In Computerized Tomography a 2D or 3D object is reconstructed from projection data (Radon transform data) from multiple directions. When the X-ray beams are sufficiently wide to fully embrace the object and when the beams from a sufficiently dense set of directions around the object can be used, this problem and its solution are well understood. When the data are more limited the image reconstruction problem becomes much more challenging; in the figure below only the region within the circle of the Field Of View is illuminated from all angles. In this talk we consider a limited data problem in 2D Computerized Tomography that gives rise to a restriction of the Hilbert transform as an operator HT from L2(a2,a4) to L2(a1,a3) for real numbers a1 < a2 < a3 < a4. We present the framework of tomographic reconstruction from limited data and the method of differentiated back-projection (DBP) which gives rise to the operator HT. The reconstruction from the DBP method requires recovering a family of 1D functions f supported on compact intervals [a2,a4] from its Hilbert transform measured on intervals [a1, a3] that might only overlap, but not cover [a2, a4]. We relate the operator HT to a self-adjoint two-interval Sturm-Liouville prob- lem, for which the spectrum is discrete. The Sturm-Liouville operator is found to commute with HT , which then implies that the spectrum of HT∗ HT is discrete. Furthermore, we express the singular value decomposition of HT in terms of the so- lutions to the Sturm-Liouville problem. We conclude by illustrating the properties obtained for HT numerically.

The Boltzmann equation models the evolution of the probability density of gas particles that interact through predominantly binary collisions. The equation consists of a transport operator and a collision operator. The latter is a bilinear integral with a non-integrable angular kernel. For a long time the equation was simplified by assuming that the kernel is integrable (so called Grad's cutoff), with a belief that such an assumption does not affect the equation significantly. Recently, however, it has been observed that a non-integrable singularity carries regularizing properties, which motivates further analysis of the equation in this setting. We study the behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime, by examining the generation and propagation in time of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. We show how the singularity rate of the angular kernel affects the order of tails that can be propagated. The result uses Mittag-Leffler functions, which are a generalization of exponential functions. This is based on joint works with Alonso, Gamba, Pavlovic and with Gamba, Pavlovic.

We consider the heat equation with a random potential in dimensions d>=3, and show that the large scale random fluctuations are described by the Edwards-Wilkinson model with the renormalized diffusivity and variance. This is based on a joint work with Lenya Ryzhik and Ofer Zeitouni.

Sea ice undergoes a marked transition in its transport properties at a critical temperature of around -5 C. Above this temperature, the sea ice is porous, allowing percolation of brine, sea water, nutrients, biomass, and heat through the ice. In the Antarctic, this critical behavior plays a particularly important role in air-sea-ice interactions, mixing in the upper ocean, in the life cycles of algae living in the sea ice, and in the interpretation of remote sensing data on the sea ice pack. Recently we have applied percolation theory to model the transition in the transport properties of sea ice. We give an overview of these results, and how they explain data we have taken in the Antarctic. We will also describe recent work in developing inverse algorithms for recovering the physical properties of sea ice remotely through electromagnetic means, and how percolation processes come into play. At the conclusion of the talk, we will show a short video on a recent winter expedition into the Antarctic sea ice pack.

Many physical problems involve interfaces. Examples include phase transition problems where the interface separates the solid and liquid regions, bubble simulation, Hele-Shaw flow, composite materials, and many other important physical phenomena. Mathematically, interface problems usually lead to differential equations whose input data and solutions have discontinuities or non-smoothness across interfaces. As a result, many standard numerical schemes do not work or work poorly for interface problems. This is an introductory talk about the interface problems and our immersed interface method. Through some simple examples, I will try to explain the problems of our interest and related background information. Then I will present our method for some typical model problems in two dimensions. Our method can handle both discontinuous coefficients and singular sources. The main idea is to incorporate the known jumps in the solution and its derivatives into the finite difference scheme, obtaining a modified scheme on the uniform grid for quite arbitrary interfaces. The second part of the talk will focus on applications of the methods combined with the the level set method for moving interface problems: including the Stokes flow with different surface tension, the simulation of Hele-Shaw flow, and computation of crystal growth.

