Topological modeling of big data in biomolecules 
1.Persistent homology analysis of big data in biomolecules 
 Persistent
homology is, for the first time, employed to quantitatively predict the
stability of the fullerene molecules. We study the groundstate
structures of fullerene molecules and the relative stability of
fullerene isomers. We find the heat of formation energy is related to
the local hexagonal cavities of small fullerenes, while the total
curvature energies of fullerene isomers are associated with their
sphericities,
which are measured by the lengths of their longlived Betti2 bars.
Persistent homology is then introduced for extracting molecular
topological fingerprints (MTFs). MTFs are utilized for protein
characterization, identification and classification. Based on the
correlation between protein compactness, rigidity and connectivity, we
propose an accumulated bar length generated from persistent topological
invariants for the quantitative modeling of protein flexibility. To
this end, a correlation matrix based filtration is developed. This
approach gives rise to an accurate prediction of the optimal
characteristic distance used in protein Bfactor analysis. Further,
MTFs are employed to characterize protein topological evolution during
protein folding and quantitatively predict the protein folding
stability. An excellent consistence between our persistent homology
prediction and molecular dynamics simulation is found.
 Mathematical modeling of biomolecules 
1.Variational multiscale models
 We
develop geometric modeling and computational algorithm for biomolecular
structures from two data sources: Protein Data Bank (PDB) and Electron
Microscopy Data Bank (EMDB) in the Eulerian (or Cartesian)
representation. Molecular surface (MS) contains nonsmooth geometric
singularities, such as cusps, tips and selfintersecting facets, which
often lead to computational instabilities in molecular simulations, and
violate the physical principle of surface free energy minimization.
Variational multiscale surface definitions are proposed based on
geometric flows and solvation analysis of biomolecular systems. The
resulting surfaces are free of geometric singularities and minimize the
total free energy of the biomolecular system. High order partial
differential equation (PDE)based nonlinear filters are employed for
EMDB data processing. After the construction of protein
multiresolution surfaces, we explore the analysis and characterization
of surface morphology by the consideration of Gaussian curvature, mean
curvature, maximum curvature, minimum curvature, shape index, and
curvedness. Based on the curvature and electrostatic analysis from our
multiresolution surfaces, we introduce a new concept, the polarized
curvature, for the prediction of protein binding sites.

 2.Protein flexibility and rigidity analysis
 Protein
structural fluctuation, typically measured by DebyeWaller factors, or
Bfactors, is a manifestation of protein flexibility, which strongly
correlates to protein function. The flexibilityrigidity index (FRI) is
a newly proposed method for the construction of atomic rigidity
functions required in the theory of continuum elasticity with atomic
rigidity, which is a new multiscale formalism for describing
excessively large biomolecular systems. The FRI method analyzes protein
rigidity and flexibility and is capable of predicting protein Bfactors
without resorting to matrix diagonalization. A fundamental assumption
used in the FRI is that protein structures are uniquely determined by
various internal and external interactions, while the protein
functions, such as stability and flexibility, are solely determined by
the structure. As such, one can predict protein flexibility without
resorting to the protein interaction Hamiltonian. Additionally, we
propose anisotropic FRI (aFRI) algorithms for the analysis of protein
collective dynamics. Eigenvectors obtained from the proposed aFRI
algorithms are able to demonstrate collective motions.


1.MIB method for multimaterial interface problem 
 Multimaterial
interface problems are omnipresent in science, engineering and daily
life. The solution to this class of problems becomes exceptionally
challenging when more than two heterogeneous materials join at one
point of the space and form a geometric singularityprimary. Based on
the MIB method, several schemes have been constructed to solve 2D
elliptic equations with discontinuous coefficients associated with
threematerial interfaces. The essential idea is to smoothly
extend functions across the interface and employ the fictitious values at irregular points. For the geometric singularities, two sets of interface conditions are considered simultaneously. Intensive
numerical experiments are carried out to validate the proposed schemes.
A second order of accuracy is obtained for complex geometric and
geometric singularities.

2.Adaptive mesh based MIB method 
 Mesh
deformation methods break down for elliptic PDEs interface
problems, as additional interface jump conditions are required to
maintain the wellposedness of the governing equation. An
interface technique based adaptively deformed mesh strategy is
introduced for resolving elliptic interface problems. We take the
advantages of the high accuracy, flexibility and robustness of MIB
method to construct an adaptively deformed mesh based interface method.
The proposed method generates deformed meshes in the physical domain
and solves the transformed governed equations in the computational
domain, which maintains regular Cartesian meshes. The mesh deformation
is realized by a mesh transformation PDE, which controls the mesh
redistribution by a source term. The source term consists of a monitor
function, which builds in mesh contraction rules. Both interface
geometry based deformed meshes and solution gradient based deformed
meshes are constructed to reduce errors in solving elliptic
interface problems. The proposed adaptively deformed mesh based
interface method is extensively validated by many numerical
experiments. Numerical results indicate that the adaptively deformed
mesh based interface method outperforms the original MIB method for
dealing with elliptic interface problems.

 A
MIB Galerkin formulation is developped for solving the elliptic
interface problem. In this approach, we build up two sets of elements
respectively on two extended subdomains which both include the
interface. As a result, two sets of elements overlap each other near
the interface. Fictitious solutions are defined on the overlapping part
of the elements, so that the differentiation operations of the original
PDEs can be discretized as if there was no interface. The extra
coeffients of polynomial basis functions, which furnish the overlapping
elements and solve the fictitious solutions, are determined by
interface jump conditions. Consequently, the interface jump conditions
are rigorously enforced on the interface. The present method utilizes
Cartesian meshes to avoid the mesh generation in conventional finite
element methods (FEMs). The accuracy, stability and robustness of the
proposed 3D MIB Galerkin are extensively validated. Near second
order accuracy has been confirmed. To our knowledge, it is the first
time for an FEM to show a near second order convergence in solvingthe
Poisson equation with realistic protein surfaces. Additionally, the
present work offers the first known near second order accurate method
for C_1 continuous or H_2 continuous solutions associated with a
Lipschitz continuous interface.

