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We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy minimisation approach. More specifically, this procedure aims to generate a sequence of numerical approximations, which results from the iterative solution of related (stabilised) linearised discrete problems, and tends to a local minimum of the underlying energy functional. Simultaneously, the finite-dimensional approximation spaces are adaptively refined; this is implemented in terms of a new mesh refinement strategy in the context of finite element discretisations, which again relies on the energy structure of the problem under consideration, and does not involve any a posteriori error indicators. In combination, the resulting adaptive algorithm consists of an iterative linearisation procedure on a sequence of hierarchically refined discrete spaces, which we prove to converge towards a solution of the continuous problem in an appropriate sense. Numerical experiments demonstrate the robustness and reliability of our approach for a series of examples.

A key advantage of isogeometric discretizations is their accurate and well-behaved eigenfrequencies and eigenmodes. For degree two and higher, however, optical branches of spurious outlier frequencies and modes may appear due to boundaries or reduced continuity at patch interfaces. In this paper, we introduce a variational approach based on perturbed eigenvalue analysis that eliminates outlier frequencies without negatively affecting the accuracy in the remainder of the spectrum and modes. We then propose a pragmatic iterative procedure that estimates the perturbation parameters in such a way that the outlier frequencies are effectively reduced. We demonstrate that our approach allows for a much larger critical time-step size in explicit dynamics calculations. In addition, we show that the critical time-step size obtained with the proposed approach does not depend on the polynomial degree of spline basis functions.

When the regression function belongs to the standard smooth classes consisting of univariate functions with derivatives up to the $(\gamma+1)$th order bounded in absolute values by a common constant everywhere or a.e., it is well known that the minimax optimal rate of convergence in mean squared error (MSE) is $\left(\frac{\sigma^{2}}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$ when $\gamma$ is finite and the sample size $n\rightarrow\infty$. From a nonasymptotic viewpoint that does not take $n$ to infinity, this paper shows that: for the standard H\"older and Sobolev classes, the minimax optimal rate is $\frac{\sigma^{2}\left(\gamma+1\right)}{n}$ ($\succsim\left(\frac{\sigma^{2}}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$) when $\frac{n}{\sigma^{2}}\precsim\left(\gamma+1\right)^{2\gamma+3}$ and $\left(\frac{\sigma^{2}}{n}\right)^{\frac{2\gamma+2}{2\gamma+3}}$ ($\succsim\frac{\sigma^{2}\left(\gamma+1\right)}{n}$) when $\frac{n}{\sigma^{2}}\succsim\left(\gamma+1\right)^{2\gamma+3}$. To establish these results, we derive upper and lower bounds on the covering and packing numbers for the generalized H\"older class where the absolute value of the $k$th ($k=0,...,\gamma$) derivative is bounded by a parameter $R_{k}$ and the $\gamma$th derivative is $R_{\gamma+1}-$Lipschitz (and also for the generalized ellipsoid class of smooth functions). Our bounds sharpen the classical metric entropy results for the standard classes, and give the general dependence on $\gamma$ and $R_{k}$. By deriving the minimax optimal MSE rates under various (well motivated) $R_{k}$s for the smooth classes with the help of our new entropy bounds, we show several interesting results that cannot be shown with the existing entropy bounds in the literature.

This paper proposes a regularization of the Monge-Amp\`ere equation in planar convex domains through uniformly elliptic Hamilton-Jacobi-Bellman equations. The regularized problem possesses a unique strong solution $u_\varepsilon$ and is accessible to the discretization with finite elements. This work establishes locally uniform convergence of $u_\varepsilon$ to the convex Alexandrov solution $u$ to the Monge-Amp\`ere equation as the regularization parameter $\varepsilon$ approaches $0$. A mixed finite element method for the approximation of $u_\varepsilon$ is proposed, and the regularized finite element scheme is shown to be locally uniformly convergent. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions $u$.

We consider the evolution of curve networks in two dimensions (2d) and surface clusters in three dimensions (3d). The motion of the interfaces is described by surface diffusion, with boundary conditions at the triple junction points/lines, where three interfaces meet, and at the boundary points/lines, where an interface meets a fixed planar boundary. We propose a parametric finite element method based on a suitable variational formulation. The constructed method is semi-implicit and can be shown to satisfy the volume conservation of each enclosed bubble and the unconditional energy-stability, thus preserving the two fundamental geometric structures of the flow. Besides, the method has very good properties with respect to the distribution of mesh points, thus no mesh smoothing or regularization technique is required. A generalization of the introduced scheme to the case of anisotropic surface energies and non-neutral external boundaries is also considered. Numerical results are presented for the evolution of two-dimensional curve networks and three-dimensional surface clusters in the cases of both isotropic and anisotropic surface energies.

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established in the paper.

The study of generalising the central difference for integer order Laplacian to fractional order is discussed in this paper. Analysis shows that, in contrary to the conclusion of a previous study, difference stencils evaluated through fast Fourier transform prevents the convergence of the solution of fractional Laplacian. We propose a composite quadrature rule in order to efficiently evaluate the stencil coefficients with the required convergence rate in order to guarantee convergence of the solution. Furthermore, we propose the use of generalised higher order lattice Boltzmann method to generate stencils which can approximate fractional Laplacian with higher order convergence speed and error isotropy. We also review the formulation of the lattice Boltzmann method and discuss the explicit sparse solution formulated using Smolyak's algorithm, as well as the method for the evaluation of the Hermite polynomials for efficient generation of the higher order stencils. Numerical experiments are carried out to verify the error analysis and formulations.

The aim of this work is to devise and analyse an accurate numerical scheme to solve Erd\'elyi-Kober fractional diffusion equation. This solution can be thought as the marginal pdf of the stochastic process called the generalized grey Brownian motion (ggBm). The ggBm includes some well-known stochastic processes: Brownian motion, fractional Brownian motion and grey Brownian motion. To obtain convergent numerical scheme we transform the fractional diffusion equation into its weak form and apply the discretization of the Erd\'elyi-Kober fractional derivative. We prove the stability of the solution of the semi-discrete problem and its convergence to the exact solution. Due to the singular in time term appearing in the main equation the proposed method converges slower than first order. Finally, we provide the numerical analysis of the full-discrete problem using orthogonal expansion in terms of Hermite functions.

We present symbolic and numerical methods for computing Poisson brackets on the spaces of measures with positive densities of the plane, the 2-torus, and the 2-sphere. We apply our methods to compute symplectic areas of finite regions for the case of the 2-sphere, including an explicit example for Gaussian measures with positive densities.

This paper studies the inverse problem of determination the history for a stochastic diffusion process, by means of the value at the final time $T$. By establishing a new Carleman estimate, the conditional stability of the problem is proven. Based on the idea of Tikhonov method, a regularized solution is proposed. The analysis of the existence and uniqueness of the regularized solution, and proof for error estimate under an a-proior assumption are present. Numerical verification of the regularization, including numerical algorithm and examples are also illustrated.