## Bruce Donald : Some mathematical and computational challenges arising in structural molecular biology

- Applied Math and Analysis ( 304 Views )Computational protein design is a transformative field with exciting prospects for advancing both basic science and translational medical research. New algorithms blend discrete and continuous mathematics to address the challenges of creating designer proteins. I will discuss recent progress in this area and some interesting open problems. I will motivate this talk by discussing how, by using continuous geometric representations within a discrete optimization framework, broadly-neutralizing anti-HIV-1 antibodies were computationally designed that are now being tested in humans - the designed antibodies are currently in eight clinical trials (See https://clinicaltrials.gov/ct2/results?cond=&term=VRC07&cntry=&state=&city=&dist= ), one of which is Phase 2a (NCT03721510). These continuous representations model the flexibility and dynamics of biological macromolecules, which are an important structural determinant of function. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. These distributions are not fully constrained by the limited information from experiments, making the problem ill-posed in the sense of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem must be regularized by making (hopefully reasonable) assumptions. I will present new ways to both represent and visualize correlated inter-domain protein motions (See Figure). We use Bingham distributions, based on a quaternion fit to circular moments of a physics-based quadratic form. To find the optimal solution for the distribution, we designed an efficient, provable branch-and-bound algorithm that exploits the structure of analytical solutions to the trigonometric moment problem. Hence, continuous conformational PDFs can be determined directly from NMR measurements. The representation works especially well for multi-domain systems with broad conformational distributions. Ultimately, this method has parallels to other branches of applied mathematics that balance discrete and continuous representations, including physical geometric algorithms, robotics, computer vision, and robust optimization. I will advocate for using continuous distributions for protein modeling, and describe future work and open problems.

## Cynthia Vinzant : Matroids, log-concavity, and expanders

- Applied Math and Analysis ( 214 Views )Matroids are combinatorial objects that model various types of independence. They appear several fields mathematics, including graph theory, combinatorial optimization, and algebraic geometry. In this talk, I will introduce the theory of matroids along with the closely related class of polynomials called strongly log-concave polynomials. Strong log-concavity is a functional property of a real multivariate polynomial that translates to useful conditions on its coefficients. Discrete probability distributions defined by these coefficients inherit several of these nice properties. I will discuss the beautiful real and combinatorial geometry underlying these polynomials and describe applications to random walks on the faces of simplicial complexes. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets of a matroid is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.

## Matthew Jacobs : A fast approach to optimal transport: the back-and-forth method

- Applied Math and Analysis ( 208 Views )Given two probability measures and a transportation cost, the optimal transport problem asks to find the most cost efficient way to transport one measure to the other. Since its introduction in 1781 by Gaspard Monge, the optimal transport problem has found applications in logistics, economics, physics, PDEs, and more recently data science. However, despite sustained attention from the numerics community, solving optimal transport problems has been a notoriously difficult task. In this talk I will introduce the back-and-forth method, a new algorithm to efficiently solve the optimal transportation problem for a general class of strictly convex transportation costs. Given two probability measures supported on a discrete grid with n points, the method computes the optimal map in O(n log(n)) operations using O(n) storage space. As a result, the method can compute highly accurate solutions to optimal transportation problems on spatial grids as large as 4096 x 4096 and 384 x 384 x 384 in a matter of minutes. If time permits, I will demonstrate an extension of the algorithm to the simulation of a class of gradient flows. This talk is joint work with Flavien Leger.

## Xiaochun Tian : Interface problems with nonlocal diffusion

- Applied Math and Analysis ( 184 Views )Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, they also come with increased difficulty in numerical analysis with nonlocality involved. In the first part of this talk, I will discuss nonlocal-to-local coupling techniques so as to improve the computational efficiency of using nonlocal models. This also motivates the development of new mathematical results -- for instance, a new trace theorem that extends the classical results. In the second part of this talk, I will describe our recent effort in computing a nonlocal interface problem arising from segregation of two species with high competition. One species moves according to the classical diffusion and the other adopts a nonlocal strategy. A novel iterative scheme will be presented that constructs a sequence of supersolutions shown to be convergent to the viscosity solution of the interface problem.

