Abstract: Grain boundaries are interfaces across which crystal orientation changes. Traditional analysis suggest that grain boundary migration is effectively motion by mean curvature. However, this view is not in accordance with what we now know as the structure of grain boundaries on an atomic level. Just as surfaces of crystals move and roughen through the dynamics of surface steps, grain boundary dynamics is controlled by the motion of line defects known as disconnections. Unlike surface steps, disconnections are sources of long range stress (i.e., they have both dislocation and step character). In this talk, I will present an approach for understanding the motion of grain boundaries via disconnection motion and the relationship between disconnections and the underlying crystal structure. Next, I will discuss the homogenization of this type of disconnection-driven motion to yield a crystal-structure specific grain boundary equation of motion. I will then show several atomistic and numerical examples of âtame" GB motion (i.e., in bicyrstals) and GB motion âin the wildâ (within polycrystals). This is very much a work in progress so I will also outline approaches for generalizations to general interface controlled microstructure evolution.
Abstract: In this talk, we will focus on a kinetic equation modeling the spatial dynamics of a set of particles subject to intra-specific competition. This equation is motivated by the study of the propagation of biological populations, such as the Escherichia coli bacterium or the cane toad Rhinella marina, for which the classical diffusion approximation underestimates the actual range expansion of the species. We will use the optics geometrics approach as well as Hamilton-Jacobi equations to study spreading results for this equation. As we will see, the multi-dimensional case engenders technical difficulties, and possible over-representation of fast individuals at the edge of the front.
Abstract: In this talk I present a general framework to model cell-cycle structured
populations living in a chemostat. The main examples are E. coli, or yeast, both
model organisms which have been intensively investigated to understand cell-cycle controls. In this simple case the cells' cell cycle influence each other only by the level of nutrients found in the culture medium. Otherwise the cell cycle in each cell behaves autonomously. As the cell-cycle depends on many cell-internal biochemical concentrations, most importantly on the cyclin protein family, the dynamical system describing the internal cell dynamics can be of arbitrary high dimension, making the model extremely complex. In order to investigate the model behaviour we decided not to use numerical time-integration, but numerical continuation and bifurcation techniques. The respective numerical algorithm is again of immense complexity, and uses a cell cohort discretisation. The plan is to refine the model in future, most importantly bringing it to a tissue level in order to describe cancer dynamics.
Abstract: We discuss several examples of inverse problems in computational super-resolution. The first one is a generalized version of well-known sparse sums-of-exponentials model, where we allow also for polynomial modulations. We derive upper bounds on the problem condition number and show that the attainable resolution exhibits HÃ¶lder-type continuity with respect to the noise level. A closely related problem is approximating piecewise-smooth functions, including jump locations, from its Fourier coefficients, with high accuracy. We can show that the asymptotic accuracy of our approach is only dictated by the smoothness of the function between the jumps. Finally we describe some on-going work on the weighted extrapolation problem on the real line for functions of finite exponential type where we abandon the sparsity assumption. It turns out that the extrapolation range scales logarithmically with the noise level, while the pointwise extrapolation error exhibits again a HÃ¶lder-type continuity.
Abstract: We consider a free boundary problem (of Hele-Shaw type) modeling tumor growth. Under certain conditions on the initial data, solutions can be obtained by passing to the stiff (incompressible) limit in a porous medium type problem with a Lotka-Volterra source term describing the evolution of the number density of cancerous cells. We will present several results concerning this derivation and the properties of the resulting free boundary problem. This is a joint work with B. Perthame and F. Quiros.
Abstract: Edges are noticeable features in images which can be extracted from noisy data using different variational models. The analysis of such models leads to the question of representing general L2-data as the divergence of uniformly bounded vector fields.
We use a multi-scale approach to construct uniformly bounded solutions of div(U)=f for general f's in the critical regularity space L2(T2). The study of this equation and related problems was motivated by recent results of Bourgain & Brezis. The intriguing critical aspect here is that although the problems are linear, construction of their solution is not. These constructions are special cases of a rather general framework for solving linear equations in critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical representations U=Î£juj which we introduced earlier in the context of image processing, yielding a multi-scale decomposition of "images" U.
