A.G. Werschulz. The Computational Complexity of Differential and Integral Equations. An Information-Based Approach. Oxford University Press, New York, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the second kind with kernel k(t; s) fip(tjs) Equations of this form arise in rational expectations theories of asset pricing. See, for example, 13] or [28] A lot is known about the complexity of computing approximate solutions to Fredholm integral equations with general kernels, see [34] and [10] for surveys of deterministic and stochastic complexity bounds for this problem. In the case of general kernels, the results in [34] show that the problem is intractable in the the worst case using deterministic algorithms. By exploiting the special structure of the kernel for the class ....

....for example, 13] or [28] A lot is known about the complexity of computing approximate solutions to Fredholm integral equations with general kernels, see [34] and [10] for surveys of deterministic and stochastic complexity bounds for this problem. In the case of general kernels, the results in [34] show that the problem is intractable in the the worst case using deterministic algorithms. By exploiting the special structure of the kernel for the class of contractive Fredholm integral equations combined with additional special structure on the functions u and p (to be defined shortly) we will ....

A. G. Werschulz, The computational complexity of differential and integral equations, Oxford University Press, New York, 1991.

.... of dimensionality: Yudin and Nemirovsky (1977) showed that randomization doesn t help in solving general multivariate nonlinear programming problems, Traub, Wasilkowski and Wo zniakowski (1988) showed that randomization doesn t help in multivariate function approximation and interpolation, and Werschulz (1991) showed that randomization doesn t help in solving multivariate elliptic partial differential equations or Fredholm integral equations of the second kind. Indeed, the fact that the general nonlinear optimization problem is a special case of the general MDP problem implies that randomization cannot ....

Werschulz, A.G. (1991) The Computational Complexity of Differential and Integral Equations Oxford University Press, New York.

....dimension remain well out of the range of todays largest computers even after the use of parallelization. Specifically, variational problems with solution in the standard Sobolev spaces H t [A75] are well out of reach in higher dimensions. This is often called the curse of dimensionality [Wer91]. The situation often looks different when only problems with solution of higher regularity such as functions with dominating mixed derivatives are considered. Under this assumption, subspaces with relatively large dimensions that contribute only little to the error reduction can be identified ....

A. G. Werschulz, The computational complexity of differential and integral equations, An information based approach, Oxford University Press, New York, 1991.

....partial differential equations, integral equations, elliptic problems 1. Introduction. In this paper we deal with the construction of finite element spaces for the approximate solution of elliptic problems in Sobolev spaces H s( Omega Gamma ; s 2 IR. It is well known that the ffl complexity [TWW88, Wer91] of solving Poissons equation in n dimensions in the Sobolev space H 1 0 H 2 on a bounded domain is O(ffl Gamman ) i.e. it is exponentially dependent on n. This dependence on n is called the curse of dimensionality. Hence, for higher dimensional problems, a direct numerical solution on ....

....is the same as in the full grid case. 11 4. Cost of resulting finite element algorithms. Now we consider the cost of solving the elliptic variational problem (1) up to some prescribed error in the worst case setting (i.e. the error of an approximation is measured in the H s norm) [Wer91]. Due to the larger supports of the wavelets in the approximation spaces V T J from coarser scales, the resulting stiffness matrices A J are rather densely populated. However, usually we need not to assemble the stiffness matrix. Instead, all that is required in an iterative scheme is the ....

A. G. Werschulz, The computational complexity of differential and integral equations, An information based approach, Oxford University Press, New York, 1991.

....conjectured to be intractable. Most problems in scientific computation which involve multivariate functions belonging to F r have been proven computationally intractable in the number of variables in the worst case setting. These include nonlinear equations [10] partial differential equations [19], function approximation [7] integral equations [19] and optimization [5] Material on the computational complexity of optimization will be presented in the second half of this article. Very high dimensional integrals occur in many disciplines. For example, problems with dimension ranging from ....

....computation which involve multivariate functions belonging to F r have been proven computationally intractable in the number of variables in the worst case setting. These include nonlinear equations [10] partial differential equations [19] function approximation [7] integral equations [19], and optimization [5] Material on the computational complexity of optimization will be presented in the second half of this article. Very high dimensional integrals occur in many disciplines. For example, problems with dimension ranging from the hundreds to the thousands occur in mathematical ....