## Dan Hu : Optimization, Adaptation, and Initiation of Biological Transport Networks

- Applied Math and Analysis ( 181 Views )Blood vessel systems and leaf venations are typical biological transport networks. The energy consumption for such a system to perform its biological functions is determined by the network structure. In the first part of this talk, I will discuss the optimized structure of vessel networks, and show how the blood vessel system adapts itself to an optimized structure. Mathematical models are used to predict pruning vessels in the experiments of zebra fish. In the second part, I will discuss our recent modeling work on the initiation process of transport networks. Simulation results are used to illustrate how a tree-like structure is obtained from a continuum adaptation equation system, and how loops can exist in our model. Possible further application of this model will also be discussed.

## Jim Nolen : Asymptotic Spreading of Reaction-Diffusion Fronts in Random Media

- Applied Math and Analysis ( 169 Views )Some reaction-advection-diffusion equations admit traveling wave solutions; these are simple models of a combustion reaction spreading with constant speed. Even in a random medium, solutions to the initial value problem may develop fronts propagating with a well-defined asymptotic speed. First, I will describe this behavior when the nonlinearity is the Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearity and the randomness comes from a prescribed random drift (a simple model of turbulent combustion). Next, I will describe propagation of fronts when the nonlinearity is a random ignition-type nonlinearity. In the latter case, there exist special solutions that generalize the notion of a traveling wave in the random setting.

## Qin Li : Stability of stationary inverse transport equation in diffusion scaling

- Applied Math and Analysis ( 149 Views )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 parameterÂ?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.

## Yat Tin Chow : An algorithm for overcoming the curse of dimensionality in Hamilton-Jacobi equations

- Applied Math and Analysis ( 144 Views )In this talk we discuss an algorithm to overcome the curse of dimensionality, in possibly non-convex/time/state-dependent Hamilton-Jacobi partial differential equations. They may arise from optimal control and differential game problems, and are generally difficult to solve numerically in high dimensions.

A major contribution of our works is to consider an optimization problem over a single vector of the same dimension as the dimension of the HJ PDE instead. To do so, we consider the new approach using Hopf-type formulas. The sub-problems are now independent and they can be implemented in an embarrassingly parallel fashion. That is ideal for perfect scaling in parallel computing.

The algorithm is proposed to overcome the curse of dimensionality when solving high dimensional HJ PDE. Our method is expected to have application in control theory, differential game problems, and elsewhere. This approach can be extended to the computational of a Hamilton-Jacobi equation in the Wasserstein space, and is expected to have applications in mean field control problems, optimal transport and mean field games.

## Shaoming Guo : Maximal operators and Hilbert transforms along variable curve

- Applied Math and Analysis ( 141 Views )I will present several results on the boundedness of maximal operators and Hilbert transforms along variable curves and surfaces, in dimension two or higher. Connections to the circular maximal operator, and the polynomial Carleson operator will also be discussed.

## Elisabetta Matsumoto : Biomimetic 4D Printing

- Applied Math and Analysis ( 134 Views )The nascent technique of 4D printing has the potential to revolutionize manufacturing in fields ranging from organs-on-a-chip to architecture to soft robotics. By expanding the pallet of 3D printable materials to include the use stimuli responsive inks, 4D printing promises precise control over patterned shape transformations. With the goal of creating a new manufacturing technique, we have recently introduced a biomimetic printing platform that enables the direct control of local anisotropy into both the elastic moduli and the swelling response of the ink.

We have drawn inspiration from nastic plant movements to design a phytomimetic ink and printing process that enables patterned dynamic shape change upon exposure to water, and possibly other external stimuli. Our novel fiber-reinforced hydrogel ink enables local control over anisotropies not only in the elastic moduli, but more importantly in the swelling. Upon hydration, the hydrogel changes shape accord- ing the arbitrarily complex microstructure imparted during the printing process.

To use this process as a design tool, we must solve the inverse problem of prescribing the pattern of anisotropies required to generate a given curved target structure. We show how to do this by constructing a theory of anisotropic plates and shells that can respond to local metric changes induced by anisotropic swelling. A series of experiments corroborate our model by producing a range of target shapes inspired by the morphological diversity of flower petals.

## Reema Al-Aifari : Spectral Analysis of the truncated Hilbert Transform arising in limited data tomography

- Applied Math and Analysis ( 133 Views )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.

## Fei Lu : Data-based stochastic model reduction for chaotic systems

- Applied Math and Analysis ( 131 Views )The need to deduce reduced computational models from discrete observations of complex systems arises in many climate and engineering applications. The challenges come mainly from memory effects due to the unresolved scales and nonlinear interactions between resolved and unresolved scales, and from the difficulty in inference from discrete data.

