Michelle Michelle

Bin Han, Michelle Michelle

This paper presents a wavelet Galerkin method for solving elliptic interface problems of the form $-\nabla\cdot(a\nabla u)=f$ in $Ω\backslash Γ$, where $Γ$ is a smooth interface within $Ω$. Since the scalar variable coefficient $a>0$ and source term $f$ are often discontinuous across $Γ$, the solution $u$ typically has discontinuous gradient $\nabla u$ across $Γ$ and hence $u\not\in H^{1.5}(Ω)$, posing significant challenges for traditional numerical methods. By utilizing a compactly supported biorthogonal wavelet for $H^1_0(Ω)$, we develop a strategy that incorporates additional wavelet elements (or basis functions) along the interface to resolve the complex geometry of the interface $Γ$ and the resulting gradient discontinuities. For the two-dimensional (2D) elliptic interface problem, the proposed method achieves near-optimal convergence rates: $\mathcal{O}(h |\log(h)|)$ in the $H^1(Ω)$-norm and $\mathcal(h^2 |\log(h)|^2)$ in the $L^{2}$-norm with respect to the approximation order. A key theoretical contribution is the use of the dual biorthogonal wavelet basis to establish the $H^1(Ω)$ convergence results. This is supported by the development of weighted Bessel properties for wavelets and several inequalities in fractional Sobolev spaces. To maintain high accuracy and robustness against high-contrast coefficients, our method leverages an augmented set of wavelet elements, similar to meshfree approaches, thereby eliminating the need for the complex re-meshing required by finite element methods. Unlike existing techniques, this wavelet Riesz basis framework captures the geometry of $Γ$ seamlessly while ensuring that the condition numbers of the coefficient matrices remain small and uniformly bounded, independent of the problem size.

Bin Han, Michelle Michelle

The solution $u$ of an elliptic interface problem in a domain $Ω$ is often smooth away from the interface $Γ\subset Ω$, but its gradient is discontinuous across $Γ$, resulting in low regularity; in particular, $u \notin H^{1.5}(Ω)$. This paper focuses on 1D elliptic interface problems using wavelet methods. We propose a Galerkin method using locally supported biorthogonal wavelet bases on bounded intervals with $m$th approximation order for any integer $m \ge 2$. Additionally, we rigorously prove that its convergence rates are of order $m-1$ in the $H^1(Ω)$-norm and order $m$ in the $L^2(Ω)$-norm, which are optimal with respect to the scheme's approximation order $m$. Our approach involves incorporating wavelet basis functions from higher scale levels to capture the singularity in the neighbourhood of the interface $Γ$. The results in this paper both complement and sharply contrast our findings in Han and Michelle (2024), where we consider a similar wavelet-based method for solving $d$-dimensional elliptic interface problems with $d\ge 2$.

Ayan Maiti, Michelle Michelle, Haizhao Yang

Recently, the authors of \cite{SYZ22} developed a neural network with width $36d(2d + 1)$ and depth $11$, which utilizes a special activation function called the elementary universal activation function, to achieve the super approximation property for functions in $C([a,b]^d)$. That is, the constructed network only requires a fixed number of neurons (and thus parameters) to approximate a $d$-variate continuous function on a $d$-dimensional hypercube with arbitrary accuracy. More specifically, only $\mathcal{O}(d^2)$ neurons or parameters are used. One natural question is whether we can reduce the number of these neurons or parameters in such a network. By leveraging a variant of the Kolmogorov Superposition Theorem, \textcolor{black}{we show that there is a composition of networks generated by the elementary universal activation function with at most $10889d + 10887$ nonzero parameters such that this super approximation property is attained. The composed network consists of repeated evaluations of two neural networks: one with width $36(2d+1)$ and the other with width 36, both having 5 layers.} Furthermore, we present a family of continuous functions that requires at least width $d$, and thus at least $d$ neurons or parameters, to achieve arbitrary accuracy in its approximation. This suggests that the number of nonzero parameters is optimal in the sense that it grows linearly with the input dimension $d$, unlike some approximation methods where parameters may grow exponentially with $d$.