Reduced basis heterogeneous multiscale methods

Numerical methods for partial differential equations with multiple scales that combine numerical homogenization methods with reduced order modeling techniques are discussed. These numerical methods can be applied to a variety of problems including multiscale nonlinear elliptic and parabolic problems or Stokes flow in heterogenenous media

MATHICSE Technical Report : A theoretical study of COmpRessed SolvING for advection-diffusion-reaction problems

We present a theoretical analysis of the CORSING (COmpRessed SolvING) method for the numerical approximation of partial differential equations based on compressed sensing. In particular, we show that the best s-term approximation of the weak solution of a PDE with respect to an orthonormal system of N trial functions, can be recovered via a Petrov-Galerkin approach using m << N orthonormal test functions. This recovery is guaranteed if the local a-coherence associated with the bilinear form and the selected trial and test bases fulfills suitable decay properties. The fundamental tool of this analysis is the restricted infsup property, i.e., a combination of the classical inf-sup condition and the well-known restricted isometry property of compressed sensing

Multiscale methods for Stokes flow in heterogeneous media

Fluid flow in porous media is a multiscale process where the effective dynamics, which is often the goal of a computation, depends strongly on the porous micro structure. Resolving the micro structure in the whole porous medium can, however, be prohibitive. Novel numerical methods that efficiently approximate the effective flow but resolve only a carefully selected reduced portion of the porous structure are of great interest. In this thesis we propose new numerical multiscale methods for Stokes flow in two- and three-scale porous media. First, we propose the Darcy--Stokes finite element heterogeneous multiscale method (DS-FE-HMM). The method is based on solving the Darcy equation on a macroscopic mesh using the finite element method with numerical quadrature, where the unknown permeability is recovered from micro finite element solutions of Stokes problems that are defined in sampling domains centered at macroscopic quadrature points. An adaptive scheme based on a posteriori error analysis is proposed, where micro-macro mesh refinement is driven by residual-based indicators that quantify both the micro and macro errors. Second, to address the increasing cost of solving the micro problems as the macroscopic mesh is refined, we combine the DS-FE-HMM with reduced basis (RB) method and propose a new multiscale method called the RB-DS-FE-HMM. Efficiency and accuracy of the method relies on a parametrization of the micro geometries and on the Petrov-Galerkin RB formulation that provides a stable and fast evaluation of the effective permeability. A residual-based adaptive mesh refinement scheme is proposed for the macroscopic problem. To achieve a conservative approximation we also combine and analyze a coupling of the RB method with a different macroscopic scheme based on the discontinuous Galerkin finite element method (DG-FEM). Finally, we consider a three-scale porous media model with macro, meso, and micro scale. At the intermediate meso scale the medium is composed of fluid and porous parts and the fluid flow is modeled with the Stokes-Brinkman equation. A three-scale numerical method is derived and an efficient algorithm based on the RB method and empirical interpolation method on the micro and meso scale is proposed

MATHICSE Technical Report : Multi-index stochastic collocation for random PDEs

In this work we introduce the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of a PDE with random data. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. We propose an optimization procedure to select the most eective mixed differences to include in the MISC estimator: such optimization is a crucial step and allows us to build a method that, provided with sufficient solution regularity, is potentially more eective than other multi-level collocation methods already available in literature. We then provide a complexity analysis that assumes decay rates of product type for such mixed differences, showing that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one dimensional problem. We show the effectiveness of MISC with some computational tests, comparing it with other related methods available in the literature, such as the Multi- Index and Multilevel Monte Carlo, Multilevel Stochastic Collocation, Quasi Optimal Stochastic Collocation and Sparse Composite Collocation methods

Psr1p interacts with SUN/sad1p and EB1/mal3p to establish the bipolar spindle

Regular Abstracts - Sunday Poster Presentations: no. 382During mitosis, interpolar microtubules from two spindle pole bodies (SPBs) interdigitate to create an antiparallel microtubule array for accommodating numerous regulatory proteins. Among these proteins, the kinesin-5 cut7p/Eg5 is the key player responsible for sliding apart antiparallel microtubules and thus helps in establishing the bipolar spindle. At the onset of mitosis, two SPBs are adjacent to one another with most microtubules running nearly parallel toward the nuclear envelope, creating an unfavorable microtubule configuration for the kinesin-5 kinesins. Therefore, how the cell organizes the antiparallel microtubule array in the first place at mitotic onset remains enigmatic. Here, we show that a novel protein psrp1p localizes to the SPB and plays a key role in organizing the antiparallel microtubule array. The absence of psr1+ leads to a transient monopolar spindle and massive chromosome loss. Further functional characterization demonstrates that psr1p is recruited to the SPB through interaction with the conserved SUN protein sad1p and that psr1p physically interacts with the conserved microtubule plus tip protein mal3p/EB1. These results suggest a model that psr1p serves as a linking protein between sad1p/SUN and mal3p/EB1 to allow microtubule plus ends to be coupled to the SPBs for organization of an antiparallel microtubule array. Thus, we conclude that psr1p is involved in organizing the antiparallel microtubule array in the first place at mitosis onset by interaction with SUN/sad1p and EB1/mal3p, thereby establishing the bipolar spindle.postprin

Removal of antagonistic spindle forces can rescue metaphase spindle length and reduce chromosome segregation defects

Regular Abstracts - Tuesday Poster Presentations: no. 1925Metaphase describes a phase of mitosis where chromosomes are attached and oriented on the bipolar spindle for subsequent segregation at anaphase. In diverse cell types, the metaphase spindle is maintained at a relatively constant length. Metaphase spindle length is proposed to be regulated by a balance of pushing and pulling forces generated by distinct sets of spindle microtubules and their interactions with motors and microtubule-associated proteins (MAPs). Spindle length appears important for chromosome segregation fidelity, as cells with shorter or longer than normal metaphase spindles, generated through deletion or inhibition of individual mitotic motors or MAPs, showed chromosome segregation defects. To test the force balance model of spindle length control and its effect on chromosome segregation, we applied fast microfluidic temperature-control with live-cell imaging to monitor the effect of switching off different combinations of antagonistic forces in the fission yeast metaphase spindle. We show that spindle midzone proteins kinesin-5 cut7p and microtubule bundler ase1p contribute to outward pushing forces, and spindle kinetochore proteins kinesin-8 klp5/6p and dam1p contribute to inward pulling forces. Removing these proteins individually led to aberrant metaphase spindle length and chromosome segregation defects. Removing these proteins in antagonistic combination rescued the defective spindle length and, in some combinations, also partially rescued chromosome segregation defects. Our results stress the importance of proper chromosome-to-microtubule attachment over spindle length regulation for proper chromosome segregation.postprin