Werschulz, A.G.: The Computational Complexity of Differential and Integral Equations: An Information-Based Approach, Oxford University

....The second author was supported in part by the National Science Foundation under Grant CCR 9420543. c fl1996 American Mathematical Society 0075 8485 96 1.00 .25 per page 1 2 KLAUS RITTER AND GRZEGORZ W. WASILKOWSKI These problems have been extensively studied in the literature (see e.g. [15] for references) However, they are mainly addressed from the worst case setting point of view. In that setting, the cost and the error of an algorithm are defined be the worst performance with respect to f from a given class F . Not surprisingly, for many classes F the results are negative. For ....

Werschulz, A. G., The Computational Complexity of Differential and Integral Equations: An Information-Based Approach, Oxford University Press, Oxford, 1991.

....the approximate solution of elliptic problems in Sobolev spaces H s( Omega Gamma ; s 2 IR. This introductory section is a summary of relevant background information on existing methods for the construction of optimized grids for discretization purposes. It is well known that the ffl complexity [TWW88, We91] of solving Poissons equation in n dimensions and in the Sobolev space H 1 0 on a bounded domain is ffl Gamman , i.e. it is exponentially dependent on n. Hence for higher dimensional problems a direct numerical solution on a regular standard triangulation is prohibitive. To overcome the ....

....numbers as well as standard arithmetic operations, such as addition and scalar multiplication and division, are performed with unit cost. Concerning the permissible information there are two approaches used in the theory of information based complexity, namely linear and standard information [TWW88, We91]. Assuming continuous linear information means that any continuous linear functional is permissible information. In contrast, standard information of cardinality R means that the only information available consists of the result of R distinct samples. As an example the information about f would ....

A. G. Werschulz, The computational complexity of differential and integral equations, An information based approach, Oxford University Press, New York, 1991.

....different settings, including the worst case, average case, probabilistic, randomized and mixed settings. The complexity is then defined as the minimal cost of computing an approximation with error at most . The reader who wants to find more about IBC is referred to the books and recent surveys [5, 9, 11, 14, 18, 20, 27]. We believe that the readers of this proceedings are mainly interested in solving scientific problems for which only partial information is available. That is why we restrict ourselves in the rest of this paper to IBC issues. To make this paper self contained, we present an abstract formulation ....

....of the functions in the class F d . Problems which suffer the curse of dimension in the worst case setting include integration, approximation, global optimization, integral and partial differential equations for classes of functions whose rth derivatives are uniformly bounded in L1 , see [1, 6, 8, 9, 13, 18, 27]. In the average case and randomized settings, the curse of dimension is present for approximation over the class of functions with r continuous derivatives which is equipped with the folded isotropic Wiener measure, see [15, 22] for the average case, and [7, 10, 21] for the randomized setting. ....

A. G. Werschulz, The Computational Complexity of Differential and Integral Equations: An Information-Based Approach Oxford University Press, Oxford, 1991.

....of the functions in the class F d . Problems which suffer the curse of dimension in the worst case setting include integration, approximation, global optimization, integral and partial differential equations for classes of functions whose rth derivatives are uniformly bounded in L1 , see [1, 4, 8, 9, 17, 24, 32]. In the average case and randomized settings, the curse of dimension is present for approximation over the class of functions with r continuous derivatives which is equipped with the folded isotropic Wiener measure, see [20, 28] for the average case and [7, 10, 27] for the randomized setting. ....

Werschulz, A. G., The Computational Complexity of Differential and Integral Equations: An Information-Based Approach, Oxford University Press, Oxford, 1991.

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A. G. Werschulz. The Computational Complexity of Differential and Integral Equations: An Information-Based Approach. Oxford University Press, New York, 1991.

.... is the complexity of surface integration A. G. Werschulz # Department of Computer and Information Sciences Fordham University, New York, NY 10023 Department of Computer Science Columbia University, New York, NY 10027 H. Wozniakowski Department of Computer Science Columbia University, New York, NY 10027 Institute of Applied Mathematics University ....