We address these challenges by introducing a discrete-time stochastic parametrization framework, through which we construct discrete-time stochastic models that can take memory into account. We show by examples that the resulting stochastic reduced models that can capture the long-time statistics and can make accurate short-term predictions. The examples include the Lorenz 96 system (which is a simplified model of the atmosphere) and the Kuramoto-Sivashinsky equation of spatiotemporally chaotic dynamics.

## Badal Joshi : A coupled Poisson process model for sleep-wake cycling

- Applied Math and Analysis ( 127 Views )Sleep-wake cycling is an example of switching between discrete states in mammalian brain. Based on the experimental data on the activity of populations of neurons, we develop a mathematical model. The model incorporates several different time scales: firing of action potentials (milliseconds), sleep and wake bout times (seconds), developmental time (days). Bifurcation diagrams in a deterministic dynamical system gives the occupancy time distributions in the corresponding stochastic system. The model correctly predicts that forebrain regions help to stabilize wake state and thus modifies the wake bout distribution.

## Jiequn Han : Deep Learning-Based Numerical Methods for High-Dimensional Parabolic PDEs and Forward-Backward SDEs

- Applied Math and Analysis ( 116 Views )Developing algorithms for solving high-dimensional partial differential equations (PDEs) and forward-backward stochastic differential equations (FBSDEs) has been an exceedingly difficult task for a long time, due to the notorious difficulty known as the curse of dimensionality. In this talk we introduce a new type of algorithms, called "deep BSDE method", to solve general high-dimensional parabolic PDEs and FBSDEs. Starting from the BSDE formulation, we approximate the unknown Z-component by neural networks and design a least-squares objective function for parameter optimization. Numerical results of a variety of examples demonstrate that the proposed algorithm is quite effective in high-dimensions, in terms of both accuracy and speed. We furthermore provide a theoretical error analysis to illustrate the validity and property of the designed objective function.

## Edmond Chow : Parallel Computing Issues in Computational Chemistry

- Applied Math and Analysis ( 113 Views )In computational mathematics and science, it is now essential to consider computer hardware issues if a new algorithm is to be deployed. One such issue is the prevalence of parallelism in almost all levels of computer hardware. We discuss some of the challenges presented by computer hardware and some potential solutions in the context on quantum chemistry algorithms. Important considerations include reducing data movement, load balance across processors, and use of SIMD (single instruction, multiple data) features in modern processors. Specific results we have obtained include efficient computations using Hartree--Fock approximations on more than 1.5 million processor cores, and a new library for computing electron repulsion integrals that is designed for SIMD operation. These results are joint work with Ben Pritchard, Xing Liu, and the Intel Parallel Computing Lab.

## John Neu : Resonances in Geometric Optics

- Applied Math and Analysis ( 112 Views )Consider wavefront propagation in the plane, in a medium whose propagation speed is doubly periodic. Think of a "wavefront" as the moving boundary between "light" and "darkness." There are "macroscopic plane wavefronts", for which the wavefront is everywhere and always bounded close to a moving line with constant normal velocity. The normal velocity depends on direction. Some nice function of angle, and we ought to compute it, no? To contemplate this calculation, visualize an infinite planar vineyard, with a square lattice of grape stakes. Gaze outward from the the grape stake at the origin. In some directions, the line of sight is blocked by a stake, and we'll call these directions "rational." The rational directions are dense, but of zero measure on the unit circle. The simplest formal asymptotics of the direction dependent speed produces a strange result: A formal series, one term for each rational direction. The graph of an individual term as a function of angle looks like the amplitude response of a forced oscillator, with a plus infinity vertical asymptote as you scan through the rational direction. Nevertheless, the series converges absolutely for almost all directions on the unit circle. (In the vineyard, almost all lines of sight escape to infinity.) At the outset, the prognosis seems: "Difficult and obscure." The problem of direction dependent speed does not LOOK like a venue for formal classical asymptotics, but that's what this talk proposes. The ingredients are standard matched asymptotic expansions, and baby number theory concerning period cells of the vineyard.