The motion of primary nodal cilia present in embryonic development resembles that of a precessing rod. Implementing regularized singularities to model this fluid flow numerically simulates a situation for which colleagues have exact mathematical solutions and experimentalists have corresponding laboratory studies on both the micro- and macro-scales. Stokeslets are fundamental solutions to the Stokes equations, which act as external point forces when placed in a fluid. By strategically distributing regularized Stokeslets in a fluid domain to mimic an immersed boundary (e.g., cilium), one can compute the velocity and trajectory of the fluid at any point of interest. The simulation can be adapted to a variety of situations including passive tracers, rigid bodies and numerous rod structures in a fluid flow generated by a rod, either rotating around its center or its tip, near a plane. The exact solution allows for careful error analysis and the experimental studies provide new applications for the numerical model. Spectral deferred correction methods are used to alleviate time stepping restrictions in trajectory calculations. Quantitative and qualitative comparisons to theory and experiment have shown that a numerical simulation of this nature can generate insight into fluid systems that are too complicated to fully understand via experiment or exact numerical solution independently.

At the beginning of this talk, We will introduce the background of tensor network states (TNS) in various areas such as quantum physics, quantum chemistry and numerical partial differential equations. Famous TNS including tensor trains (TT), matrix product states (MPS), projected entangled pair states (PEPS) and multi-scale entanglement renormalization ansatz (MERA). Then We will explain how to define TNS by graphs and we will define tensor network ranks which can be used to measure the complexity of TNS. We will see that the notion of tensor network ranks is an analogue of tensor rank and multilinear rank. We will discuss basic properties of tensor network ranks and the comparison among tensor network ranks, tensors rank and multilinear rank. If time permits, we will also discuss the dimension of tensor networks and the geometry of TNS. This talk is based on papers joined with Lek-Heng Lim.

We investigate the effect of finite horizontal boundary properties on the critical Rayleigh and wave numbers for controlled Rayleigh-Benard convection in an infinite horizontal domain. Specifically, we examine boundary thickness, thermal diffusivity and thermal conductivity. Our control method is through perturbation of the lower boundary heat flux. A linear differential-proportional control law uses the local amplitude of a shadowgraph to actively distribute the lower boundary heat flux. Realistic boundary conditions for laboratory experiments are selected. Through linear stability analysis and experiment we examine the important boundary properties and make predictions of the properties necessary for successful control experiments. A surprising finding of this work is that for certain realistic parameter ranges, one may find an isola to time-dependent convection as the primary bifurcation.

A general consumer--resource model assuming discrete consumers and a continuously structured resource is examined. We study two foraging behaviors, which lead to fixed and flexible patch residence times, in conjunction with a simple consumer energetics model linking resource consumption, foraging behavior, and metabolic costs. Results indicate a single evolutionarily stable foraging strategy for fixed and flexible foraging in a nonspatial environment, but flexible foraging in a spatial environment leads to consumer grouping, which affects the resource distribution such that no single foraging strategy can exclude all other strategies. This evolutionarily stable coexistence of multiple foraging strategies may help explain a dichotomous pattern observed in a wide variety of natural systems.

The Hilbert space dimension of quantum-many body systems grows exponentially with the system size. Fortunately, nature does usually not explore this monstrous number of degrees of freedom and we have a chance to describe quantum systems with much smaller sets of effective degrees of freedom. A very precise description for systems with one spatial dimension is based on so-called matrix product states (MPS). With such a reduced parametrization, the computation cost, needed to achieve a certain accuracy, is determined by entanglement properties (quantum non-locality) in the system.
I will give a short introduction to the notion of entanglement entropies and their scaling behavior in typical many-body systems. I will then employ entanglement entropies to bound the required computation costs in MPS simulations. This will lead us to the amazing conclusion that 1D quantum many-body systems can usually be simulated efficiently on classical computers, both for zero and finite temperatures, and for both gapless and critical systems.
In these considerations, we will encounter a number of mathematical concepts such as the theorem of typical sequences (central limit theorem), concentration of measure (Levy's lemma), singular value decomposition, path integrals, and conformal invariance.

