### Welcome to the website for Jr. NAP!

#### 1/30: Sarah

Title: Simultaneous reconstruction of wave speed and optical absorption in quantitative photoacoustic tomography Abstract: There is a well developed theory for inverse problems in photoacoustic tomography (PAT) which relies on complete knowledge of the acoustic wave speed within a medium of interest. In practical applications, however, this is not possible. When an incorrect estimate of this parameter is used in the reconstruction algorithms, the images contain artifacts. This motivates the search for reconstruction methods which do not require a priori knowledge of the wave speed. In this talk I will present a one-step reconstruction algorithm for PAT which recovers simultaneously the wave speed and optical parameters of a medium, and which takes advantage of multiple data sets to stabilize the reconstruction. |

#### 10/31: Sarah

Title: Numerical Methods for the Wave Equation in Free Space Abstract: In order to be computationally feasible, a problem must be finite. For PDEs posed in infinite domains this typically involves some sort of truncation. However, boundary conditions must be carefully chosen for the computational domain in order to minimize numerical error. In this talk I will present a few methods for numerically solving the wave equation posed in free space, namely truncating the domain and applying absorbing boundary conditions (ABCs) or perfectly matched layers (PML). I will discuss the ideas behind each method and give examples of them in action. |

#### 9/12: Soledad

Title: Relax, don't need to round: integrality of clustering formulations. Abstract: When dealing with combinatoric NP-hard problems one can either find an efficient algorithm that works in every case but its solution is not optimal (this is the philosophy behind approximation algorithms). Or efficiently compute a probably optimal solution of the original problem under more restrictive hypothesis (which is the spirit of compressed sensing recovery guarantees). In this talk I am going to explain what that means exactly, and how we use the second approach for data clustering. This talk has no prerequisites and with high probability cookies will be offered to the audience. |

#### 4/17: Yimin

Topic: Point Source Recovery for Helmholtz Equation Abstract: In this talk I shall give a sketchy idea on how to prove the uniqueness and stability of recovering point sources in unknown media associated with Helmholtz equation with multiple full Neumann data. |

#### 4/3: Soledad

Title: Approximating solutions of NP-hard problems. Abstract: We will study different convex optimization ideas to approximate solutions of NP-hard problems. We will illustrate the ideas through examples: the set cover problem and clustering problems. No prerequisites are needed to follow this talk. |

#### 3/20: Chris

Title: Randomized Methods in Numerical Linear Algebra Abstract: In this talk we will introduce the basics of random matrix theory, and use these tools to develop randomized techniques for computing the eigenvalues and eigenvectors of large matrices. In particular, we will sketch the proofs of some error bounds for the Nystr\"om method and for some simple SVD algorithms applied to low-rank matrices. |

#### 2/27: Tian

Title: Algorithms and Computational Complexity Abstract: An algorithm is any well-defined computational procedure that takes some value, or set of values, as input and produces some value, or set of values as output. In other words, algorithms are like road maps for accomplishing a given, well-defined task. One of the most important aspects of an algorithm is how fast it is. It is often easy to come up with an algorithm to solve a problem, but if the algorithm is too slow, it may be not worth trying at all. Since the exact speed of an algorithm depends on where the algorithm is run, as well as the exact details of its implementation, typically runtime relative to the size of the input is talked. We will introduce the notations for computational complexity and also how to perform an algorithm analysis. |

#### 2/20: Charlie

Title: Mullins-Sekerka Problem and LaplaceEquation on Multiply Connected Regions Abstract: Mullins-Sekerka problem is a free boundary problem commonly used in modeling crystal growth or solidification and liquidation where material movement is governed by diffusion and no surface tension. The talk will focus on why Laplace Equation in non-conventional regions is of interest, and the peculiarity of existence, uniqueness, and numerical methods under this circumstance. |

#### 2/13: Christina

Title: Numerical methods for multiscale inverse problems Abstract: We will consider inverse problems for multiscale partial differential equations of the form $-\div \left(\aeps\nabla u^\epsilon\right)+b^{\epsilon}u^{\epsilon} = f$ in which solution data is used to determine coefficients in the equation. Such problems contain both the general difficulty of finding an inverse and the challenge of multiscale modeling, which is hard even for forward computations. The problem in its full generality is typically ill-posed and one approach is to reduce the dimensionality of the original problem by just considering the inverse of an effective equation without microscale $\epsilon$. We will here include microscale features directly in the inverse problem. In order to reduce the dimension of the unknowns and avoid ill-posedness, we will assume that the microscale can be accurately parametrized by piecewise smooth coefficients. We indicate in numerical examples how the technique can be applied to medical imaging and exploration seismology. |

#### 11/14: Rohit

Title: Regularity of the solution for Obstacles exhibiting non-local behavior Abstract: Starting from an optimal cash management problem and its formulation as a stochastic impulse control problem we will derive an obstacle problem where the obstacle exhibits non-local behavior. Generalizing the 1-d situation we will discuss some properties of the solution as well as introduce the notion of a quasi-variational inequality. The objective of the talk will be to relate the probabilistic interpretation of the problem to its analytic reformulation and discuss some applications. |

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