Applied Math
PhD Seminar

2024 Spring

The seminar of this semester is organized by Hengzhun Chen and Guiyun Xiao, and co-organized by the graduate student union in the School of Mathematical Sciences at Fudan. This section is partially sponsored by Shuqin Zhang.

Past Presentations

2024-03-07 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand The energy landscape theory has been widely applied to study the stochastic dynamics of biological systems. Different methods have been developed to quantify the energy landscape for gene networks, e.g., using Gaussian approximation (GA) approach to calculate the landscape by solving the diffusion equation approximately from the first two moments. However, how high-order moments influence the landscape construction remains to be elucidated. Also, multistability exists extensively in biological networks. So, how to quantify the landscape for a multistable dynamical system accurately, is a paramount problem. In this work, we prove that the weighted summation from GA (WSGA), provides an effective way to calculate the landscape for multistable systems and limit cycle systems. Meanwhile, we proposed an extended Gaussian approximation (EGA) approach by considering the effects of the third moments, which provides a more accurate way to obtain probability distribution and corresponding landscape. By applying our generalized EGA approach to two specific biological systems: multistable genetic circuit and synthetic oscillatory network, we compared EGA with WSGA by calculating the KL divergence of the probability distribution between these two approaches and simulations, which demonstrated that the EGA provides a more accurate approach to calculate the energy landscape.

2024-03-14 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand The Kaczmarz method is a classical and popular iterative method for solving the system of linear equations, which cyclically projects the estimation onto each of the solution spaces defined by a single equation. In this paper, a fast greedy block Kaczmarz method combined with general greedy strategy and averaging technique is proposed for solving large linear systems. Theoretical analysis of the convergence of the proposed method is given in details. Numerical experiments show that the proposed method is efficient and faster than the existing methods.

2024-03-21 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, we'll explore the parameterization of many-electron wave functions with ACE and computation of ground state for atom and molecule systems using the Variational Monte Carlo method, the novelty of our method lies in (i) a convenient and accurate linear polynomial expansion; (ii) a hierarchical structure that applies naturally to a multigrid variation; and (iii) possible revealing the correlation of the system by increasing the body-order.

2024-03-28 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, we will discuss how to solve inverse problems by identifying unknown coefficients in coupled PDE systems. We focus on non-negative solutions that are relevant to biology and ecology. Our new scheme involves controlling the injection of different source terms to obtain multiple sets of mean flux data, leading to unique identifiability results. We use monotonicity properties related to the system's structure and connect our findings to practical biological applications.

2024-04-11 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this work, we show the implementation of Møller-Plesset second order perturbation (MP2) static polarizability, with approximation of resolution-of-identity (RI, also known as density-fitting). Technically, this task is second derivative to RI-MP2 energy w.r.t. perturbation of external dipole electric field, which involves intense tensor derivatives with constraints. We try to optimize floating point operations (FLOPs) and perform efficiency analysis. We expect this work will be incorporated with doubly hybrid density functional approximations to produce accurate static polarizability with affordable computational cost.

2024-04-18 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand A non-local mean-field control (MFC) is developed for a prototype model of accidental spill of a hazardous contaminant in subsurface porous media. Numerical experiments are presented to investigate the performance of the MFC, which show that the MFC determines an optimal flow pattern to ensure clean groundwater supply during and beyond the time period of contaminant spill with minimal cost.

2024-04-25 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Electronic structure theory for strongly correlated systems (SCSs) poses a long-standing challenge in quantum chemistry and has attracted tremendous efforts in recent decades. Among various theoretical methods that have been developed for strongly correlated molecular systems, multi-configurational self-consistent field (MCSCF) theory has played a particularly important role as it provides a systematic approach to static correlation, which is the most challenging part of SCSs. However, due to the exponential scaling of MCSCF-type methods with respect to the size of the system under study, a direct application of those approaches to complex systems is computationally demanding and becomes even prohibitive for extended systems like solids and surfaces. Density matrix embedding theory (DMET), which combines low-level (e.g., Hartree-Fock approximation) and high-level correlated quantum chemistry methods, provides a systematic framework to reduce the computational cost for treating SCSs. In this work, we present an efficient quantum embedding approach that combines DMET with the complete active space self-consistent field and subsequent state interaction treatment of spin-orbit coupling (CASSI-SO) and apply it to an efficient ab initio study of strongly correlated single-impurity systems.