The derivation of nonlinear dispersive PDE, such as the nonlinear Schr\"{o}dinger (NLS) from many particle quantum dynamics is a central topic in mathematical physics, which has been approached by many authors in a variety of ways. In particular, one way to derive NLS is via the Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear non-homogeneous PDE that describes the dynamics of a gas of infinitely many interacting bosons, while at the same time retains some of the features of a dispersive PDE. We will discuss the process of going from a quantum many particle system of bosons to the NLS via the GP. The most involved part in such a derivation of NLS consists in establishing uniqueness of solutions to the GP. In the talk we will focus on approaches to the uniqueness step that are motivated by the perspective coming from nonlinear dispersive PDE, including the approach that we developed with Chen, Hainzl and Seiringer based on the quantum de Finetti's theorem. Also we will look into what else the nonlinear PDE such as the NLS can tell us about the GP hierarchy, and will present recent results on infinitely many conserved quantities for the GP hierarchy that are obtained with Mendelson, Nahmod and Staffilani.

Most dangerous cardiac arrhythmia, ventricular fibrillation (VF), is characterized by chaotic electrical behavior of the tissue. At the onset of the first, more organized stage of VF waves of electric activity in the heart become reentrant leading to fast irregular contraction. Better understanding of the mechanisms underlying early VF events will lead to more efficient treatment. Reentry induction has been performed in several experiments. We devise a model of cardiac tissue and use it to obtain a close match to the experimental results. The model combines macroscopic and microscopic properties of cardiac tissue.

In this talk we investigate the relationship between Hele-Shaw evolution with a drift and a transport equation with a drift potential, where the density is transported with a constraint on its maximum. The latter model, in a simplified setting, describes the congested crowd motion with a density constraint. When the drift potential is convex, the crowd density is likely to aggregate, and thus if the initial density starts as a patch (i.e. if it is a characteristic function of some set) then it is expected that the density evolves as a patch. We show that the evolving patch satisfies a Hele-Shaw type equation. This is joint work with Damon Alexander and Yao Yao.

Adaptive methods for solving systems of partial differential equations have become widespread. Robust adaptive software for solving parabolic systems in one and two space dimensions is now widely available. Three spatial adaptive strategies and combinations thereof are frequently employed: mesh refinement (h-refinement); mesh motion (r-refinement); and order variation (p-refinement). These adaptive strategies are driven by a priori and a posteriori error estimates. I will present an adaptive h-refinement finite element code in three dimensions on structured grids. These structured grids contain irregular nodes. Solution values at these nodes are determined by continuity requirements across element boundaries rather than by the differential equations. The differential-algebraic system resulting from the spatial discretization is integrated using Linda Petzold's multistep DAE code DASPK. The large linear systems resulting from Newton's method applied to nonlinear system of differential algebraic equations is solved using preconditioned GMRES. In DASPK the matrix-vector products needed by GMRES are approximated by a ``directional derivative''. Thus, the Jacobian matrix need not be assembled. However, this approach is inefficient. I have modified DASPK to compute the matrix-vector product using stored Jacobian matrix. As in the earlier version of DASPK, DASSL, this matrix is kept for several time steps before being updated. I will discuss appropriate preconditioning strategies, including fast-banded preconditioners. In three dimensions when using multistep methods for time integration it is crucial to use a ``warm restart'', that is, to restart the dae solver at the current time step and order. This requires interpolation of the history information. The interpolation must be done in such a way that mode irregularity is enforced on the new grid. A posteriori error estimates on uniform grids can easily be generalized from two-dimensional results (Babuska and Yu showed that in the case of odd order elements, jumps across elemental boundaries give accurate estimates, and in the case of even order elements, local parabolic systems must be solved to obtain accurate estimates). Babuska's work can even be generalized to meshes with irregular modes but now they no longer converge to the true error (in the case of odd order elements). I have developed a new set of estimates that extend the work of Babuska to irregular meshes and finite difference methods. These estimates provide a posteriori error indicators in the finite element context. Several examples that demonstrate the effectiveness of the code will be given.

