Tractability of quasilinear problems. II: Second-order elliptic problems

In a previous paper, we developed a general framework for establishing tractability and strong tractability for quasilinear multivariate problems in the worst case setting. One important example of such a problem is the solution of the Helmholtz equation −Δu + qu = f in the d-dimensional unit cube, in which u depends linearly on f , but nonlinearly on q. Here, both f and q are d-variate functions from a reproducing kernel Hilbert space with finite-order weights of order ω. This means that, although d can be arbitrary large, f and q can be decomposed as sums of functions of at most ω variables, with ω independent of d. In this paper, we apply our previous general results to the Helmholtz equation, subject to either Dirichlet or Neumann homogeneous boundary conditions. We study both the absolute and normalized error criteria. For all four possible combinations of boundary conditions and error criteria, we show that the problem is tractable. That is, the number of evaluations of f and q needed to obtain an ε- approximation is polynomial in ε −1 and d, with the degree of the polynomial depending linearly on ω. In addition, we want to know when the problem is strongly tractable, meaning that the dependence is polynomial only in ε −1 , independently of d. We show that if the sum of the weights defining the weighted reproducing kernel Hilbert space is uniformly bounded in d and the integral of the univariate kernel is positive, then the Helmholtz equation is strongly tractable for three of the four possible combinations of boundary conditions and error criterion, the only exception being the Dirichlet boundary condition under the normalized error criterion

What is the complexity of volume calculation?

We study the worst case complexity of computing ε-approximations of volumes of d-dimensional regions g([0, 1]d ), by sampling the function g. Here, g is an s times continuously differentiable injection from [0, 1]d to R d , where we assume that s ≥ 1. Since the problem can be solved exactly when d = 1, we concentrate our attention on the case d ≥ 2. This problem is a special case of the surface integration problem studied in [12]. Let c be the cost of one function evaluation. The results of [12] might suggest that the ε- complexity of volume calculation should be proportional to c(1/ε) d/s when s ≥ 2. However, using integration by parts to reduce the dimension, we show that if s ≥ 2, then the complexity is proportional to c(1/ε) (d−1)/s . Next, we consider the case s = 1, which is the minimal smoothness for which our volume problem is well-defined. We show that when s = 1, an ε-approximation can be computed with cost proportional to at most c(1/ε) (d−1)d/2 . Since a lower bound proportional to c(1/ε) d−1 holds when s = 1, it follows that the complexity in the minimal smoothness case is proportional to c(1/ε) when d = 2, and that there is a gap between the lower and upper bounds when d ≥ 3

What is the complexity of surface integration?

We study the worst case complexity of computing ε-approximations of surface integrals. This problem has two sources of partial information: the integrand f and the function g defining the surface. The problem is nonlinear in its dependence on g. Here, f is an r times continuously differentiable scalar function of l variables, and g is an s times continuously differentiable injective function of d variables with l components. We must have d ≤ l and s ≥ 1 for surface integration to be well-defined. Surface integration is related to the classical integration problem for functions of d variables that are min{r,s − 1} times continuously differentiable. This might suggest that the complexity of surface integration should be 2((1/ε)d/ min{r,s−1} ). Indeed, this holds when d < l and s = 1, in which case the surface integration problem has infinite complexity. However, if d ≤ l and s ≥ 2, we prove that the complexity of surface integration is O((1/ε)d/ min{r,s} ). Furthermore, this bound is sharp whenever d < l

Tractability of multivariate approximation over a weighted unanchored Sobolev space: Smoothness sometimes hurts

We study d-variate approximation for a weighted unanchored Sobolev space having smoothness m ≥ 1. Folk wisdom would lead us to believe that this problem should become easier as its smoothness increases. This is true if we are only concerned with asymptotic analysis: the nth minimal error is of order n^-(m-δ) for any δ > 0. However, it is unclear how long we need to wait before this asymptotic behavior kicks in. How does this waiting period depend on d and m? We prove that no matter how the weights are chosen, the waiting period is at least m^d, even if the error demand ε is arbitrarily close to 1. Hence, for m ≥ 2, this waiting period is exponential in d, so that the problem suffers from the curse of dimensionality and is intractable. In other words, the fact that the asymptotic behavior improves with m is irrelevant when d is large. So, we will be unable to vanquish the curse of dimensionality unless m = 1 , i.e., unless the smoothness is minimal. We then show that our problem can be tractable if m = 1. That is, we can find an ε-approximation using polynomially-many (in d and ε^-1) information operations, even if only function values are permitted. When m = 1, it is even possible for the problem to be strongly tractable, i.e., we can find an ε-approximation using polynomially-many (in ε^-1) information operations, independent of d. These positive results hold when the weights of the Sobolev space decay sufficiently quickly or are bounded finite-order weights, i.e., the d-variate functions we wish to approximate can be decomposed as sums of functions depending on at most ω variables, where ω is independent of d