Abstract: We prove rigorous convergence properties for a semi-discrete, moment-based approximation of a model kinetic equation in one dimension. This approximation is equivalent to a standard spectral method in the velocity variable of the kinetic distribution and, as such, is accompanied by standard algebraic estimates of the form Nâq, where N is the number of modes and q depends on the regularity of the solution. However, in the multiscale setting, we show that the error estimate can be expressed in terms of the scaling parameter Îµ, which measures the ratio of the mean-freepath to the characteristic domain length. In particular, we show that the error in the spectral approximation is O(ÎµN+1). More surprisingly, for isotropic initial conditions, the coefficients of the expansion satisfy super convergence properties. In particular, the error of the lth coefficient of the expansion scales like O(Îµ2N ) when l = 0 and O(Îµ2N+2âl) for all 1 â¤ l â¤ N. This result is significant, because the low-order coefficients correspond to physically relevant quantities of the underlying system. All the above estimates involve constants depending on N, the time t, and the initial condition. We investigate specifically the dependence on N, in order to assess whether increasing N actually yields an additional factor of Îµ in the error. Numerical tests will also be presented to support the theoretical results.
Abstract: In this talk, I focus on current biological problems and on how to use mathematical modeling to analyze a variety of pressing questions arising from oncology, developmental pattern formation and population ecology. I first discuss novel mathematical models for cancer growth dynamics and heterogeneity. These studies rely on evolutionary principles and shed light on 3D hepatic tumor dynamics, spatial heterogeneity and tumor invasion, and single cancer cell responses to antimitotic therapies. We also develop mathematical models that quantitatively demonstrate how the interplay between non-genetic instability, stress-induced adaptation, and selection leads to the transient and reversible phenotypic evolution of cancer cell populations exposed to therapy. Finally, we study control techniques for optimal therapeutic administration.
Abstract: Semi-supervised learning refers to machine learning algorithms that make use of both labeled data and unlabeled data for learning tasks. Examples include problems such as speech recognition, website classification, and discovering folding structure of proteins. In many problems there is an abundance of unlabeled data, while labeled data often requires expert labeling and is expensive to obtain. This has led to a resurgence of semi-supervised learning techniques, which use the topological or geometric properties of large amounts of unlabeled data to aid the learning task. In this talk, I will discuss some new rigorous PDE scaling limits for existing semisupervised learning algorithms and their practical implications. I will also discuss how these scaling limits suggest new ideas for fast algorithms for semi-supervised learning.
Abstract: A common feature of systems of conservation laws of continuum physics is that they are endowed with natural companion laws which are in such case most often related to the second law of thermodynamics. This observation easily generalizes to any symmetrizable system of conservation laws. They are endowed with nontrivial companion conservation laws, which are immediately satisfied by classical solutions. Not surprisingly, weak solutions may fail to satisfy companion laws, which are then often relaxed from equality to inequality and overtake a role of a physical admissibility condition for weak solutions. We want to discuss what is a critical regularity of weak solutions to a general system of conservation laws to satisfy an associated companion law as an equality. An archetypal example of such result was derived for the incompressible Euler system by Constantin et al. () in the context of the seminal Onsager's conjecture. This general result can serve as a simple criterion to numerous systems of mathematical physics to prescribe the regularity of solutions needed for an appropriate companion law to be satisfied.
The second part of the talk will concern the problem of uniqueness. Strong solutions are unique, and as it has been observed for many systems, not only in the class of strong solutions, but also in a wider class of entropy weak or even entropy/dissipative measure-valued solutions. These properties are referred as weak-strong or measure-valued-strong uniqueness correspondingly. We will discuss dissipative measure-valued solutions to hyperbolic systems and we do not assume that a priori solutions satisfy any bounds, in particular, that a solution consists only of a classical Young measure. We do not exclude possibilties of concentration measures.