Given a number of labeled and unlabeled images, it is possible to determine the class membership of each unlabeled image by creating a sequence of such image transformations that connect it, through other unlabeled images, to a labeled image. In order to measure the total transformation, a robust and reliable metric of the path length is proposed, which combines a local dissimilarity between consecutive images along the path with a global connectivity-based metric. For the local dissimilarity we use a symmetrized version of the zero-order image deformation model (IDM) proposed by Keysers et al. For the global distance we use a connectivity-based metric proposed by Chapelle and Zien in [2]. Experimental results on the MNIST benchmark indicate that the proposed classifier out-performs current state-of-the-art techniques, especially when very few labeled patterns are available.

This talk will survey my efforts with coworkers to develop and analyze Monte Carlo sampling algorithms for complex (usually high dimensional) probability distributions. These sampling problems are typically difficult because they have multiple high probability regions separated by low probability regions and/or they are badly scaled in the sense that there are strong unknown relationships between variables. I'll begin the talk by discussing a simple modification of the standard diffusion Monte Carlo algorithm that results in a more efficient and much more flexible tool for use, for example, in rare event simulation. If time permits I'll discuss a few other ensemble based sampling tools designed to directly address energy barriers and scaling issues.

One of the pressing problems in the analysis of reaction-diffusion equations is obtaining accurate and reliable estimates of the error of numerical solutions. Recently, we made significant progress using a new approach that at the heart is computational rather than analytical. I will describe a framework for deriving and analyzing a posteriori error estimates, discuss practical details of the implementation of the theory, and illustrate the error estimation using a variety of well-known models. I will also briefly describe an application of the theory to the class of problems that admit invariant rectangles and discuss the preservation of invariant rectangles under discretization.

A plethora of biological and chemical pattern formation problems are modeled using coupled reaction-diffusion equations of activator-inhibitor type. In 1993, the new phenomenon of self- -replicating spots and pulses was discovered in the Gray-Scott model and in a ferrocyanide reaction it models, by John Pearson and Harry Swinney and collaborators. In this talk, we present an analysis of pulse splitting. Furthermore, it turns out that the perturbation theory developed for the Gray-Scott analysis can be extended in a natural way to analyze a general class of coupled activator-inhibitor systems, including the Gierer-Meinhart and Schnakenberg models, in order to determine whether pulses attract or repel each other, and if they repel, whether they also the undergo self-replication. The work may be classified as a treatment of the moderately strong and strong pulse interaction problem. This work is part of a larger collaborative project with Arjen Doelman, Wiktor Eckhaus, Rob Gardner and my student Dave Morgan. We will conclude with some open questions.

We present several versions of non-overlapping Domain Decomposition Methods (DDM) for the solution of Helmholtz transmission problems for (a) multiple scattering configurations, (b) bounded composite scatterers with piecewise constant material properties, and (c) layered media. We show that DDM solvers give rise to important computational savings over other existing solvers, especially in the challenging high-frequency regime.

Geometric problems - such as finding corresponding points over a collection of shapes, or computing shape deformation under geometric constraints - pose various computational challenges. I will show that despite the very different nature of these two highly non-convex problems, Semidefinite Programming (SDP) can be leveraged to provide a tight convex approximation in both cases. A different approach is used for each problem, demonstrating the versatility of SDP: (i) For establishing point correspondences between shapes, we devise an SDP relaxation. I will show it is a hybrid of the popular spectral and doubly-stochastic relaxations, and is in fact tighter than both. (ii) For the computation of piecewise-linear mappings, we introduce a family of maximal SDP restrictions. Solving a sequence of such SDPs enables the optimization of functionals and constraints expressed in terms of singular values, which naturally model various geometry processing problems.

After the sun, the oceans are the most significant contributor to our climate. Oceanic surface gravity waves are thought to have no influence on the global circulation of the oceans. However, oceanic surface gravity waves have a mean Lagrangian motion, the Stokes drift. This talk will present preliminary results that suggest that the dynamics of basin-scale oceanic currents are modified by the presence of the Stokes drift. In places where the Stokes drift is significant, it is possible that the ocean circulation, and hence climate, is not entirely well captured by present day models of the general circulation.

