Efficient sum-of-exponentials approximations for the heat kernel and their applications

In this paper, we show that efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. In one space dimension, the heat kernel admits an approximation involving a number of terms that is of the order $O(\log(\frac{T}{\delta}) (\log(\frac{1}{\epsilon})+\log\log(\frac{T}{\delta})))$ for any x\in\bbR and $\delta \leq t \leq T$, where $\epsilon$ is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only $O(\log^2(\frac{T}{\delta}))$ terms for fixed accuracy $\epsilon$. These approximations can be used to accelerate integral equation-based methods for boundary value problems governed by the heat equation in complex geometry. The resulting algorithms are nearly optimal. For $N_S$ points in the spatial discretization and $N_T$ time steps, the cost is $O(N_S N_T \log^2 \frac{T}{\delta})$ in terms of both memory and CPU time for fixed accuracy $\epsilon$. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.Comment: 23 pages, 5 figures, 3 table

Applying an Improved MRPS-GMM Method to Detect Temporal Patterns in Dynamic Data System

The purpose of this thesis is to introduce an improved approach for the temporal pattern detection, which is based on the Multivariate Reconstructed Phase Space (MRPS) and the Gaussian Mixture Model (GMM), to overcome the disadvantage caused by the diversity of shapes among different temporal patterns in multiple nonlinear time series. Moreover, this thesis presents an applicable software program developed with MATLAB for users to utilize this approach. A major study involving dynamic data systems is to understand the correspondence between events of interest and predictive temporal patterns in the output observations, which can be used to develop a mechanism to predict the occurrence of events. The approach introduced in this thesis employs Expectation-Maximization (EM) algorithm to fit a more precise distribution for the data points embedded in the MRPS. Furthermore, it proposes an improved algorithm for the pattern classification process. As a result, the computational complexity will be reduced. A recently developed software program, MATPAD, is presented as a deliverable application of this approach. The GUI of this program contains specific functionalities so that users can directly implement the procedure of MRPS embedding and fit data distribution with GMM. Moreover, it allows users to customize the related parameters for specific problems so that users will be able to test their own data

Efficient high-order integral equation methods for the heat equation

Efficient high-order integral equation methods have been developed for solving the boundary value problems of the heat equation with complex geometries in two and three dimensions. First of all, the classical heat potential theory is applied to convert such problems to Volterra integral equations of the second kind via the heat layer potentials. Some advantages of the integral formulation as compared with standard finite difference and finite element methods include reduction of the dimension of the problem by one, high order accuracy, unconditional stability, insensitivity to different geometries, and elimination of truncating the computational domain and the need of artificial boundary conditions for exterior problems. However, the heat layer potentials contains convolution integrals in both space and time whose direct evaluation requires O(NS2NT2) work and O(NSNT) storage, where NS is the total number of discretization points in the spatial boundary and NT is the total number of time steps. This is excessively expensive even for problems of modest size, especially for three-dimensional problems. In order to evaluate the heat layer potentials accurately and efficiently, they are split into two parts - the local part which contains the temporal integration from t - δ to t and the history part which contains the temporal integration from 0 to t - δ. For the local part, Product integration is applied on the temporal integral to convert it to a sum of several spatial convolution integrals where the so-called local kernels have logarithmic singularity in two dimensions and 1r singularity in three dimensions. These weakly singular integrals are discretized via high-order quadratures and the resulting discrete summations can then be evaluated via fast algorithms such as the fast multipole method and its descendants. For the history part, efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. Specifically, in one space dimension, the heat kernel admits an approximation involving a number of terms that is of the order O(log(T δ )(log(1 ) + log log(T δ ))) for any x Î R and δ ≤ t ≤ T, where E is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only O (log2(Tδ )) terms for fixed accuracy E. These approximations can be used to accelerate the evaluation of the history part of the heat layer potentials for stationary geometries. For two-dimensional problems with complex stationary geometries, the sum-of-exponentials approximation is used for the heat kernel and all local and history kernels are compressed only once. The resulting algorithm is very efficient with quasilinear complexity in both space and time for both interior and exterior problems. For two-dimensional problems with complex moving geometries, the spectral Fourier approximation is applied for the heat kernel and NUFFT is used to speed up the evaluation of the history part of the heat potentials. The complexity of the algorithm is again quasilinear in both space and time, albeit only for the interior problem. For three-dimensional problems, the sum-of-exponentials approximation is applied to speed up the evaluation of the history part. The singular surface integrals in the local kernels are treated with a spectrally accurate integrator. The algorithm is applicable for both interior and exterior problems and has quasilinear complexity with respect to the temporal variable. All these algorithms can be parallelized in a straightforward manner and their performance is demonstrated with extensive numerical experiments