Tractability of the Fredholm problem of the second kind

We study the tractability of computing ε-approximations of the Fredholm problem of the second kind: given f ∈ Fd and q ∈ Q2d, find u ∈ L2(Id) satisfying u(x)− q(x,y)u(y)dy=f(x) ∀x∈Id =[0,1]d. Id Here, Fd and Q2d are spaces of d-variate right hand functions and 2d-variate kernels that are continuously embedded in L2(Id) and L2(I2d), respectively. We consider the worst case setting, measuring the approximation error for the solution u in the L2 (I d )-sense. We say that a problem is tractable if the minimal number of information operations of f and q needed to obtain an ε- approximation is sub-exponential in ε−1 and d. One information operation corresponds to the evaluation of one linear functional or one function value. The lack of sub-exponential behavior may be defined in various ways, and so we have various kinds of tractability. In particular, the problem is strongly polynomially tractable if the minimal number of information operations is bounded by a polynomial in ε−1 for all d. We show that tractability (of any kind whatsoever) for the Fredholm problem is equivalent to tractability of the L2-approximation problems over the spaces of right-hand sides and kernel functions. So (for example) if both these approximation problems are strongly polynomially tractable, so is the Fredholm problem. In general, the upper bound provided by this proof is essentially non-constructive, since it involves an interpolator algorithm that exactly solves the Fredholm problem (albeit for finite-rank approximations of f and q). However, if linear functionals are permissible and that Fd and Q2d are tensor product spaces, we are able to surmount this obstacle; that is, we provide a fully-constructive algorithm that provides an approximation with nearly-optimal cost, i.e., one whose cost is within a factor ln ε−1 of being optimal

Is Gauss quadrature optimal for analytic functions?

We consider the problem of optimal quadratures for integrands f: [ -1 , 1 ] → R which have an analytic extension f to an open disk D, of radius r about the origin such that |f| ≤ 1 on Dr. If r = 1, we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degree n rather than at optimal points, tends to infinity with n. In particular there is an "infinite" penalty for using Gauss quadrature. On the other hand, if r > l, Gauss quadrature is almost optimal. These results hold for both the worst-case and asymptotic settings

Herpesvirus Telomerase RNA (vTR) with a Mutated Template Sequence Abrogates Herpesvirus-Induced Lymphomagenesis

Telomerase reverse transcriptase (TERT) and telomerase RNA (TR) represent the enzymatically active components of telomerase. In the complex, TR provides the template for the addition of telomeric repeats to telomeres, a protective structure at the end of linear chromosomes. Human TR with a mutation in the template region has been previously shown to inhibit proliferation of cancer cells in vitro. In this report, we examined the effects of a mutation in the template of a virus encoded TR (vTR) on herpesvirus-induced tumorigenesis in vivo. For this purpose, we used the oncogenic avian herpesvirus Marek's disease virus (MDV) as a natural virus-host model for lymphomagenesis. We generated recombinant MDV in which the vTR template sequence was mutated from AATCCCAATC to ATATATATAT (vAU5) by two-step Red-mediated mutagenesis. Recombinant viruses harboring the template mutation replicated with kinetics comparable to parental and revertant viruses in vitro. However, mutation of the vTR template sequence completely abrogated virus-induced tumor formation in vivo, although the virus was able to undergo low-level lytic replication. To confirm that the absence of tumors was dependent on the presence of mutant vTR in the telomerase complex, a second mutation was introduced in vAU5 that targeted the P6.1 stem loop, a conserved region essential for vTR-TERT interaction. Absence of vTR-AU5 from the telomerase complex restored virus-induced lymphoma formation. To test if the attenuated vAU5 could be used as an effective vaccine against MDV, we performed vaccination-challenge studies and determined that vaccination with vAU5 completely protected chickens from lethal challenge with highly virulent MDV. Taken together, our results demonstrate 1) that mutation of the vTR template sequence can completely abrogate virus-induced tumorigenesis, likely by the inhibition of cancer cell proliferation, and 2) that this strategy could be used to generate novel vaccine candidates against virus-induced lymphoma