Special day: Thursday, 4pm, Room 120 Physics

We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales char- acterized by the magnitude of a dimensionless parameter—the Knudsen number (Kn). In the diffusive scaling (Kn ≪ 1), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed as compared to the well- posedness of inverse transport equation, which raises the question of how uniqueness being lost as Kn → 0. We tackle this problem by examining the stability of inverse problem with varying Kn. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of Knp (p = 1 or 2), and as a result lead to ill-posedness in the zero limit of Kn. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings.

Single-cell RNA sequencing (scRNA-Seq) has emerged as a powerful tool to sample the complexity of large populations of cells and observe biological processes at unprecedented molecular resolution. This offers the exciting prospect of understanding the molecular programs that guide cellular differentiation during development. Here, we introduce Waddington-OT: a mathematical framework for understanding the temporal dynamics of development based on snapshots of expression profiles. The central challenge in analyzing these data arises from the fact that scRNA-Seq is destructive, which means that one cannot directly measure the trajectory of any given cell over time. We model the population of developing cells mathematically with a time-varying probability distribution (i.e. stochastic process) on a high-dimensional gene expression space, and we propose to recover the temporal coupling of the process with optimal transport. We demonstrate the power of Waddington-OT by applying the approach to study 315,000 scRNA-seq profiles collected at 40 time points over 16 days during reprogramming of fibroblasts to induced pluripotent stem cells. We construct a high-resolution map of reprogramming that rediscovers known features; uncovers new alternative cell fates including neural- and placental-like cells; predicts the origin and fate of any cell class; and implicates regulatory models in particular trajectories. Of these findings, we highlight Obox6, which we experimentally show enhances reprogramming efficiency. Our approach provides a general framework for investigating cellular differentiation.

Experiments on knotted DNA molecules suggest that certain physical properties of DNA knots can be predicted from a corresponding ideal shape. Intuitively, when a given knot in a piece of string is pulled tight, it always achieves roughly the same geometrical configuration, with a minimum length of string within the knot. Such a configuration is called an ideal shape for the knot, and approximations of ideal shapes in this sense have been found via a series of computer experiments. These shapes have intriguing physical features and have been shown to capture average properties of knotted polymers. But when does a shape satisfy the intuitive geometrical definition for ideality? In this talk I show that ideal shapes can be understood using only elementary (but new!) mathematics. In particular, I show that global curvature, a very natural and simple generalization of the classic concept of local curvature, leads to a simple characterization of an ideal shape and to a necessary condition for ideality. Another application of global curvature can be found in characterizing the equilibria of knotted curves or rods, which may exhibit self-contact after sufficient twisting. Here global curvature provides a simple way to formulate the constraint that prevents a rod from passing through itself.

We discuss a new class of cubic interpolating and approximating "L₁ splines" that preserve the shape both of smooth data and of data with abrupt changes in magnitude or spacing. The coefficients of these splines are calculated by minimizing the L₁ norm of the second derivatives. These splines do not require constraints, penalties, a posteriori filtering or interaction with the user. Univariate and multivariate cases are treated in one and the same framework. L₁ splines are implemented using efficient interior-point methods for linear programs.

We study the large-time behavior of hydrodynamic model which describes the collective behavior of continuum of agents, driven by pairwise alignment interactions with additional external potential forcing. The external force tends to compete with alignment which makes the large time behavior very different from the original Cucker-Smale (CS) alignment model, and far more interesting. Here we focus on uniformly convex potentials. In the particular case of \emph{quadratic} potentials, we are able to treat a large class of admissible interaction kernels, $\phi(r) \gtrsim (1+r^2)^{-\beta}$ with `thin' tails $\beta \leq 1$ --- thinner than the usual `fat-tail' kernels encountered in CS flocking $\beta\leq\nicefrac{1}{2}$: we discover unconditional flocking with exponential convergence of velocities \emph{and} positions towards a Dirac mass traveling as harmonic oscillator. For general convex potentials, we impose a necessary stability condition, requiring large enough alignment kernel to avoid crowd dispersion. We prove, by hypocoercivity arguments, that both the velocities \emph{and} positions of smooth solution must flock. We also prove the existence of global smooth solutions for one and two space dimensions, subject to critical thresholds in initial configuration space. It is interesting to observe that global smoothness can be guaranteed for sub-critical initial data, independently of the apriori knowledge of large time flocking behavior.