2024-05-09 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand The problem of basis set incompleteness error(BSIE) in correlation calculation methods (such as RPA, GW, MP2, etc.) has received widespread attention. Essentially, BSIE originates from the perturbation expansion, which requires a complete set of eigenfunctions for non interacting Hamiltonian. Although using correlated consistent basis sets to extrapolate to the complete basis set (CBS) limit can effectively eliminate BSIE, accurate basis set limits are still difficult to obtain. The accuracy of the extrapolation results also needs to be confirmed. Our research aims to provide the RPA correlation energy without BSIE, in order to provide the correct CBS. This can provide us with some reference results and also evaluate the accuracy of common correlation calculation methods in quantum chemistry.

2024-05-16 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this work, we address a sufficient and necessary continuity condition for the construction of $C^r$ conforming finite element spaces on general triangulations. It has been commonly conjectured that such spaces can be generated using the piecewise polynomials with degrees $\ge 2^{d} r + 1$ and an additional $C^{2^{d - s} r}$ smoothness on $s$-dimensional subsimplices. Under these conditions, three authors first provided a rigorous construction for any continuity in any dimension. We will prove that this condition is a tight condition for finite element construction. Specifically, we introduce the concept of extendability for the pre-element space, a generalization of (super)spline spaces and finite element spaces. We show that the superspline space is extendable if and only if such a condition holds, while the finite element space is always extendable under mild conditions. The theory is then established by combining both directions. This concept of extendability not only clarifies the essential connection between spline methods and finite element methods, but also provides valuable insights into the fundamental requirements for constructing conforming finite element spaces on general triangulations.

2024-05-23 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, we investigate the energy minimization model arising in the ensemble Kohn-Sham density functional theory for metallic systems, in which a pseudo-eigenvalue matrix and a general smearing approach are involved. We study the invariance of the energy functional and the existence of the minimizer of the ensemble Kohn-Sham model. We propose an adaptive two-parameter step size strategy and the corresponding preconditioned conjugate gradient methods to solve the energy minimization model. Under some mild but reasonable assumptions, we prove the global convergence for the gradients of the energy functional produced by our algorithms. Numerical experiments show that our algorithms are efficient, especially for large scale metallic systems. In particular, our algorithms produce convergent numerical approximations for some metallic systems, for which the traditional self-consistent field iterations fail to converge. This talk is based on a joint work with Xiaoying Dai, Stefano de Gironcoli, and Aihui Zhou.

2024-05-30 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this work, we focus on the physical observables of continuum Schr\"odinger operators for incommensurate systems. We characterize the density of states in both real and reciprocal spaces. Specifically, we justify the thermodynamic limit of the density of states, and propose efficient numerical schemes based on planewave approximation. We present both rigorous analysis and numerical simulations to support the reliability and efficiency of our numerical algorithms.

2024-06-06 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Quantum computation is an emerging technology with important potential for solving certain problems pivotal in various scientific and engineering disciplines. This paper introduces an efficient quantum protocol for the explicit construction of the block-encoding for an important class of Hamiltonians. Using the Schr\"odingerisation technique -- which converts non-conservative PDEs into conservative ones -- this particular class of Hamiltonians is shown to be sufficient for simulating any linear partial differential equations that have coefficients which are polynomial functions. The class of Hamiltonians consist of discretisations of polynomial products and sums of position and momentum operators. This construction is explicit and leverages minimal one- and two-qubit operations. The explicit construction of this block-encoding forms a fundamental building block for constructing the unitary evolution operator for this Hamiltonian. The proposed algorithm exhibits polynomial scaling with respect to the spatial partitioning size, suggesting an exponential speedup over classical finite-difference methods. This work provides an important foundation for building explicit and efficient quantum circuits for solving partial differential equations.

2024-06-13 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this presentation, we investigate the endpoint geodesic problem on the Stiefel manifold that seeks a constrained rotation joining two orthonormal $\mathbb{R}^{n\times k}$ matrices. A local diffeomorphism between the special orthogonal group and the skew symmetric matrices is constructed to characterize a quotient structure. Based on this geometric insight, a novel and robust Newton solver is proposed. Data structures and routines are developed to guarantee the superior performance of the proposed algorithm over the state-of-the-art algorithm. For further separated matrices, the speedup is at the scale of $8\sim 10$.