....to be well defined, and so the integrand appearing in the transformed problem cannot be approximated arbitrarily closely. This is similar to many other IBC problems, for which minimal smoothness implies infinite complexity, see, e.g. the many instances in Traub and Wozniakowski (1980) and Werschulz (1991). 2 We now suppose that d # l and s # 2. Then we prove that the # complexity is at most proportional to (1 #) d min r,s . Note that we have s in the denominator of the exponent, rather than the s 1 that we would expect. This surprising result holds because the surface integral can ....

A. G. Werschulz (1991), The Computational Complexity of Differential and Integral Equations: An Information-BasedApproach, Oxford University Press, New York.

....2m and f belongs to a class F of problem elements. For most problems that arise in practice, F is a space of functions defined on a d dimensional domain. Most work on the computational complexity of such problems has assumed that F has been the unit ball of a Sobolev space H r , see, e.g. [13]. Suppose that the only information we have about any f 2 F is the values of a finite number of continuous linear functionals of f , and that the cost of each of these evaluations is c(d) Let us further suppose that we This research was supported in part by the National Science Foundation under ....

....algorithm is a finite element method using only function evaluations. Hence, standard information consisting of function values is as powerful as arbitrary continuous linear information for our problem. This result should be contrasted to the results for spaces of fixed smoothness reported in [13], in which we found that there was a heavy penalty associated with using function values instead of more general continuous linear functionals. 1 We choose the H m norm mainly for the sake of specificity. However, the usual reason for looking at error estimates in this norm is that it is ....

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Werschulz, A. G., The Computational Complexity of Differential and Integral Equations: An Information -Based Approach, Oxford University Press, Oxford, 1991.

....to state an error bound for U n : Theorem 4.1. Let r # 1. Then e(U n ) 4 1 n min r,s 1 . Before proving Theorem 4.1, we first establish 8 Lemma 4.2. If g (s) is of bounded variation, then #g P D g# L 1 (I) 4 1 n s 1 Var # g (s) Proof. Since k # s, we may use Werschulz (1991, Lemma 5.4.3) to see that there exists C 0, independent of n, such that for any w #W s 1,1 (I) the inequality #w P D w# L 1 (I) #Cn (s 1) #w (s 1) # L 1 (I) holds. From DeVore and Lorentz (1993, Chapter 7, Theorem 5.2) it then follows that #g P D g# L 1 (I) 4 w s 1 (g,n 1 ) 1 ....

....n r . 4.11) Moreover, g and P D g agree at the endpoints of I, and so an integration by parts yields I 2 = Z 1 0 [g(x) P D g) x) d(P D f ) Hence I 2 # #g P D g# L 1 (I) #(P D f ) # # L (I) From Lemma 4. 2, we have #g P D g# L 1 (I) 4 n (s 1) A second application of Werschulz (1991, Lemma 5.4.3) yields #( f P D f ) # # L (I) 4 # f # # L (I) so that #(P D f ) # # L (I) 4 # f # # L (I) # 1. Thus I 2 4 n (s 1) 4.12) The result now follows from (4.10) 4.11) and (4.12) Combining the results of Theorems 3.1 and 4.1, and using Lemma 4.1, we ....

Werschulz, A. G. (1991), The Computational Complexity of Differential and Integral Equations: An Information-Based Approach, Oxford University Press, New York.

....than knowing that the problem elements have a bounded derivative in the L 2 sense. 1. INTRODUCTION Most work on the complexity of elliptic boundary value problems Lu D f has assumed that the class F of problem elements f consisted of functions whose smoothness was fixed and known, see, e.g. [6]. In particular, if F is the unit ball of a Sobolev space, then comp. is a power of #1 ; moreover, we found conditions that are necessary and sufficient for a finite element h method 1 to be (almost) optimal. Unfortunately, assuming that F is the unit ball of a Sobolev space of fixed ....

....two subcases of whether or not the number of breakpoints is known. 2. PROBLEM DESCRIPTION Let I D .#1; 1 . In what follows, we use the standard notations and definitions for Sobolev spaces of functions defined on I , as well as Sobolev norms, seminorms, and inner products. See the appendix of [6] and the references found therein for further details. We use one slightly nonstandard notation; namely, we define kf k H #1 .I D sup v2H 1 .I hf; vi L 2 .I kvkH 1 .I : That is, we consider H #1 .I to be the dual space of H 1 .I , rather than of H 1 0 .I . See [6, pg. 127] ....