 P. Constantin, W. E, and E. S. Titi. Onsagerâs conjecture on the energy conservation for solutions of Eulerâs equation. Comm. Math. Phys., 165(1):207â209, 1994.
 E. Feireisl, P. Gwiazda, A. Swierczewska-Gwiazda, and E. Wiedemann. Regularity and Energy Conservation for the Compressible Euler Equations.
Arch. Ration. Mech. Anal., 223(3):1â21, 2017
 P. Gwiazda, M. MichÃ¡lek, A. Swierczewska-Gwiazda. A note on weak solutions of conservation laws and energy/entropy conservation, arXiv:1706.10154
Abstract: Tyrosine kinase inhibitors such as imatinib (IM), have significantly improved treatment of chronic myelogenous leukemia (CML). Yet, most patients are not cured for undetermined reasons. In this talk we will describe our recent work on modeling the autologous immune response to CML. We will also discuss our previous results on cancer vaccines, drug resistance, and the dynamics of hematopoietic stem cells.
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.
Abstract: Studying the demographic histories of humans or other species and understanding their effects on contemporary genetic variability is one of the central tasks of population genetics. In recent years, a number of methods have been developed to infer demographic histories, especially historical population size changes, from genomic sequence data. Coalescent Hidden Markov Models have proven to be particularly useful for this type of inference. Due to the Markovian structure of Coalescent Hidden Markov Models, an essential building block is the joint distribution of local genealogies, or statistics of these genealogies, in populations of variable size. This joint distribution of local genealogies has received little attention in the literature, especially under variable population size. In this talk, we present a novel method to compute the joint distribution of the total length of the genealogical trees at two linked loci for samples of arbitrary size. We show that the joint distribution can be obtained by solving a system of hyperbolic PDEs and present a numerical algorithm that can be used to efficiently and accurately solve the system and compute this distribution. Our flexible method can be straightforwardly extended to other statistics and structured populations. This is a joint work with Matthias Steinrucken (UChicago).
Abstract: Anyone who has ever gotten stuck in traffic knows how the superiority of a GPS-based map or app over a traditional print map comes not necessarily just from the formerâs access to more information, but more so its access to dynamical information, for eg. the flow of traffic. Similarly, in the last decade, a consensus view is emerging that we should not view proteins, and materials in general, as static entities but instead account for their ever-fluctuating dynamic nature. In this talk, we will describe how we are trying to amalgamate traditional statistical mechanics with recent developments in predictive artificial intelligence (AI) and deep learning to construct and use âdynamical mapsâ for molecular systems. These low-dimensional âdynamical mapsâ go beyond traditional static molecular maps (also called potential or free energy landscapes) by incorporating information also about dynamic quantifiables. We will then illustrate their usefulness with the fundamentally important problem of drug unbinding from proteins. A very important feature of drug efficacy is the drugâs residence time in the target protein. Structural details of the unbinding process are in general hard to capture in experiments, while the relevant timescales are far beyond the most powerful supercomputers. Here we will show how by constructing appropriate dynamical maps we are able to elucidate with unprecedented spatio-temporal resolution and statistical reliability the entire unbinding process of a variety of ligand-protein systems, shedding light on the role of protein conformations and of water molecules as molecular determinants of unbinding.
Abstract: The design and optimization of the next generation of materials and applications strongly hinge on our understanding of the processing-microstructure-performance relations; and these, in turn, result from the collective behavior of materialsâ features at multiple length and time scales. Although the modeling and simulation techniques are now well developed at each individual scale (quantum, atomistic, mesoscale and continuum), there remain long-recognized grand challenges that limit the quantitative and predictive capability of multiscale modeling and simulation tools. In this talk we will discuss three of these challenges and provide solution strategies in the context of specific applications. These comprise (i) the homogenization of the mechanical response of materials in the absence of a complete separation of length and/or time scales, for the simulation of metamaterials with exotic dynamic properties; (ii) the collective behavior of materialsâ defects, for the understanding of the kinematics of large inelastic deformations; and (iii) the upscaling of non-equilibrium material behavior for the modeling of anomalous diffusion processes.