Circulator based on spoof surface plasmon polaritons

Circulators based on spoof surface plasmon polaritons are designed and analyzed. In the letter, we use blade structure to realize the propagation of SSPPs wave and a matching transition is used to feed energy from coplanar waveguide to the SSPPs. And the circulator shows good nonreciprocal transmission characteristics. The simulation results indicate that in the frequency band from 5 to 6.6 GHz, the isolation degree and return loss basically reaches 15dB and the insertion loss is less than 0.5dB. Moreover, the use of confinement electromagnetic waves can decrease the size of the ferrite and show a broadband characteristic.Comment: 3 pages, 6 figures, submitted to IEEE antennas and wireless propagation letters on 27-Mar-201

Antitumor effect of salidroside on mice bearing HepA hepatocellular carcinoma

Salidroside, a phenylpropanoid glycoside extracted from Rhodiola rosea L., has antiproliferative effects on tumour cells in mice. However it’s antitumor mechanism remains largely unknown. In this study, 4 groups of mice bearing hepatocarcinoma cells were given treatment with vehicle alone, cyclophosphamide (25 mg/kg, i.p.) and salidroside, either 100 or 200 mg/kg (p.o.) for 14 days. The morphology of tumour specimens was analysed by transmission electron microscopy. Apoptotic cells in sections of mouse tumour tissue were analysed using an in situ apoptosis kit. The expression of Bcl-2, Bax and caspase 3 mRNA were examined with RT-PCR. The results showed that the tumour weights in groups 100 or 200 mg/kg/day of salidroside were reduced significantly (45.34 and 52.48% respectively), compared to vehicle groups. Salidroside increased apoptotic cells index, e.g. in 200 mg/kg group, it was four times higher compared to the control group. Even more, treatment with salidroside decreased Bcl-2 mRNA expression and increased Bax and caspase 3 mRNA expressions. These indicated that the antitumor mechanism of salidroside may induce tumour cell apoptosis in mice by triggering the mitochondrial-dependent pathway and activation of caspase 3

Inhibition effects of paeonol on mice bearing EMT6 breast cancer through inducing rumor cell apoptosis

Paeonol, a phenolic component from the root bark of Paeonia moutan, has been identified to possess antitumor effects on mice bearing EMT6 breast cancer in our previous studies. However, the underlying mechanisms remain unknown. In the present study the molecular mechanisms of paeonol were further investigated in EMT6 mice model. The results showed that treatment of mice with 175 and 350 mg/kg/day of paeonol significantly inhibited the growth of the EMT6 tumor in mice, and induced tumor cell apoptosis which were demonstrated by light microscopy after hematoxylin and eosin staining and apoptosis analysis by flow cytometry. In addition, compared with the control group, paeonol increased the number of tumor cells in G0/G1 phase but decreased the number of cells in S and G2/M phase. Paeonol treatment (350 mg/kg body weight) also resulted in a decrease of Bcl-2 and an increase in Bax and caspase-3 expressions, which were demonstrated by immunohistochemical and western blot analysis. These results indicate that the antitumor effects of paeonol might be associated with arresting tumor cells in the G0/G1 phase, inducing cell apoptosis and regulation of the expression of Bcl-2, Bax and activation of caspase-3