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Werschulz, A. G., The Computational Complexity of Differential and Integral Equations: An Information-Based Approach, Oxford University Press, Oxford, 1991.

....an approximation at (nearly) minimal cost. Most earlier work on the complexity of elliptic problems has assumed that exact information is available, i.e. we calculate the values these coefficient functions or of the right hand side exactly. An extensive treatment of the topic may be found in [12]. What happens if the information is contaminated by noise Suppose that we can only guarantee that these calculated function values are accurate to within some positive noise level ffi. What can we say about the complexity of such problems The systematic study of complexity for problems with ....

Werschulz, A. G., The Computational Complexity of Differential and Integral Equations: An InformationBased Approach, Oxford University Press, Oxford, 1991.

....by L. Plaskota, and is described in his monograph [7] on the complexity of problems with partial information that is contaminated by noise. In this paper, we study the complexity of elliptic partial differential equations Lu = f , with noisy partial information. Most previous work (see, e.g. [10], 11] and [12] as well as the references cited therein) on the complexity of elliptic PDEs has assumed that we have complete information about the coefficients of L, and exact (but partial) information about the right hand side f . As a typical result, consider the 2mth order elliptic boundary ....

....equations. One such result is the following, from [10, pp. 110 111] Assume that we can compute f and the coefficients a of L a (or their derivatives) at points in Omega Gamma Then the nth minimal error is Theta(n Gammar=d ) this error being achieved by an FEMQ using n evaluations. Although [10] does not derive the complexity from this minimal error result, it is not too difficult to show that the complexity is still Theta Gamma (1= d=r Delta . Indeed, we can use multigrid techniques (see [2] especially Chapter 7) to get a sufficiently good approximation to the FEMQ, in time ....

Werschulz, A. G., The Computational Complexity of Differential and Integral Equations: An InformationBased Approach, Oxford University Press, Oxford, 1991.

....for U n : Theorem 4.1. Let r 1. Then e(U n ) 4 1 n minfr;s 1g : Before proving Theorem (4.1) we first establish Lemma 4.2. If g (s) is of bounded variation, then kg Gamma Pi Delta gk L 1 (I) 4 1 n minfk 1;s 1g Var g (s) Proof. Let = minfs 1; k 1g. From Werschulz (1991, Lemma 5.4.3) we know that there exists C 0, independent of n, such that for any w 2 W s 1;1 (I) the inequality kw Gamma Pi Delta wk L 1 (I) Cn Gamma kw (s 1) k L 1 (I) holds. From DeVore and Lorentz (1993, Chapter 7, Theorem 5.2) it then follows that kg Gamma Pi Delta gk ....

....of I, and so an integration by parts yields I 2 = Z 1 0 [g(x) Gamma ( Pi Delta g) x) d( Pi Delta f) Hence jI 2 j kg Gamma Pi Delta gk L1 (I) k( Pi Delta f) 0 k L1 (I) From Lemma 4. 2, we have kg Gamma Pi Delta gk L1 (I) 4 n Gamma minfk 1;s 1g : A second application of Werschulz (1991, Lemma 5.4.3) yields k(f Gamma Pi Delta f) 0 k L1 (I) 4 kf 0 k L1 (I) so that k( Pi Delta f) 0 k L1 (I) 4 kf 0 k L1 (I) 1: Thus jI 2 j 4 n Gamma minfk 1;s 1g : 4.12) The result now follows from (4.10) 4.11) and (4.12) Combining the results of Theorems 3.1 and 4.1, and ....

Werschulz, A. G. (1991), The Computational Complexity of Differential and Integral Equations: An Information-Based Approach, Oxford University Press, New York.

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Werschulz, A. G., The Computational Complexity of Differential and Integral Equations: An InformationBased Approach, Oxford University Press, Oxford, 1991.

No context found.

A.G. Werschulz. The Computational Complexity of Differential and Integral Equations. An Information-Based Approach. Oxford University Press, New York, 1991.

No context found.

A.G. Werschulz, The Computational Complexity of Differential and Integral Equations: An Information-Based Approach, Oxford Univ. Press, (1991).

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