation, and preservation operators for the multi-variables, we construct again an interacting Fock space (IFS). Thereby we extend the theory of orthogonal polynomials in the 1-dimensional space to that in the multi-dimensional space. As a byproduct we show the relationship between the support of the measure and the deficiency rank of the form generator, which appears in the construction of the IFS. We finish with some open problems. This lecture is based on the joint work with A. Dhahri (Chungbuk) and N. Obata (Tohoku).

- Date & Time : 7/21, 13:30~14:30, 15:00~16:00

We start with the standard construction of generalized white noise functionals as infinite dimensional distributions and we study the analytic characterization theorem for S-transform of generalized white noise functionals. Then we study basic concepts and results on white noise operators which is necessary for the study of quantum white noise calculus. The analytic characterization of operator symbols and the Fock expansion theorem are of particular importance.

In the first part of the lectures we will discuss 1-dimensional orthogonal polynomials. Main topics that will be discussed are the followings.

- Three-term recurrence relation and the Jacobi coefficients

- Examples

- Graph spectrum

- Interacting Fock spaces

- Accardi-Bozejko formula

The main reference for this lecture is <Quantum probability and spectral analysis of graphs>, Springer, 2007, by A. Hora and N. Obata.

‣ Date & Time : 7/20, 13:30~14:30, 15:00~16:00

I will discuss some aspects of the algebraic structure of finitely generated groups of diffeomorphisms of compact one-manifolds. In particular, we show that if G is not virtually metabelian then (G x Z)*Z cannot act faithfully by C^2 diffeomorphisms on a compact one-manifold. Among the consequences of this result is a completion of the classification of right-angled Artin groups which admit faithful C^{\infty} actions on the circle, a program initiated together with H. Baik and S. Kim. This represents joint work with S. Kim.

3-d printing gives us unprecedented ability to tailor microstructures to achieve desired goals. From the mechanics perspective one would like, for example, to know how to design structures that guide stress, in the same way that conducting fibers are good for guiding current. In that context the natural question is: what are the possible pairs of (average stress, average strain) that can exist in the material. A more grand question is: what are the possible effective elasticity tensors that can be achieved by structuring a material with known moduli. This is a highly non-trivial problem: in 3-dimensions elasticity tensors have 18 invariants and even an object as simple as a distorted hypercube in 18 dimensions requires about 4.7 million numbers to specify it. Here we review some of the progress that has been made on this question. This is joint work with Marc Briane and Davit Harutyunyan

The behavior of shape memory materials, the response of composites, and inverse problems (where one seeks to determine what is inside a body from boundary measurements) would at first sight seem to have little in common. However there are unifying mathematical themes that underlie them all. Finding composites that have the best properties, for design applications, often reduces to a type of energy minimization problem with a non-linear energy, even though the underlying physical equations may be linear. The same sort of energy minimization problems govern the response of shape memory materials. Similarly, the response of inhomogenous bodies, as governed by the appropriately defined Dirichlet to Neumann map, is similar in many respects to an effective tensor in the theory of composites, and this connection can be made more mathematically explicit. As a result of these connections, mathematical tools developed in one area can be applied to problems in the other areas.

In this talk I will introduce tensor networks for breaking the curse-of-dimensionality in large scale optimizaition problems. I will mainly focus on the tensor train (TT) format, which is one of the simplest tensor networks. I will show how large-scale optimization problems,

which are intractable by standard numerical methods, can be solved by using the concept of tensorization and TT format. In addition, I will discuss several state-of-the art numerical schemes for tensor networks including truncated iteration scheme, alternating linear scheme, and Riemannian optimization approach.

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E6-1 Room 1409
Discrete Math
권창현 (Industrial and Management Systems Engineering, Uni)
On the Price of Satisficing in Network User Equilibria

When network users are satisficing decision-makers, the resulting traffic pattern attains a satisficing user equilibrium, which may deviate from the (perfectly rational) user equilibrium. In a satisficing user equilibrium traffic pattern, the total system travel time can be worse than in the case of the PRUE. We show how bad the worst-case satisficing user equilibrium traffic pattern can be, compared to the perfectly rational user equilibrium. We call the ratio between the total system travel times of the two traffic patterns the price of satisficing, for which we provide an analytical bound. Using the sensitivity analysis for variational inequalities, we propose a numerical method to quantify the price of satisficing for any given network instance.

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E6-1 Room 1409
Discrete Math
Edgardo Roldán-Pensado (Instituto de Matemáticas, UNAM, Mexico)
On the colourful Helly theorem

Let F be a family of convex sets in R^d coloured using d+1 colours. Lovasz’s Colourful Helly Theorem states that if any colourful subfamily of convex sets is intersecting, then one of the monochromatic families is intersecting. We study what happens with the rest of the families.

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E6-1 Room 1409
Discrete Math
이의웅 (Computer Science Department, Carnegie Mellon Unive)
FPT Approximation Algorithms for Graph Problems

Approximation algorithms and fixed-parameter tractable (FPT) algorithms have been two major ways to deal with NP-hardness of combinatorial optimization problems. The notion of FPT approximation can be naturally defined, and it is getting significant attention recently. Starting from gentle introductions to approximation algorithms and FPT algorithms, I will talk about my three recent results on FPT approximability.

– Given a graph G = (V, E) and an integer k, we study k-Vertex Separator, where the goal is to remove the minimum number of vertices such that each connected component in the resulting graph has at most k vertices. We give an O(log k)-FPT approximation algorithm for k-Vertex Separator. Our result improves the best previous graph partitioning algorithms.

– We also study k-Path Transversal, where the goal is to remove the minimum number of vertices such that there is no simple path of length k. We present an O(log k)-FPT approximation algorithm for k-Path Transversal. There was no nontrivial approximation algorithm for k > 4 before this work.

– Finally, k-cut is the problem where we want to remove the minimum number of edges such that the graph has at least k connected components. We give a (2 – ε)-FPT approximation algorithm for some epsilon > 0, improving upon a (non-FPT) 2-approximation.

The third result is joint work with Anupam Gupta and Jason Li.