Abstract: In the first part of the talk we will investigate a Keller-Segel model with quorum sensing and a fractional diffusion operator. This model describes the collective cell movement due to chemical sensing with flux limitation for high cell densities and with anomalous media represented by a nonlinear, degenerate fractional diffusion operator. The purpose here is to introduce and prove the existence of a properly defined entropy solution. In the second part of the talk we will analyze an equation that is gradient flow of a functional related to Hardy-Littlewood-Sobolev inequality in whole Euclidean space R^d, d \geq 3. Under the hypothesis of integrable initial data with finite second moment and energy, we show local-in-time existence for any mass of ``free-energy solutions", namely weak solutions with some free energy estimates. We exhibit that the qualitative behavior of solutions is decided by a critical value. The motivation for this part is to generalize Keller-Segel model to higher dimensions.
This is a joint work with K. H. Karlsen and E. A. Carlen.
Abstract: Quadrature by Expansion, or `QBX', is a systematic, high-order approach to
singular quadrature that applies to layer potential integrals with general
kernels on curves and surfaces. The efficient and accurate evaluation of
layer potentials, in turn, is a key building block in the construction of
solvers for elliptic PDEs based on integral equation methods.
I will present a new fast algorithm incorporating QBX that evaluates layer
potentials on and near surfaces in two and three dimensions with user-specified
accuracy, along with supporting theoretical and empirical results on complexity
and accuracy. A series of examples on unstructured geometry across a variety of applications in two and three dimensions demonstrates the applicability of
Abstract: In this talk, we will review the state of the art for the synchronization problem of the Kuramoto model at the kinetic and particle level. Additionally, we will introduce new developments and variational techniques for the dynamics of this model and some of its variants and generalization.
Abstract: Signal fragmentation is a method for transmitting a low frequency signal over a collection of small antennas through a modal expansion (similar to one level of a wavelet expansion), in which the mode has compact support in time. We analyze the spectral leakage and optimality of signal fragmentation. For a special choice of mode, the spectral leakage can be eliminated for sinusoidal signals and minimized for bandlimited or AM signals. We derive the optimal mode for either support size or for energy efficiency. The derivation of these results uses the Poisson summation formula and the Shannon Interpolation Formula.
Abstract: Info-metrics is the science of modeling, reasoning, and drawing inferences under conditions of noisy and insufficient information. It is at the intersection of information theory, statistical inference, and decision-making under uncertainty. It plays an important role in helping make informed decisions even when there is inadequate or incomplete information because it provides a framework to process available information with minimal reliance on assumptions that cannot be validated; it gets the data to confess with minimal torture under careful supervision from the judge and jury.
My talk will be based on my new book âFoundations of Info-Metrics: Modeling, Inference, and Imperfect Information,â http://info-metrics.org/ that develops and examines the theoretical underpinning of info-metrics and provides extensive interdisciplinary applications. I will present the basic framework and theory via graphical illustrations and will then discuss several interdisciplinary examples.
Abstract: The interface between basic ecology and applied mathematics is robust, and results from this interface are often critical to effective conservation. In my CSCAMM seminar, I will focus on one part of this interface whereby ecological observations and datasets have created new opportunities for a variety of mathematical tools and approaches. For instance, datasets derived from efforts to track the movements of wild animals (e.g., using GPS-satellite collars) have presented new opportunities to investigate the roles of learning and memory on animal performance. Likewise, to understand better how animals interact with their landscapes, we need to know more about how they perceive and react to landscape features, creating new opportunities for modeling work that goes beyond classical, diffusion-based formulations of spatial spread. Together such mathematical applications are revealing relationships among individual movements, landscape dynamics, and ecological processes, strengthening the interdisciplinary bridge linking mathematics, ecology, and conservation.
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