## Alexander Kiselev : Regularity and blow up in ideal fluid

- Applied Math and Analysis ( 106 Views )The incompressible Euler equation of fluid mechanics has been derived in 1755. It is one of the central equations of applied analysis, yet due to its nonlinearity and non-locality many fundamental properties of solutions remain poorly understood. In particular, the global regularity vs finite time blow up question for incompressible three dimensional Euler equation remains open. In two dimensions, it has been known since 1930s that solutions to Euler equation with smooth initial data are globally regular. The best available upper bound on the growth of derivatives of the solution has been double exponential in time. I will describe a construction showing that such fast generation of small scales can actually happen, so that the double exponential bound is qualitatively sharp. This work has been motivated by numerical experiments due to Hou and Luo who propose a new scenario for singularity formation in solutions of 3D Euler equation. The scenario is axi-symmetric. The geometry of the scenario is related to the geometry of 2D Euler double exponential growth example and involves hyperbolic points of the flow located at the boundary of the domain. If time permits, I will discuss some recent attempts to gain insight into the three-dimensional fluid behavior in this scenario.

## Jeremy L. Marzuola : The relaxation of a family of broken bond crystal surface models

- Applied Math and Analysis ( 104 Views )With Jon Weare (Chicago), we study the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. With computational experiments and theoretical arguments we are able to derive several partial differential equation limits identified (or nearly identified) in previous studies and to clarify the correct choice of surface tension appearing in the PDE and the correct scaling regime giving rise to each PDE. We also provide preliminary computational and analytic investigations of a number of interesting qualitative features of the large scale behavior of the models. The PDE models involved are fully non-linear Fourth order diffusion type equations with many interesting geometric features. We will given time discuss recent progress analyzing properties of solutions to such PDE.

## Yoshiaki Teramoto : Benard-Marangoni problem of heat convection with free surface

- Applied Math and Analysis ( 103 Views )When a fluid layer is heated from below with temperature larger than a certain critical value, the convective motion appears in the fluid. The convection caused by the thermocapillary effect is called Benard-Marangoni heat convection. The thermocapillary effect is the dependence of the surface tension on the temperature. Near a hot spot on a free surface of fluid a thermocapillary tangential stress generates a fluid motion. In this talk the mathematical model system for this convection is explained. The Oberbeck-Boussinesq approximation is used for the system and the upper boundary is a free surface with surface tension which depends on the temperature. After formulating the linearized problem around the conductive state, we derive the resolvent estimates which guarantee the sectorial property. Stationary and Hopf bifurcations (periodic solutions) are proved to exist depending on the parameters (Raylegh and Marangoni numbers).

## Sijue Wu : On two dimensional gravity water waves with angled crests

- Applied Math and Analysis ( 100 Views )In this talk, I will survey the recent understandings on the motion of water waves obtained via rigorous mathematical tools, this includes the evolution of smooth initial data and some typical singular behaviors. In particular, I will present our recently results on gravity water waves with angled crests.

## Xiu Yang : Enhancing Sparsity of Hermite Polynomial Expansions by Iterative Rotations

- Applied Math and Analysis ( 98 Views )Compressive sensing has become a powerful addition to uncertainty quantification in recent years. This paper identifies "new" bases for random variables through linear mappings such that the representation of the quantity of interest is more sparse with new basis functions associated with the new random variables. This sparsity increases both the efficiency and accuracy of the compressive sensing-based uncertainty quantification method. Specifically, we consider rotation-based linear mappings which are determined iteratively for Hermite polynomial expansions. We demonstrate the effectiveness of the new method with applications in solving stochastic partial differential equations and high-dimensional problems.

## Sanchit Chaturvedi : Phase mixing in astrophysical plasmas with an external Kepler potential

- Applied Math and Analysis ( 85 Views )In Newtonian gravity, a self-gravitating gas around a massive object such as a star or a planet is modeled via Vlasov Poisson equation with an external Kepler potential. The presence of this attractive potential allows for bounded trajectories along which the gas neither falls in towards the object or escape to infinity. We focus on this regime and prove first a linear phase mixing result in 3D outside symmetry with exact Kepler potential. Then we also prove a long-time nonlinear phase mixing result in spherical symmetry. The mechanism is phenomenologically similar to Landau damping on a torus but mathematically the situation is quite a lot more complex. This is based on an upcoming joint work with Jonathan Luk at Stanford.

## Ken Golden : Mathematics of Sea Ice

- Applied Math and Analysis ( 28 Views )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.